arX

iv:m

ath/

0401

114v

3 [

mat

h.PR

] 2

Jul

200

4

The Skorokhod embedding problem and its

offspring

Jan Ob loj∗

June, 2004

Abstract

This is a survey about the Skorokhod embedding problem. It presents

all known solutions together with their properties and some applications.

Some of the solutions are just described, while others are studied in detail

and their proofs are presented. A certain unification of proofs, thanks

to real potential theory, is made. Some new facts which appeared in a

natural way when different solutions were cross-examined, are reported.

Azema and Yor’s and Root’s solutions are studied extensively. A possible

use of the latter is suggested together with a conjecture.

Keywords: Skorokhod embedding problem, Chacon and Walsh embedding,Azema-Yor embedding, Root embedding, stopping times, optimal stopping.

Mathematics Subject Classification: 60G40, 60G44, 60J25, 60J45, 60J65

∗Laboratoire de Probabilites et Modeles Aleatoires, Universite Paris 6,4 pl. Jussieu, Boite 188, 75252 Paris Cedex 05, France

Wydzia l Matematyki, Uniwersytet Warszawskiul. Banacha 2, 02-097 Warszawa, Poland

email: obloj@mimuw.edu.pl

WWW: www.mimuw.edu.pl/∼obloj/

Work partially supported by French government scholarship N 20022708,and Polish Academy of Science scholarship.

CONTENTS 2

Contents

1 Introduction 4

2 The problem and its methods 6

3 A guided tour through solutions 83.1 Skorokhod (1961) . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Dubins (1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Root (1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Hall (1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Rost (1971) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 Monroe (1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Heath (1974) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 Chacon and Walsh (1974-1976) . . . . . . . . . . . . . . . . . . . 123.9 Azema-Yor (1979) . . . . . . . . . . . . . . . . . . . . . . . . . . 143.10 Falkner (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.11 Bass (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.12 Vallois (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.13 Perkins (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.14 Jacka (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.15 Bertoin and Le Jan (1993) . . . . . . . . . . . . . . . . . . . . . . 163.16 Falkner, Grandits, Pedersen and Peskir (2000 - 2002) . . . . . . . 183.17 Roynette, Vallois and Yor (2002) . . . . . . . . . . . . . . . . . . 183.18 Hambly, Kersting and Kyprianou (2002) . . . . . . . . . . . . . . 193.19 Cox and Hobson (2002) . . . . . . . . . . . . . . . . . . . . . . . 193.20 Ob loj and Yor (2004) . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Embedding measures with finite support 214.1 Brownian embedding . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Around the Azema-Yor solution 255.1 An even simpler proof of the Azema-Yor formula . . . . . . . . . 255.2 Nontrivial initial laws . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Minimizing the minimum - reversed AY solution . . . . . . . . . 295.4 Martingale theory arguments . . . . . . . . . . . . . . . . . . . . 305.5 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 The H1-embedding 336.1 A minimal H1-embedding . . . . . . . . . . . . . . . . . . . . . . 336.2 A maximal H1 embedding . . . . . . . . . . . . . . . . . . . . . . 34

CONTENTS 3

7 Root’s solution 357.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 The reversed barrier solution of Rost . . . . . . . . . . . . . . . . 397.4 Certain generalizations . . . . . . . . . . . . . . . . . . . . . . . . 40

8 Shifted expectations and uniform integrability 42

9 Working with real diffusions 44

10 Links with optimal stopping theory 4510.1 The maximality principle revisited . . . . . . . . . . . . . . . . . 4610.2 The optimal Skorokhod embedding problem . . . . . . . . . . . . 4810.3 Investigating the law of the maximum . . . . . . . . . . . . . . . 49

11 Some applications 5111.1 Embedding processes . . . . . . . . . . . . . . . . . . . . . . . . . 5111.2 Invariance principles . . . . . . . . . . . . . . . . . . . . . . . . . 5211.3 A list of some other applications . . . . . . . . . . . . . . . . . . 53

1 Introduction 4

1 Introduction

The so called Skorokhod stopping problem or Skorokhod embedding problem wasfirst formulated and solved by Skorokhod in 1961 (English translation in 1965[81]). For a given centered probability measure µ with second moment, and aBrownian motion B, one looks for an integrable stopping time T such that thedistribution of BT is µ. This original formulation has been changed, generalizedor narrowed, a great number of times. The problem has stimulated research inthe probability theory for over 40 years now. A full account of it is impossible,still we believe there is a point in trying to gather various solutions and appli-cations in one place, to discuss and compare them and to see where else we cango with the problem. This was the basic goal set for this survey and no matterhow far from it we have ended up, we hope that it will prove of some help toother researchers in the domain.

Let us start with some history of the problem. Skorokhod’s solution requireda randomization external to B. Three years later another solution was proposedby Dubins [23], which did not require any external randomization. Around thesame time, a third solution was proposed by Root. It was part of his Ph.D.thesis and was then published in an article [74]. This solution has some specialminimal properties, which will be discussed in Section 7.

Soon after, another doctoral dissertation was written on the subject: Monroedeveloped a new approach using additive functionals. His results were publishedin 1972 [58]. Although he did not have any explicit formulae for the stoppingtimes, his ideas proved fruitful as can be seen from elegant solutions by Vallois(1983) and Bertoin-Le Jan (1992) that use additive functionals as well.

The next landmark was set in 1971 with the work of Herman Rost [76]. Hegeneralized the problem looking at any Markov process and not just Brownianmotion. He gave an elegant necessary and sufficient condition for the existenceof a solution to the embedding problem. Rost made extensive use of potentialtheory. This approach was also used a few years later by Chacon and Walsh [15],who proposed a new solution to the original problem which included Dubins’solution as a special case.

The next important development of the theory came in 1979 from the Frenchschool with a solution proposed by Azema and Yor [3]. Unlike Rost they madeuse of martingale theory rather than Markov and potential theory and theirsolution was carried out in the setup of any continuous local martingale (Mt)with < M,M >∞= ∞.1 We will see in Section 5 that the solution can beobtained as a limit case of Chacon and Walsh’s one. Azema and Yor’s solutionhas interesting properties which are discussed as well. In particular the solutionmaximizes stochastically the law of the supremum up to the stopping time. Thisdirection was continued by Perkins [65], who proposed his own solution in 1985,which in turn minimizes the law of the maximum and maximizes the law of theminimum.

Finally, yet another solution was proposed in 1983 by Richard Bass [4].

1Actually they solved the problem for any recurrent, real diffusion.

1 Introduction 5

He used stochastic calculus apparatus and defined the stopping time througha time-change procedure. This solution is also reported in a recent book byStroock [83].

Further development was at least twofold. On one hand, authors tried toextend older results to wider classes of processes or to give some new explicitformulae for the stopping times. On the other hand, there has been a consider-able interest in some particular solutions and their optimal properties. In thefirst category we need to mention three developments. Firstly, works followingMonroe’s approach: solution with local times by Vallois (1983) [84] and the pa-per by Bertoin and Le Jan (1992) [6], where they develop explicit formulae fora wide class of Markov processes. Secondly, Azema and Yor’s construction wastaken as a starting point by Grandits and Falkner [32] and then Pedersen andPeskir [64] who worked with non-singular diffusions. Finally Roynette, Valloisand Yor [79] used Rost criteria to investigate Brownian motion and its localtime [79]. There were also some older works on n-dimensional Brownian motion(Falkner [26]) and right processes (Falkner-Fitzsimmons [27]).

The number of works in the second main category is overwhelming and wewill not try to describe it for now as it will be done progressively through thissurvey. We want to mention, however, that the emphasis was placed on onehand on the solution of Azema and Yor and its accurate description, and on theother hand, following Perkins’ work, on the control of maximum and minimumof the stopped martingale (Kertz and Rosler [46], Hobson [39], Cox and Hobson[17]).

All of the solutions mentioned above, along with some others, are discussedin Section 3, some of them in more detail, some very briefly. The exceptionsare Perkins’, Azema and Yor’s and Root’s solutions. The first one is discussedin Section 6 and the second one in Section 5. It is discussed in detail with anemphasis on a real-potential theoretic approach, following Chacon and Walsh.This is introduced, for measures with finite support, in Section 4. The sectionfollowing the one on Azema and Yor’s solution is devoted to Root’s solution. Itcontains a detailed description of his results, those of Loynes and famous resultsof Rost. The two sections that follow, Sections 8 and 9, focus on certain exten-sions of the classical case. The former treats the case of non-centered measuresand uniform integrability questions and the latter extends the problem and itssolutions to the case of real diffusions. The last two sections are concernedwith applications: Section 10 develops the link between the Skorokhod embed-ding problem and optimal stopping theory and the last section shows how theembedding extends to processes; it also discusses some other applications.

To end this introduction, we need to make one remark about the terminology.Skorokhod’s name is associated with several “problems”. In this work, we areconcerned solely with the Skorokhod embedding problem, which we call also“Skorokhod’s stopping problem” or simply “Skorokhod’s problem”. This shouldnot generate any confusion.

2 The problem and its methods 6

2 The problem and its methods

Let us first state the problem formally in its classical form. If not mentionedotherwise, (Bt)t≥0 is always a Brownian motion, defined on a probability space(Ω,F ,P) and Ft = σ(Bs : s ≤ t) is its natural filtration taken completed.

Problem (The Skorokhod embedding problem). For a given centeredprobability measure µ on R,

∫

Rxdµ(x) = 0,

∫

Rx2dµ(x) = v ∈ [0,+∞], find

a stopping time T , such that BT ∼ µ, ET = v and (BT∧t)t≥0 is a uniformlyintegrable martingale.

Actually, in the original work of Skorokhod [81], the variance was assumedto be finite v <∞, but it was not crucial for his reasoning and it was soon notedthat the condition of uniform integrability of (BT∧t) is the correct way to expressthe idea that T should be “small” (see Section 8). We point out that without thissupplementary condition on T , there is a trivial solution to the above problem.We believe it was first proposed by Doob. For any probability measure µ,we denote its distribution function Fµ(x) = µ((−∞, x]) and F−1

µ its right-continuous inverse. The distribution function of a standard normal variableis denoted Φ. Then the stopping time T = inft ≥ 2 : Bt = F−1

µ (Φ(B1)),embeds µ in Brownian motion BT ∼ µ. However, we always have ET = ∞.

As noted in the Introduction, the Skorokhod embedding problem has beengeneralized in various manners. Still, even if we stick to the original formulation,it has seen a number of different solutions, developed using almost all the majortechniques which have flourished in the theory of stochastic processes duringthe last forty years. It is this methodological richness which we want to stressfrom the beginning. It has at least two sources. Firstly, different applicationsmotivated, in the most natural way, the development of solutions formulatedwithin different frameworks and, which is often connected, proved in differentmanners. Secondly, the same solutions, as the Azema-Yor solution for example,were understood better as they were studied using various methodologies.

Choosing the most important members from a family, let alone orderingthem, is always a risky procedure. Yet, we believe it has some interest and it isworthwhile trying. We venture to say that the three most important theoreticaltools in the history of Skorokhod embedding have been: potential theory, thetheory of excursions and local times and martingale theory. We will try toargue for this statement briefly, by pointing to later sections where particularembeddings which use these tools are discussed.

Potential theory has two aspects for us: the simple, real-potential aspectand the more sophisticated aspect of general potential theory for stochasticprocesses. The former starts with the study of the potential function µU(x) =−∫

|x − y|dµ(y). It was used first by Chacon and Walsh (see Section 3.8) andproved to be a simple yet very powerful tool in the context of the Skorokhodproblem. It lies at the heart of the studies performed by Perkins (Section 6)and more recently by Hobson [39] and Cox and Hobson (Section 3.19). We showalso how it can yield the Azema-Yor solution (see Section 5.1).

2 The problem and its methods 7

General potential theory was used by Rost to establish his results on theexistence of a solution to the Skorokhod embedding problem in a very generalsetup. Actually, they are better known as results in potential theory itself underthe name of “continuous time filling-scheme” (see Section 3.5). They allowedRost not only to formulate his own solution but also to understand well thesolution developed by Root and its deep optimal properties (see Section 7).In a sense, Root’s solution relies on potential theory, even if he doesn’t use itexplicitly in his work. Recently, Rost’s results were used by Roynette, Valloisand Yor to develop an embedding for Brownian motion and its local time atzero (see Section 3.17).

Probably the single most fruitful idea in the history of solving the embeddingproblem is the following. Compare the realization of a Brownian trajectory (ormore generally its functional in which we want to embed the given measure)with the realization of a (function of) some well-controlled increasing process.Use the latter to decide when to stop the former. We have two such “well-controlled” increasing processes to hand: the one-sided supremum of Brownianmotion and the local time (at zero). And it comes as no surprise that to realizethis idea we need to rely on either martingale theory or the theory of local timesand excursions. The one-sided supremum of Brownian motion appears in thecelebrated Azema and Yor’s solution and their original proof in [3] used solelymartingale-theory arguments (see Section 5.4). Local times were first applied tothe Skorokhod embedding problem by Monroe (see Section 3.6) and then usedby Vallois (see Section 3.12). Bertoin and Le Jan used additive functionals ina very general setup (Section 3.15). Excursion theory was used by Rogers [72]to give another proof of the Azema-Yor solution. It is also the main tool in arecent paper by Ob loj and Yor [61].

We speak about a solution to the Skorokhod embedding problem when theembedding recipe works for any centered measure µ. Yet, if we look at a givendescription of a solution, it was almost always either developed to work withdiscrete measures and then generalized, or on the contrary, it was thought towork with diffuse measures and only later generalized to all measures. Thisbinary classification of methods happens to coincide nicely with a tool-basedclassification. And so it seems to us that the solutions which start with atomicmeasures rely on analytic tools which, more or less explicitly, involve potentialtheory and, vice versa, these tools are well-suited to approximate any measureby atomic measures. On the other hand, the solutions which rely on martingaletheory or the theory of excursions and local times, are in most cases easilydescribed and obtained for diffuse measures. A very good illustration for ourthesis is the Azema-Yor solution as it has been proven using all of the above-mentioned tools. And so, when discussing it using martingale arguments as inSection 5.4, or arguments based on excursion theory (as in Rogers [72]), it ismost natural to concentrate on diffuse measures, while when relying on the real-potential theory as in Section 5.1, we approximate any measure with a sequenceof atomic measures.

3 A guided tour through solutions 8

3 A guided tour through solutions

In the previous section, we tried to give a certain classification of known solu-tions to the Skorokhod embedding problem based mainly on their methodology.We intended to describe the field and give the reader some basic intuition. Weadmit that our discussion left out some solutions, as for example the one pre-sented by Bass (see Section 3.11), which did not fit into our line of reasoning.We will find them now, as we turn to the chronological order. We give a brief de-scription of each solution, its properties and relevant references. The referencesare summarized in Table 1.

3.1 Skorokhod (1961)

In a paper in 1961 (translated from the Russian in 1965), Skorokhod [81] definedthe problem under consideration and proposed the first solution. He introducedit to obtain some invariance principles for random walks. Loosely speaking, theidea behind his solution is to observe that any centered measure with two atomscan be embedded by the means of first exit time. One then describes the measureµ as a mixture of centered measures with at most two atoms and the stoppingrule consists in choosing independently one of these measures, according to themixture, and then using the appropriate first exit time. More rigorously hissolution can be described as follows (after Perkins [65]). For Brownian motion(Bt) and a centered probability measure µ on R, define for any λ > 0

−ρ(λ) = inf

y ∈ R :

∫

1x≤y∪x≥λxdµ(x) ≤ 0

.

Let R be an independent random variable with the following distribution func-tion:

P(R ≤ x) =

∫

1y≤x

(

1 +y

ρ(y)

)

dµ(y).

Then, the stopping time defined by

TS = inft : Bt /∈ (−ρ(R), R),

satisfies BTS∼ µ and (Bt∧TS

) is a uniformly integrable martingale. In his orig-inal work Skorokhod assumed moreover that the second moment of µ was finitebut this is not necessary. The work of Skorokhod was somewhat generalized inStrassen [82] and Breiman [8].

3.2 Dubins (1968)

Three years after the translation of Skorokhod’s original work, Dubins [23] pre-sented a different approach which did not require any random variable indepen-dent of the original process itself (the variable R in the above construction).The idea of Dubins is crucial as it will be recovered in various solutions later on.

3.3 Root (1968) 9

It is best described in Meyer [55]. Given a centered probability measure µ, weconstruct a sequence of probability measures µn with finite supports, convergingto µ. We start with µ0 - the measure concentrated in the expectation of µ (i.e.0). We look then at µ− and µ+ - the measure µ restricted (and renormalized)to two resulting intervals ((−∞, 0], (0,+∞)) - and take its expectations, m+

and m−. Choosing the weights in a natural way we obtain a centered measureµ1 (concentrated on 2 points m+ and m−). The three points considered so far(m+,m−, 0) cut the real line into 4 intervals and we again look at µ restrictedand renormalized on these 4 intervals. We take its expectations and chooseappropriate weights and obtain a centered measure µ2. We continue this con-struction, in the nth step we pass from a measure µn concentrated in 2n−1 pointsand a total of 2n−1 points cutting the line to the measure µn+1 concentrated in2n new points (we have therefore a total of 2n+1−1 points cutting the real line).Not only does µn converge to µ but the construction embeds easily in Brownianmotion using the successive hitting times (T k

n )1≤k≤2n−1 of pairs of new points(nearest neighbors in the support of µn+1 of a point in the support of µn). T is

the limit of T 2n−1

n and it is straightforward to see that it solves the Skorokhodproblem.

Bretagnolle [9] has investigated this construction and in particular the ex-ponential moments of Dubins’ stopping time.

3.3 Root (1968)

The third solution to our problem came from Root. This solution receivedrelatively little attention up to this moment even though it has a remarkableproperty - it minimizes the variance (or equivalently the second moment) of thestopping time, as shown by Rost [78]. This is probably due to the fact that thesolution is not explicit and is actually hard to describe even for simple terminaldistributions. Still, as this solution was the starting point of our work, we willtalk about it separately in Section 7.

3.4 Hall (1968)

Hall described his solution to the original Skorokhod problem in a technicalreport at Stanford University [35] and, to our best knowledge, it stayed un-published. He did however publish an article on applications of the Skorokhodembedding to sequential analysis [36], where he also solved the problem forBrownian motion with drift. This is quite remarkable, as the problem wastreated again over 30 years later by Falkner, Grandits, Pedersen and Peskir (cf.Section 3.16) and this work is never cited. Hall’s solution to the Skorokhod em-bedding for Brownian motion with a drift was an adaptation of his method forregular Brownian motion, in very much the same spirit that the just mentionedauthors were adapting Azema and Yor’s solution somewhat 30 years later.

We want to discuss Hall’s basic solution as it pictures well the way peoplewere thinking and dealing with the embedding problem at that time. Hall lookedat Skorokhod’s and Dubins’ solutions and saw that one could start as Dubins

3.5 Rost (1971) 10

does but then proceed in a way similar to Skorokhod’s, only this time makingthe randomization internal.

More precisely, first he develops an easy randomized embedding based onan independent, two-dimensional random variable. For µ, an integrable proba-bility measure on R,

∫

xdµ(x) = m < ∞, define a distribution on R2 through

dHµ(u, v) = (u − v)( ∫∞

m (x −m)dµ(x))−1

1v≤m≤udµ(v)dµ(u). Then, for a cen-tered measure (m = 0), the stopping time T = inft ≥ 0 : Bt /∈ (U, V ), where(U, V ) ∼ H independent of B, embeds µ in B, i.e. BT ∼ µ.

The second step consists in starting with an exit time, which will give usenough randomness in B itself to make the above randomization internal. Thecrucial tool is the classical Levy result about the existence of a measurabletransformation from a single random variable with a continuous distributionto a random vector with an arbitrary distribution. Define µ+ and µ−, asabove, to be the renormalized restrictions of µ to R+ and R− respectively,and denote their expectations m+ and m− (i.e. m+ = 1

µ((0,∞))

∫

(0,∞)xdµ(x),

m− = 1µ((−∞,0])

∫

(−∞,0]xdµ(x)). Take T1 = Tm+∧m−

to be the first exit time

from [m−,m+]. It has a continuous distribution. There exist therefore mea-surable transformations f+ and f− such that (U i, V i) = f i(T1) has distribu-tion Hµi

, i ∈ +,−. Working conditionally on BT1 = mi, the new starting(shifted) Brownian motion Wt = Bt+T1 , and the stopping time T1 are indepen-dent. Therefore, the stopping time T2 = inft ≥ 0 : Wt /∈ (U i, V i) embedsµi in the new (shifted) Brownian motion W . Finally then, the stopping timeTH = T1 + T2 embeds µ in B.

As stated above, in [36] this solution was adapted to serve in the case of Bdt =

Bt+dt and Hall identified correctly the necessary and sufficient condition on theprobability measure µ for the existence of an embedding, namely

∫

e−2dxµ(dx) ≤1. His method is very similar to that of Falkner and Grandits (cf. Section 3.16and Section 9). As a direct application, Hall also described further the class ofdiscrete submartingales that can be embedded in (Bd

t ) by means of an increasingfamily of stopping times.

3.5 Rost (1971)

The first work [75], written in German, was generalized in [76]. Rost’s main workon the subject [76], does not present a solution to the Skorokhod embeddingproblem in itself. It is actually best known for its potential theoretic results,namely for the “filling scheme” in continuous time. It answers however a generalquestion of existence of a solution to the embedding problem. The formulationin terms of a solution to the Skorokhod embedding problem is found in [77].

Consider a Markov process (Xt)t≥0 on a locally compact space (E, E) witha denumerable base and a Borel σ-field. Denote U the potential kernel of X .Suppose that η and µ are positive measures on E and that µU and ηU are σ-finite. Then, for initial distribution X0 ∼ η, there exists a stopping time T suchthat XT ∼ µ if and only if

µU ≤ ηU. (3.1)

3.6 Monroe (1972) 11

Actually, Rost gives a more general condition that works for any finite mea-sures but its description requires some more space and the one given here con-tains the basic intuition that is important for us. Let us examine the conditiongiven above. It can be rephrased as “µ happens after η”. It says that

µU(f) ≤ ηU(f),

for any positive, Borel, bounded function f : E → R, where νU(f) = Eν

[

∫∞0f(Xt)dt

]

and Eν denotes the expectation with respect to the initial measure ν. The right

hand can be written as: Eη

[

∫ T

0f(Xt)dt+

∫∞Tf(Xt)dt

]

. One sees that in order

to be able to define T condition 3.1 is needed. On the other hand if it is satisfied,

one can also hope that Eη

[

∫ T

0 f(Xt)dt]

= ηU(f) − µU(f), defines a stopping

time. Of course this is just an intuitive justification of a profound result andthe proof makes a considerable use of potential theory.

Rost developed further the subject and formulated it in the language of theSkorokhod embedding problem in [77]. We note that the stopping rules obtainedmay be randomized as in the original work of Skorokhod.

An important continuation of Rost’s work is found in two papers of Fitzsim-mons. In [28] (with some minor correction recorded in [29]) an existence theoremfor embedding for quite general Markov process is obtained as a bi-product ofinvestigation of a natural extention of Hunt’s balayage LBm. The second paper[30], is devoted to the Skorokhod problem for general right Markov processes.Suppose that X0 ∼ ν and the potentials of ν and µ are σ-finite and µU ≤ νU .It is shown that there exists then a monotone family of sets C(r); 0 ≤ r ≤ 1such that if T is the first entry time of C(R), where R is independent of X anduniformly distributed over [0, 1], then XT ∼ µ. Moreover, in the case whereboth potentials are dominated by some multiple of an excessive measure m,then the sets C(r) are given explicitly in terms of Radon-Nikodym derivativesd(νU)dm and d(µU)

dm .

3.6 Monroe (1972)

We want to examine this work only very briefly. Monroe [58] doesn’t have anyexplicit solutions but again what is important is the novelty of his methodology.Let (La

t )t≥0 be the local time at level a of Brownian motion or more generally,of a continuous martingale (Xt). Then the support of dLa

t is contained in theset Xt = a almost surely. We can use this fact and try to define the stoppingtime T through inequalities involving local times at levels a ranging throughthe support of the desired terminal distribution µ. The distribution of BT willthen have the same support as µ. Monroe shows that we can carry out this ideaand define T in such way that BT ∼ µ and (Bt∧T ) is uniformly integrable. Wewill see an explicit and more general construction using this approach in 3.15.To a lesser extent also works of Jeulin-Yor [45] and Vallois [84] may have beeninspired by the presented method.

3.7 Heath (1974) 12

3.7 Heath (1974)

Heath [38] considered the Skorokhod embedding problem for n-dimensionalBrownian motion. More precisely, he considers Brownian motion killed at thefirst exit time from the unit ball. This comes from the fact that he relies onthe results of Rost [76], who worked with transient processes. The constructionis an adaptation of a potential theoretic construction of Mokobodzki. The re-sults generalize to processes satisfying the “right hypotheses”. We note that thestopping times are randomized.

3.8 Chacon and Walsh (1974-1976)

Making use of potential theory on the real line, Chacon and Walsh [15] gave anelegant and simple description of a solution that happens to be quite general. Inparticular, Dubins’ solution is a special case of it. Actually, Chacon and Baxter[5] worked earlier with a more general setup and obtained results for examplefor n-dimensional Brownian motion, but it is the former paper which interestsus more here.

For a centered probability measure µ on R, its potential function is definedvia µU(x) = −

∫

R|x − y|dµ(y). The name “potential” refers to the Newtonian

potential on R and not to the potential kernel of Brownian motion. We wantto stress this, as it can generate certain confusion. We can speak about thepotential of µ without any process and indeed, one of the advantages of theapproach we are about to develop, is that it can be applied to various processeswithout any changes.

Informations about the potential µU can be found, for example, in Chacon[13]. The following inequality between potential kernels is fairly obvious:

µU(x) = −∫ ∞

−∞|x− y|dµ(y) ≤ δ0U(x) = −|x|,

furthermore, |µU(x) + |x|| → 0 as |x| → ∞. Potential µU is a continuous,concave function which break at atoms of µ. It also encodes information aboutthe weak convergence of probability measures. Precisely, for µn, µ centeredprobability measure µn ⇒ µ if and only if µnU → µU pointwise. Finally, thearea between µU and δ0U corresponds to the second moment of µ:

∫

R(|x| +

µU(x))dx = −∫

Rx2dµ(x). The crucial property of potentials for us, is that, if

B0 ∼ ν, then the law of BTa,bhas the potential equal to νU outside of (a, b)

and linear on (a, b).We are ready to describe the solution proposed by Chacon and Walsh. Write

µ0 = δ0. Choose a point between the curves µ0U and µU , and draw a line l1through this point which stays above µU (actually in the original constructiontangent lines are considered, which is natural but not necessary). This linecuts the potential δ0U in two points a1 < 0 < b1. We consider new potentialµ1U given by µ0U on (−∞, a1] ∪ [b1,∞) and linear on [a1, b1]. We iterate theprocedure. The choice of lines which we use to produce potentials µnU is notimportant. It suffices to see that we can indeed choose lines in such a way that

3.8 Chacon and Walsh (1974-1976) 13

µnU → µU (and therefore µn ⇒ µ). This is true, as µU is a concave functionand it can be represented as an infimum of a countable number of affine func-tions. The stopping time is obtained therefore through a limit procedure. Ifwe write Ta,b for the exit time of [a, b] and θ for the standard shift operator,then T1 = Ta1,b1 , T2 = T1 + Ta2,b2 θT1 , ..., Tn = Tn−1 + Tan,bn θTn−1 andT = lim Tn. It is fairly easy to show that the limit is almost surely finite and fora measure with finite second moment, E T =

∫

x2dµ(x). The solution is easilyexplained with a series of drawings:

µU

µ0U

µU

µ0U

µU

µ1U

b1a1

µU

µ2U

b2a2

What is really being done on the level of Brownian motion? The procedurewas summarized very well by Meilijson [52]. Given a centered measure µ expressit as a limit of µn - measures with finite supports. Do it in such a way that thereexists a martingale (Xn)n≥1, Xn ∼ µn which has almost surely dichotomoustransitions (i.e. conditionally on Xn = a there are only two possible values forXn+1). Then embed the martingale Xn in Brownian motion by successive firsthitting times of one of the two points.

Dubins’ solution is a special case of this one. What Dubins proposes isactually a simple method of choosing the tangents. To obtain potential µ1Udraw tangent at (0, µU(0)), which will cut the potential δ0U in two pointsa < 0 < b. Then draw tangents in (a, µU(a)) and (b, µU(b)). The lines will cutthe potential µ1U in four points yielding potential µ2U . Draw tangents in thosefour points obtaining 8 intersections with µ2U which give new coordinates fordrawing tangents. Iterate.

Chacon and Walsh solution remains true in a more general setup. It isvery important to note that the only property of Brownian motion we haveused here is that for any a < 0 < b, (Bt∧Ta,b

)t≥0 is a uniformly integrable

martingale and so EBTa,b= 0, which implies P(BTa,b

= a) = bb−a . This is the

property which makes the potentials of µn piece-wise linear. However, this istrue not only for Brownian motion but for any continuous martingale (Mt) with

3.9 Azema-Yor (1979) 14

< M,M >∞= ∞ a.s.2 The solution presented here is therefore valid in thismore general situation.

We think that Chacon and Walsh solution has received too little attentionin the literature. We will try to demonstrate how it allows to recover intuitivelyother solutions. As a matter of fact, in the case of a measure µ with finitesupport, this solution contains as a special case not only the solution of Dubinsbut also Azema and Yor’s one. We will devote Section 4 to talk about it.

3.9 Azema-Yor (1979)

This solution is probably the single best known solution to the Skorokhod em-bedding problem. It has received a lot of attention in the literature and its prop-erties were investigated in detail. In consequence we won’t be able to presentall of the relative material. We will first discuss the solution for measures withfinite support in Section 4, and then the general case in Section 5.

3.10 Falkner (1980)

Embedding a measure into a n-dimensional Brownian motion is discussed [26].This was then continued for right processes by Falkner and Fitzsimmons [27].Even though we do not intend to discuss this topic here we want to pointhowever at some difficulties which arise in the multi-dimensional case. They aremainly due to the fact the Brownian motion does not visit points any more. Ifwe consider measures concentrated on the unit circle it is quite easy to see thatonly one of them, namely the uniform distribution, can be embedded by meansof an integrable stopping time. Distributions with atoms cannot be embeddedat all.

3.11 Bass (1983)

Bass’ solution is basically an application of Ito’s formula and Dubins-Schwarztheorem. It uses also some older results of Yershov. Let F be the distributionfunction of a centered measure µ, Φ the distribution function of N (0, 1) andpt(x) the density of N (0, t). Define function g(x) = F−1(Φ(x)). Take a Brow-nian motion (Bt), then g(B1) ∼ µ. Using Ito’s formula (or Clark’s formula) wecan write

g(B1) =

∫ 1

0

a(s,Bs)dBs, where a(s, y) = −∫

∂p1−s(z)

∂zg(z + y)dz.

We put a(s, y) = 1 for s ≥ 1 and define a local martingale M by Mt =∫ t

0a(s,Bs)dBs. Its quadratic variation is given by R(t) =

∫ t

0a2(s,Bs)ds. De-

note by S the inverse of R. Then the process Nt = MS(t) is a Brownian mo-tion. Furthermore NR(1) = M1 ∼ µ. Bass [4] then uses a differential equationargument (following works by Yershov [88]) to show that actually R(1) is a

2One can also consider real diffusions on a natural scale.

3.12 Vallois (1983) 15

stopping time in the natural filtration of N . The argument allows to con-struct, for an arbitrary Brownian motion (βt) a stopping time T such that(NR(1), R(1)) ∼ (βT , T ), which is the desired solution to our problem.

In general the martingale M is hard to describe. However in simple casesit is straightforward. We give two examples. First consider µ = 1

2 (δ−1 + δ1).Then

Mt = sgn(Bt)(

1 − 2Φ( −|Bt|√

1 − t

))

, t ≥ 1.

Write g1 for “the last 0 before time 1,” g1 = supu < 1 : Bu = 0. This is nota stopping time. The associated Azema supermartingale (cf. [89] p. 41-43) isdenoted Zg1

t = P(g1 > t|Ft), where (Ft) is the natural filtration of (Bt). Wecan rewrite M in the following way

Mt = sgn(Bt)(1 − 2Zg1t ) = E

[

sgn(Bt)(1 − 21g1>t)|Ft

]

.

Consider now µ = 34δ0 + 1

8 (δ−2 + δ2) and write ξ = −Φ−1(18 ). Then

Mt = sgn(Bt)2(

Φ(−ξ + |Bt|√

1 − t

)

− Φ(−ξ − |Bt|√

1 − t

))

= E[

1B1>ξ − 1B1<−ξ|Ft

]

.

What really happens in the construction of the martingale M should now beclearer: we transform the standard normal distribution of B1 into µ by a mapR → R and hit this distribution by readjusting Brownian paths.

3.12 Vallois (1983)

In their work Jeulin and Yor [45] studied stopping times T h,k = infB+t h(Lt) +

B−t k(Lt) = 1, where B and L are respectively Brownian motion and its local

time at 0 and k, h are two positive Borel functions which satisfy some additionalconditions. They described the distributions of XT and (XT , T ). This, usingthe Levy theorem, can be seen as a generalization of results of Azema and Yor.Vallois [84] follows the framework of Jeulin and Yor and develops a completesolution to the embedding problem for Brownian motion. For any probabilitymeasure on R he shows there exist two functions h+ and h− and δ > 0 suchthat

TV = TLδ ∧ inft : B+

t = h+(Lt) or B−t = h−(Lt),

where TLδ = inft : Lt = δ. His method allows to calculate the functions h+

and h− as is shown in several examples.Furthermore, Vallois proves that (Bt∧T ) is a uniformly integrable martingale

if and only if µ has first moment and is centered. In case of a measure withfinite second moment we have also that EB2

T = ET .The formulae obtained by Vallois are quite complicated. In the case of

symmetric measures they can simplified considerably as noted in our paperwith Marc Yor [61].

3.13 Perkins (1985) 16

3.13 Perkins (1985)

Perkins [65] investigated the problem of H1-embedding and developed a solutionto the Skorokhod problem that minimizes stochastically the distribution of themaximum of the process (up to the stopping time) and maximizes stochasticallythe distribution of the minimum. We will present his work in Section 6.

3.14 Jacka (1988)

Jacka [43] was interested in what can be seen as a converse to Perkins’ goal. Helooked for a maximal H1-embedding, that is an embedding which maximizesstochastically the distribution of the maximum of the absolute value of theprocess. It seems that his work was mainly motivated by applications in theoptimal stopping theory. In his subsequent paper, Jacka [44] used his embeddingto recover the best constants Cp appearing in

E[

supt≥0

Mt

]

≤ Cp

(

E |M∞|p)

1p

(p > 1). (3.2)

The interplay between Skorokhod embeddings and optimal stopping theory, ofwhich we have an example above, is quite rich and diverse. We will come backto this matter in Section 10.

The basic idea behind Jacka’s work is the following. The Azema-Yor solutionmaximizes the maximum. If one reverses this solution, it can be seen to minimizethe minimum (see Section 5). All one has to do then is wait as long as possibleto see weather Brownian motion becomes rather positive or negative and thenuse the appropriate, Azema-Yor type, embedding. More precisely, first wait tillthe first exit time T−k,k, from some interval [−k, k]. This corresponds to cuttingthe potential δ0U with a horizontal line at level −k. The lowest we can get isthe horizontal tangent to µU . That is we have k such that the line y = −k istangent to µU in some (x∗, µU(x∗), such that dµU

dx (x∗) = 0.If Brownian motion exists at the top, BT−k,k

= k, then use the standardAzema-Yor embedding for the measure µ1 defined through µ1(x∗) = 1 andµ1(x) = 2µ(x) for x > x∗. If Brownian motion exists at the bottom BT−k,k

=−k, then use the reversed Azema-Yor embedding (cf. Section 5.3) to embed themeasure µ2 defined through µ2(x∗+) = 0 and µ2((−∞, x]) = 2µ((−∞, x]). Acareful reader might have noticed that we imply that x∗ is the median for µ. Thisindeed is true. We will comment more on this construction in Section 6.2, givethe formulae and justify them using the real potential theoretic approach. Wewill also indicate there some important inequalities for submartingales, whichprove optimal thanks to this embedding.

3.15 Bertoin and Le Jan (1993)

Tools used by Bertoin and Le Jan in [6] for constructing a solution to theSkorokhod problem are similar to these used by Monroe [58]. Authors however

3.15 Bertoin and Le Jan (1993) 17

do not use Monroe’s work. They consider a much general setting and obtainexplicit formulae.

Let E be a locally compact space with a countable base and X a Huntprocess on E, with a regular recurrent starting point 0 ∈ E. Local time at 0is denoted by L. Authors use the correspondence between positive continuousadditive functionals and their Revuz measures. The invariant measure λ whichis used to define the correspondence is given by

∫

fdλ =

∫

dη

∫ ζ

0

f(Xs)ds+ cf(0),

where η is the characteristic measure of the excursion process, c is the delaycoefficient3 and ζ is the first hitting time of 0.

Take a Revuz measure µ with µ(0) = 0, µ(E) ≤ 1 and Aµ the positivecontinuous additive functional associated with it. Then T = inft : Aµ

t > Ltembeds µ in X . However this solution is not a very useful one as we haveELT = ∞.

We consider an additional hypothesis:

There exists a bounded borel function Vµ, such that for every Revuz

measure ν:

∫

Vµdν =

∫

Ex(Aνζ )dµ(x). (3.3)

Then, if µ is a probability measure and γ0 = ||Vµ||∞, for any γ ≥ γ0, thefollowing stopping time embeds µ in X :

T γBLJ = inf

t ≥ 0 : γ

∫ t

0

(γ − Vµ(Xs))−1dAµ

s > Lt

. (3.4)

Furthermore this solution is optimal in the following sense. Consider any Revuzmeasure ν with Aν not identically 0. Then E0A

νTγBLJ

=∫

(γ − Vµ)dν and for

any solution S of the embedding problem for µ, E0AνS ≥

∫

(γ0 − Vµ)dν.Authors then specify to the case of symmetric Levy processes. Let us in-

vestigate this solution in the case of X a real Brownian motion. The functionVµ exists and its supremum is given by γ0 = ||Vµ||∞ = 2 max

∫

x+dµ,∫

x−dµ.Hypothesis 3.3 is equivalent to µ having a finite first moment. In this caseintroduce

ρ(x) =

2∫

a>x(a− x)dµ(a) , for x ≥ 0

2∫

a<x(x− a)dµ(a) , for x < 0.

Then

T γ0

BLJ = inf

t : γ0

∫

Lxt

ρ(x)dµ(x) > L0

t

.

Note that if supp(µ) ⊂ [a, b] the ρ(a) = ρ(b) = 0, so in particular TBLJ ≤ Ta,b,the first exit time from [a, b].

3That is the holding time in 0 has exponential distribution with parameter c.

3.16 Falkner, Grandits, Pedersen and Peskir (2000 - 2002) 18

3.16 Falkner, Grandits, Pedersen and Peskir (2000 - 2002)

We want to mention, exceptionally, several solutions in this one subsection.The reason being that they are all closely connected and subsequent ones werethought as generalizations of the preceeding. The solution developed by Azemaand Yor works for recurrent real diffusions. A natural question consists intaking into account a simple transient diffusion and asking about the existenceof a solution to the embedding problem.

The first problem considered was to develop a solution to the embeddingproblem for Brownian motion with drift. This was first done by Hall backin 1969, however his work doesn’t seem to be well known. Consequently, thesubject was treated again (in somewhat similar way) by Grandits and Falkner[32] and Peskir [68]. They show that for a probability measure µ there ex-ists a stopping time T such that XT = BT + dT ∼ µ (d > 0) if and only if∫

e−2dxdµ(x) ≤ 1. Authors work with a martingale Yt = e−2dXt − 1 rather thanwith the process X itself and they adapt the arguments used by Azema and Yor.Grandits and Falkner gave an explicit formula for T when

∫

e−2dxdµ(x) = 1,while Peskir obtained an explicit formula valid in all cases and showed the so-lution is pointwise the smallest possible stopping time satisfying BT + dT ∼ µ,which maximizes stochastically the supremum of the process up to the stoppingtime. The result coincides with the standard Azema-Yor solution when d → 0.The arguments used were extended by Pedersen and Peskir [64], to treat anynon-singular diffusion on R starting at 0.

The methodology of these embeddings can be summarized as follows. Givena diffusion compose it with its scale function. The result is a continuous localmartingale. Use the Dambis, Dubins-Schwarz theorem to work with a Brownianmotion. Then use the Azema-Yor embedding and follow the above steps in areversed direction. Some care is just needed as the local martingale obtainedfrom the diffusions might converge. We develop this subject in Section 9.

3.17 Roynette, Vallois and Yor (2002)

In the article [79], authors consider Brownian motion with its local time at 0as a Markov process X and apply Rost’s results to derive conditions on µ - aprobability measure on R×R+ - under which there exists an embedding of µin X . They give an explicit form of the condition and of the solution if thereexists one.

Consider µ(dx, dl) = ν(dl)M(l, dx) a probability measure on R×R+, whereν is a probability measure on R+ and M is a Markov kernel. Then there existsa stopping time T , such that (BT , LT ) ∼ µ and (Bt∧T ) is a uniformly integrablemartingale if and only if

1.∫

R×R+|x|µ(dx, dl) <∞,

2. m(l) :=∫

Rx+M(l, dx) =

∫

Rx−M(l, dx) <∞,

3. m(l)ν(dl) = 12ν[l,∞)dl.

3.18 Hambly, Kersting and Kyprianou (2002) 19

Moreover, if ν is absolutely continuous with respect to the Lebesgue measure,then T is a stopping time with respect to Ft = σ(Bs : s ≤ t).

An explicit construction of T is given. It is carried out in two steps. Firstlya stopping time T1 is constructed which embeds v in L - it follows argumentsof Jeulin and Yor [45]. Secondly, using Azema-Yor embedding, T > T1 is con-structed such that conditionally on LT1 = l, BT ∼M(l, dx) and Bt has no zerosbetween T1 and T , so that LT1 = LT .

This construction yields two interesting studies. It enables authors to in-vestigate the possible laws of LT , where T is a stopping time such that (Bt∧T )is a uniformly integrable martingale, thus generalizing the work of Vallois [85].It also gives the possibility to look at question of independence of T and BT

(primarily undertaken in [20], [21]).

3.18 Hambly, Kersting and Kyprianou (2002)

These authors [37] develop a Law of Iterated Logarithm for random walks condi-tioned to stay nonnegative. One of the ingredients of their proof is an embeddingof a given distribution into the Bessel(3) process. This distribution is actuallythe distribution of a renewal function of the initial distribution under the lawof random walk conditioned to stay nonnegative. Authors use properties of thelatter to realize their embedding. It is done in two steps - first they stop theprocess in an intermediate distribution. Both parts of the construction requireexternal randomization, independent of the process. This construction is notextremely interesting from our point of view. It is interesting for its directrelationship and applications in the study of random walks.

3.19 Cox and Hobson (2002)

The work of Cox and Hobson [17] can be seen as a generalization of Perkins’solution to the case of regular diffusions and measures which are not necessarilycentered. Indeed, consider any probability measure µ on R. Authors define twofunctions γ− and γ+, which for a centered probability measure µ coincide withthe ones defined by Perkins (cf. Section 6.1). For a continuous local martingale(Mt) define its maximum and minimum processes through SM

t = sups≤tMs

and sMt = − infs≤tMs. Then, given that < M,M >∞= ∞, the stopping timeTCH = inft > 0 : Mt /∈ (−γ−(SM

t ), γ+(sMt )) embeds µ in M . Furthermore, itminimizes stochastically the supremum and maximizes stochastically the min-imum (cf. (6.2), (6.3)). The setup is then extended to regular diffusions usingscale functions. For regular diffusions scale functions exist and are continuous,strictly increasing. This implies that the maximum and minimum are preserved.Conditions on the diffusion (through its scale function) and on µ for the exis-tence of an embedding are given.

Further, Cox and Hobson investigate the existence of Hp-embeddings. Con-ditions are given that in important cases are necessary and sufficient.

3.20 Ob loj and Yor (2004) 20

3.20 Ob loj and Yor (2004)

Almost all explicit solutions discussed so far dealt with continuous processes.The solution proposed by Bertoin and Le Jan (cf. Section 3.15) is the only ex-ception, and even this solution is explicit only if we have easy access to additivefunctionals and can identify the function Vµ given by (3.3). Stopping discontin-uous processes is in general harder, as we have less control over the value at thestopping time. Yet, in some cases, we can develop tools in discontinuous setupinspired by the classical framework. This is the case of an embedding for “nice”functionals of Brownian excursions, described in our joint work with Marc Yor[61]. This solution was inspired by the Azema-Yor solution, as we explain below.For the sake of simplicity we will not present the reasoning found in Ob loj andYor [61] in all generality but we will rather restrain ourselves to the case of theage process. We are interested therefore in developing an embedding for thefollowing process: At = t − gt, where gt = sups ≤ t : Bs = 0 is the last zerobefore time t. In other words, At is the age, at time t, of the Brownian excursionstraddling time t. This is a process with values in R+, which is discontinuousand jumps down to 0 at points of discontinuity. More precisely, we will focuson an equivalent process given by At =

√

π2 (t− gt). This process is, in some

sense, more natural to consider, as it is just the projection of the absolute valueof Brownian motion on the filtration generated by the signs.

Starting with the standard Azema-Yor embedding, let us rewrite their stop-ping time using the Levy theorem.

T = inft ≥ 0 : St ≤ Ψµ(Bt) = inft ≥ 0 : St − Ψ−1µ (St) ≥ St −Bt

law= inft ≥ 0 : Ψµ(Lt) ≥ |Bt|, where Ψµ = Id− Ψ−1

µ . (3.5)

The idea of replacing the supremum process by the local time process comesfrom the fact that the latter, unlike the former, is measurable with respect tothe filtration generated by the signs. Even though the function Ψµ is in generalhard to describe, the above formula suggests that it might be possible to developan embedding for the age process of the following form: T = inft > 0 : At ≥ϕ(Lt), where the function ϕ should be a function of the measure µ we want toembed. This proves to be true. For any measure µ on R+ with µ(0), let [a, b]be its support, µ(x) = µ([x,∞)) its tail, and define

ψµ(x) =

∫

[0,x]

y

µ(y)dµ(y), for a ≤ x < b, (3.6)

ψµ(x) = 0 for 0 ≤ x < a and ψµ(x) = ∞ for x ≥ b. The function ψµ is right-continuous, non-decreasing and ψµ(x) < ∞ for all x < b. We can then defineits right inverse ϕµ = ψ−1

µ . Then the stopping time

TOY = inft > 0 : At ≥ ϕµ(Lt)

embeds µ in A, ATOY∼ µ.

4 Embedding measures with finite support 21

The proof is carried out using excursions theory (even though a differentone, relying on martingale arguments, is possible too). The crucial step is thecalculation of the law of LTOY

, which we give here. Note (τl) the inverse of thelocal time at 0 and (el) the excursion process and V (el) = Aτl− the rescaledlifetime of the excursion.

P(LTOY≥ l) = P(TOY ≥ τl)

= P(

on the time interval [0, τl] for every excursion es, s ≤ l,

its (rescaled) lifetime did not exceed ϕµ(s))

= P(

∑

s≤l

1V (es)≥ϕµ(s) = 0)

= exp

(

−∫ l

0

ds

ϕµ(s)

)

, (3.7)

where the last equality follows from the fact that the processNl =∑

s≤l 1F (es)≥ϕ(s),l ≥ 0, is an inhomogeneous Poisson process with appropriate parameter. Thecore of the above argument remains valid for much more general functionals ofBrownian excursions, thus establishing a methodology of solving the Skorokhodembedding in a number of important setups.

4 Embedding measures with finite support

We continue here the discussion of Chacon and Walsh’s solution from Section3.8. The taget measure µ, which is a centered probability measure on R, issupposed to have a finite support, µ =

∑ni=1 αiδxi

, where αi > 0,∑n

i=1 αi = 1and

∑ni=1 αixi = 0, xi 6= xj for 1 ≤ i 6= j ≤ n. Potential µU is then a piece-

wise linear function breaking at the atoms x1, . . . , xn. It is interesting thatin this simple setting several solutions are easily seen to be special cases of theChacon-Walsh solution.

4.1 Brownian embedding

Let (Mt) be a continuous martingale with < M,M >∞= ∞ a.s. and considerthe embedding problem of µ in M . First we can ask how many solutions to theembedding problem exist? The answer is simple: except for the trivial case, aninfinite number. Recall that n is the number of points in the support of µ. Wesay that two solutions T and S that embed µ are different if P(T = S) < 1.

Proposition 4.1. If n ≥ 3, then there exists an infinite, non-countable numberof solutions T that embed µ: MT ∼ µ, and such that (Mt∧T ) is a uniformlyintegrable martingale.

Proof. This is an easy consequence of the Chacon and Walsh embedding de-scribed in Section 3.8. We describe only how to obtain an infinity of solutionsfor n = 3, the ideas for higher values of n being the same. Our reasoning iseasiest explained with a series of drawings:

4.1 Brownian embedding 22

Solution Further developments

Skorokhod (1961) [81] Strassen [82], Breiman [8], Perkins [65]Meilijson [53]

Dubins (1968) [23] Meyer [55], Bretagnolle [9]Chacon R.-Walsh [15]

Hall, W.J. (1968) [35]Root (1969) [74] Loynes [50], Rost [78]Rost (1971) [76] Azema-Meyer [1], Baxter, Chacon P. [12],

Fitzsimmons [28], [29], [30]Monroe (1972) [58] Vallois [84], Bertoin-Le Jan [6]Heath (1974) [76]Chacon-Walsh (1974-76) [5], [15] Chacon-Ghoussoub [14]Azema-Yor (1979) [3] Azema-Yor [2], Pierre [70], Jeulin-Yor [45]

Rogers [72], Meilijson [52], Zaremba [90]van der Vecht [86], Pedersen-Peskir [63]

Falkner (1980) [26] Falkner-Fitzsimmons [27]Bass (1983) [4]Vallois (1983) [84]Perkins (1985) [65] Cox-Hobson[17]Jacka (1988) [43]Bertoin-Le Jan (1992) [6]Grandits-Falkner [32] Peskir (2000) [68], Pedersen-Peskir [64]Pedersen-Peskir [64]Roynette-Vallois-Yor (2002) [79]Hambly-Kersting-Kyprianou (2002) [37]Cox-Hobson (2002) [17]Ob loj-Yor (2004) [61]

Table 1: Genealogy of solutions to the Skorokhod embedding

4.1 Brownian embedding 23

δ0U

µU

b1a1

❳❳❳❳❳❳

µU

b2a2

PP

b1a1

µU

b2a2

b1a1

µU

b3a3

Note that x1 = a2, x2 = a3 and x3 = b3. Consider the set of paths ω whichare stopped in a2 = x1: AT = ω : M(ω)T (ω) = x1. It can be described asthe set of paths that hit a1 before b1 and then a2 before b2. It is clear fromour construction that we can choose these points in an infinite number of ways,each of which yields a different embedding (as paths stopped at x1 differ).

However in the original work Chacon and Walsh allowed only to take linestangent to the graph of µU . In this case we have to chose the order of drawingn− 1 tangent lines and this can be done in (n − 1)! ways. Several of them areknown as particular solutions. Note that we could just as well take a measurewith a countable support and the constructions described below remain true.

• Azema-Yor - solution consists in drawing the lines from left to right.This was first noticed, for measures with finite support, by Meilijson [52].Azema-Yor solution can be summarized in the following way. Let X ∼ µ.Consider a martingale Yi = E[X |1X=x1, . . . ,1X=xi

], Y0 = 0. Note that Yihas dichotomous transitions, it increases and then becomes constant andequal to X as soon as it decreases, Yn−1 = X . Embed this martingaleinto (Mt) by consecutive first hitting times. It suffices to remark nowthat the hitting times correspond to ones obtained by taking tangent linesin Chacon and Walsh construction, from left to right. Indeed, the lowerbounds for the consecutive hitting times are just atoms of µ in increasingorder from x1 to xn−1.

The other way to see all this is to use the property that the Azema-Yor solution maximizes stochastically the maximum of the process. If wewant the supremum of the process to be maximal we have to take carenot to “penalize big values” or in other words “not to stop too early in asmall atom” - this is just to say that we want to take the bounds of ourconsecutive stopping times as small as possible, which is just to say wetake the tangent lines from left to right.

Section 5 is devoted to the study of this particular solution in a generalsetting.

4.2 Random walk 24

• Reversed Azema-Yor - this solution consists in taking tangents fromright to left. It minimizes the infimum of the stopped process. We discussit in Section 5.3.

• Dubins - this solution is always a particular case of the Chacon and Walshsolution. In a sense it gives a canonical method of choosing the tangents.We described it in Section 3.8.

4.2 Random walk

When presenting above, in Section 3.8, the Chacon and Walsh solution, wepointed out that it only takes adventage of the martingale property of Brownianmotion. We could ask therefore if it can also be used to construct embeddingsfor processes with countable state space? And it is very easy to see the answeris affirmative, under hypothesis which garantee that no over-shoot phenomenoncan take place.

Let (Mn)n≥0 be a discrete martingale taking values in Z with M0 = 0 a.s.Naturally, a symmetric random walk provides us with a canonical example andactually this section was motivated by a question rasied by Takahiko Fujitaconcerning prcesiely the extension of Azema-Yor embedding to random walks.

Denote the range of M by R = z ∈ Z : infn ≥ 0 : Mn = z <∞ a.s. andlet µ be a centered probability measure on R. It is then an atomic measure4. Wecan still use the Chacon and Walsh methodology under one condition. Namely,we can only use such exit times Ta,b that both a, b ∈ R. Otherwise we wouldface over-shoot phenomena and the method would not work. It seems probableto us that for any measure µ there exists such a sequence of pairs (ak, bk)that ak, bk ∈ R, and T = limk→∞ Tk embeds µ, XT ∼ µ, where T0 = 0,Tk = infn ≥ Tk−1 : Mn /∈ [ak, bk], k ≥ 1. However, we were not able to provethis so far.

Naturally in the Azema-Yor method, the left end points, (ak), are just theconsequtives atoms of µ, which by definition are in R. The right-end points aredetermined by tangents to µU as is described in detail in Section 5.1 below.They are given by Ψµ(ak), where Ψµ is displayed in (5.3), and they might notbe in R.

Proposition 4.2. In the above setting suppose that the measure µ is such thatΨµ(R) ⊂ R. Then the Azema-Yor stopping time, T = infn ≥ 0 : Sn ≥Ψµ(Mn), where Sn = maxk≤nMk, embedds the measure µ in M , MT ∼ µ.

The same Proposition, for a random walk, was also obtained by TakahikoFujita [31]. He used discrete martingales, which are closely realted to the mar-tingales used by Azema and Yor [3] in their orignal proof, and which we discussin Section 5.4.

We want to point out the the condition Ψµ(R) ⊂ R is a necessary onefor the existence of a solution to the Skorokhod embedding problem. We give

4The support is not neccesarry finite but this is not crucial for the method, so we decidedto place this discussion in this section.

5 Around the Azema-Yor solution 25

a simple example. Consider a standard, symmetric random walk, Xn, andtake µ = 2

9δ−3 + 49δ0 + 1

3δ2. Then Ψµ(0) = 67 /∈ Z, but one can easily point

out a good embedding. It suffices to take tangents to µU from right to left,that is to consider the reversed Azema-Yor solution. Then the stopping timeT = infn ≥ T−1,2 : Mn /∈ [−3, 0] embeds µ in M .

5 Around the Azema-Yor solution

In this section we present various results concerning the solution proposed byAzema and Yor [3]. We first provide a very simple proof of the solution in Section5.1. Then, in 5.5 we discuss some of its properties. The proof encouraged ustoo to give a special, reversed form of the solution in 5.3. Finally, the proof hastwo advantages that are described in sections 5.2 and 9 respectively. Firstly, itgeneralizes easily to the case of non-trivial initial laws and secondly it generalizesto the case of real regular diffusions.

Let us give first a brief history of this solution. It was presented in the13th “Seminaire de Probabilites” in [3] together with a follow up [2], whichexamined some maximizing property (see Section 5.5 below) of the solution.The construction was carried out for a continuous martingale and extendedeasily to recurrent, real diffusions. The original proof was then simplified byPierre [70] and Vallois [84]. A proof using excursion theory was given by Rogers[72]. An elementary proof was given by Zaremba [90]. This work was in ouropinion underestimated. Not only does it give a “bare-handed” proof, but doingso it actually provides a convenient limiting procedure that allows us to see thatAzema and Yor’s solution is a special case of Chacon and Walsh’s solution. Thisfact, for measures with finite support has been observed by Meilijson [52], whoalso described some other properties of the Skorokhod problem [51] (see alsovan der Vecht [86]). The Azema-Yor solution has been recently generalized byseveral authors - we described their papers together above (see Section 3.16).Finally, using the identity in law (S−B,S) ∼ (|B|, L), works using the local timeat 0 to develop a solution to the embedding problem can be seen as generalizingAzema and Yor’s construction, and we mention therefore articles by Jeulin andYor [45] and by Vallois [84].

5.1 An even simpler proof of the Azema-Yor formula

In this section for simplicity of notation we write “AY solution” for Azema andYor’s solution TAY .

We saw in Section 4 how the AY construction for a measure with finitesupport is just a special case of the method described by Chacon and Walsh,and consists in taking as tangent lines the extentions of the linear pieces of thepotential from left to right. In this section we will see how to generalize thisidea.

Let µ be a centered probability measure on R and x a point in the interior ofits support, such that µ has no atom at x: µ(x) = 0. We are interested in the

5.1 An even simpler proof of the Azema-Yor formula 26

tangent line lx to µU at (x, µU(x)) and its intersection with the line (t,−t) :t ∈ R. Denote this intersection by (−Ψµ(x),Ψµ(x)). The following drawing

summarizes these quantities:

x

lx

Ψµ(x)

It is straightforward to calculate the equation of lx. Its slope is given by

dµU(t)dt |t=x and so we can write lx = (x, y) : y = dµU(t)

dt |t=x · x+ β.

µU(t) = −∫

R

|t− y|dµ(y) = −∫

(−∞,t)

(t− y)dµ(y) −∫

[t,∞)

(y − t)dµ(y)

= tµ([t,∞)) − tµ((−∞, t)) − 2

∫

[t,∞)

ydµ(y), (5.1)

where, to obtain the third equality, we used the fact that µ is centered and so∫

[t,∞)ydµ(y) = −

∫

(−∞,t)ydµ(y).

dµU(t)

dt= µ([t,∞)) − µ((−∞, t)) − tµ(dt) − tµ(dt) + 2tµ(dt)

= µ([t,∞)) − µ((−∞, t)) = 2µ([t,∞)) − 1. (5.2)

We go on to calculate β. Using the fact that (x, µU(x)) ∈ lx and (5.1)-(5.2)we obtain β = −2

∫

[x,∞)ydµ(y). We are ready to determine the value of Ψµ(x)

writing the equation of the intersection of lx with (t,−t) : t ∈ R.

−Ψµ(x) =(

2µ([x,∞)) − 1))

Ψµ(x) − 2

∫

[x,∞)

ydµ(y)

Ψµ(x) =1

µ([x,∞))

∫

[x,∞)

ydµ(y), (5.3)

which is just the barycentre function defined by Azema and Yor. We still haveto deal however with atoms of µ. Clearly in atoms the potential “breaks” andthere is no uniqueness of the tangent. It is easy to see from the constructionof Ψµ(x) that it is a non-decreasing, continuous function (for x in the supportand µ(x) = 0). Fix an x such that µ(x) > 0. To assign value to Ψµ(x) wefollow the case of measures with finite support described in Section 4. We seethat, as a consequence of taking the tangents from left to right, we treat thepoint (x, µU(x)) as belonging rather to the linear piece of the potential on theleft than the one on the right. In other words we take Ψµ to be left continuous.The formula (5.3) remains valid for all x in the interior of support of µ then.

We want finally to describe the function Ψµ outside of the support of µ(and on its boundaries). Suppose that the support of µ is bounded from below:

5.1 An even simpler proof of the Azema-Yor formula 27

aµ = infx : x ∈ supp(µ). For all x ≤ aµ we have simply that µU(x) = xand so the tangent is just the line (t, t) : t ∈ R, which intersects the line(t,−t) : t ∈ R in (0, 0), so we put Ψµ(x) = 0.

Suppose now that the support of µ is bounded from above: bµ = supx :x ∈ supp(µ). Then for all x ≥ bµ we have µU(x) = −x and so the verypoint (x, µU(x)) lies on the line (t,−t) : t ∈ R and we put Ψµ(x) = x. Thiscompletes the definition of the function Ψµ, which coincides with the one givenin [3].

We want now to define the stopping time that results from this setup. Atfirst glance we could have a small problem: following closely the case of measureswith finite support would end up in a composition of an uncountable infinity offirst hitting times of intervals (x,Ψµ(x)) for x in the support of µ. This wouldnot make much sense. However there is an easy way to overcome this difficulty- we can use the fact that Ψµ is increasing. So St = supMs : s ≤ t describeswhich values of type Ψµ(x) have been reached so far - and so where to put the“left” boundary. We want to stop the first time when we reach an x such thatthe supremum so far is equal to Ψµ(x). In other words we take:

TAY = inft ≥ 0 : St ≥ Ψµ(Mt). (5.4)

Theorem 5.1 (Azema and Yor [3]). Let (Mt)t≥0 be a continuous martingalewith < M,M >∞= ∞ a.s. For any centered probability measure µ on R, TAY

defined through (5.4), is a stopping time in the natural filtration of (Mt) andMTAY

∼ µ. Furthermore, (Mt∧TAY)t≥0 is a uniformly integrable martingale.

Proof. From (5.4) it is evident that TAY is a stopping time in the natural filtra-tion of (Mt). The proof is carried out through a limit procedure using approx-imation of µU by piece-wise linear functions. We define measures µn throughtheir potentials. Let A = a1, a2 . . . be the set of atoms of µ (i.e. µ(x) > 0iff ∃! i : x = ai) and Pn = k

n : k ∈ Z. Define µnU through:

• ∀ x ∈ A ∪ Pn µnU(x) = µU(x),

• µnU is linear between points of A ∪ Pn.

It is straightforward that µnU(x) → µU(x) for all x ∈ R, which in turn impliesµn ⇒ µ. For each of the measures µn we have the stopping time Tn - theAY stopping time, which embeds µn as shown in Section 4. Finally from thedefinition of a tangent, we see that lines defining Tn converge to tangents of µU

and Tn → T = TAY almost surely. Then MTn→ MT a.s. and since MTn

d−→ µ,we obtain MTAY

∼ µ. To see that (Mt∧TAY) is a uniformly integrable martingale

a classical argument is used.

It has to be noted that what we did in this section is essentially a simplifiedreading of Zaremba’s work [90] - simplified through the work of Chacon andWalsh.

5.2 Nontrivial initial laws 28

5.2 Nontrivial initial laws

One of the advantages of the approach presented here is that in fact the onlycondition imposed on the initial law, is that its potential stays above the one ofthe measure µ we want to embed. We can still give an explicit formulae for thestopping time TAY , for nontrivial initial laws.

Let (MT ) be a continuous martingale with < M,M >∞= ∞ a.s. and withinitial law M0 ∼ ν,

∫

|x|dν(x) < ∞. Let µ be any probability measure onR such that µU ≤ νU . This, by Lemmas 2.2 and 2.3 in [13], implies that∫

|x|dµ(x) <∞,∫

xdν(x) =∫

xdµ(x) = m, and

lim|x|→∞

(νU(x) + |x−m|) = lim|x|→∞

(µU(x) + |x−m|) = 0. (5.5)

Then µ can be embedded in M by a stopping time, which is a generalization ofAzema and Yor’s one.

Consider a tangent to the potential µU in a point (x, µU(x)) (which is notan atom of µ). We want to describe the points of intersection of this line withthe potential of ν. Studying (5.1), (5.2) and (5.3) it is easily seen that the pointsare of form (t, νU(t)), where t satisfies

νU(t) =(

µ([x,∞)) − µ((−∞, x)))

t+

∫ x

−∞ydµ(y) −

∫ ∞

x

ydµ(y). (5.6)

We want to define a generalized barycentre function Ψνµ(x) as a certain solution

of the above and the stopping time will be given as usual through

T νAY = inft ≥ 0 : St ≥ Ψν

µ(Bt). (5.7)

Let us look first at a simple example. Let B be a Brownian motion startingfrom zero and ν be centered, concentrated in a and b, in other words ν ∼ BTa,b

.Condition (5.5) implies that µ is also centered. Let aµ = inft : t ∈ supp(µ) ≤ abe the lower bound of the support of µ and Ψµ be the standard barycentrefunction defined through (5.3). Writing the explicit formula for νU in (5.6) weobtain

Ψνµ(x) =

a , for x ≤ aµ∫

[x,∞)ydµ(y)+ ba

b−a

µ([x,∞))+ ab−a

, for aµ < x < b

Ψµ(x) , for x ≥ b.

Consider a general setup now. The equation (5.6) has either two solutionst1x < t2x or solutions form an interval t ∈ [t−x , t

+x ], t−x ≤ t+x , see Figure 1. Let us

denote these two situations respectively with I and II. The latter happens ifand only if the two potentials coincide in x. Finally, for a probability measureρ let aρ = inft : t ∈ supp(ρ), bρ = supx : x ∈ supp(ρ) be respectively thelower and the upper bound of the support of ρ. Define:

Ψνµ(x) =

aν , for x ≤ aµt2x , for aµ < x < bµ if Ix for (aµ < x < bµ if II) and for (x ≥ bµ).

(5.8)

5.3 Minimizing the minimum - reversed AY solution 29

µU νU

t+xt−x

µU

t2yt1yyx

Figure 1: Possible intersections of tangents for nontrivial initial law.

A simple observation is required to see that the definition agrees with intuition.Since µU ≤ νU ≤ δmU and µU , νU coincide with δmU outside their supports,the support of ν is contained in the support of µ, [lν , uν ] ⊂ [lµ, uµ]. This grantsthat Ψν

µ(x) ≥ x.It is also worth mentioning that the function Ψν

µ lost some of the propertiescharacterizing Ψµ. In particular we can have Ψν

µ(x) = x, Ψνµ(x1) > x1 for some

x1 > x.

Proposition 5.2. Let (Mt) be a continuous real martingale with < M,M >∞=∞, M0 ∼ ν, and µ a probability measure on R with µU ≤ νU . Define Ψν

µ

and T νAY through (5.8) and (5.7) respectively. Then (Mt∧Tν

AY) is a uniformly

integrable martingale and MTνAY

∼ µ.

The proof of this result is analogous to the proof of Theorem 5.1, taking intoaccount the above discussion of Ψν

µ.We want to remark that the construction presented above might be used

to mix or refine some of the known solutions to the embedding problem. Forexample if µ([−a, a]) = 0 we can first stop at T−a,a and then use T ν

AY withν = 1

2 (δ−a + δa).

5.3 Minimizing the minimum - reversed AY solution

In Section 5.1 we saw how to derive Azema and Yor’s solution to the Skorokhodproblem using potential theory. It easy to see that the same arguments allow usto recover an another solution, which we call the reversed Azema-Yor solution.It was first studied by van der Vecht [86] (see also Meilijson [53]). Recall thatthe lx was the tangent to the potential µU in (x, µU(x)). We considered itsintersection with the line (t,−t) : t ∈ R, which was denoted (−Ψµ(x),Ψµ(x)).Consider now the intersection of lx and the line (t, t) : t ∈ R and denote it by(Θµ(x),Θµ(x)). A simple calculation (analogous to derivation of (5.3) above)shows that:

Θµ(x) =1

µ((−∞, x))

∫

(−∞,x)

ydµ(y). (5.9)

5.4 Martingale theory arguments 30

This is a valid formula for x that are not atoms. Looking at the discrete casewe see that we want Θµ to be right continuous, so we finally need to define:

Θµ(x) =1

µ((−∞, x])

∫

(−∞,x])

ydµ(y). (5.10)

Recall from 6.1, that we denote st = − infMs : s ≤ t the infimum process.Define the following stopping time

TrAY = inft ≥ 0 : −st ≤ Θµ(Mt). (5.11)

Proposition 5.3. Let (Mt)t≥0 be a continuous martingale with < M,M >∞=∞ a.s. and let µ be a centered probability measure on R. Define Θµ through(5.10). Then TrAY is a stopping time and Bt∧TrAY

∼ µ. Furthermore, (Bt∧TrAY)t≥0

is a uniformly integrable martingale. In particular, if∫

Rx2dµ(x) = v <∞ then

ETrAY = v.

Actually we do not need any proof as this solution is still an Azema-Yorsolution. Indeed if X ∼ µ then TrAY is just TAY for ν the law of −X and acontinuous martingale Nt = −Mt. Speaking informally then, in the originalAzema-Yor solution one takes tangents from left to right and in the reversedversion one takes the tangents from right to left. We will see below that it yieldssome symmetric optimal properties.

5.4 Martingale theory arguments

We chose above to discuss the Azema-Yor solution using the real potential the-ory. This methodology allows to see well what is really being done and what isreally being assumed, opening itself easily to certain generalizations, as we sawabove and as we will see again in Section 9. One can choose however a differentmethodology, which will certainly have its own advantages as well. One of thegreat things about the Azema-Yor solution is that it has been studied using al-most all methodologies ever applied to the Skorokhod embedding problem. Wewant to sketch now the heart of the proof that goes through martingale theory.These are the ideas that actually led to the original derivation of the solution.

Let f be an integrable function and denote its primitive by F (x) =∫ x

−∞ f(y)dy.Then

Nft = F (St) − (St −Mt)f(St) : t ≥ 0 (5.12)

is a martingale. This can be easily seen when f is continuously differentiable,since integrating by parts we obtain

NFt = F (St) −

∫ t

0

f(Su)dSu +

∫ t

0

f(Su)dMu −∫ t

0

(Su −Mu)f ′(Su)dSu

= F (St) − F (St) +

∫ t

0

f(Su)dMu =

∫ t

0

f(Su)dMu, (5.13)

5.5 Some properties 31

as (Su −Mu) is zero on the support of dSu. The result extends to more generalfunctions f through a limit procedure5. We have a centered probability measureµ and we want to find T , which embeds µ, of the form T = inft ≥ 0 : St ≥Ψ(Mt). Our goal is therefore to specify the function Ψ. Suppose we can applythe stopping theorem to the martingale NF and the stopping time T . We obtain

EF (ST ) = E(ST −MT )f(ST ). (5.14)

Note that MT ∼ µ and ST = Ψ(MT ). We suppose also that Ψ is strictlyincreasing. If we note ρ the law of ST and ρ = ρ([x,+∞)) its tail, the aboveequation can be rewritten as

∫

F (x)dρ(x) =

∫

(x− Ψ−1(x))f(x)dρ(x), integrating by parts

∫

f(x)ρ(x)dx =

∫

(x− Ψ−1(x))f(x)dρ(x), which we rewrite as

dρ(x) = − ρ(x)

x− Ψ−1(x)dx from the arbitrary character of f .

We know that ST = Ψ(MT ), so that µ(x) = P(MT ≥ x) = P(ST ≥ Ψ(x)) =ρ(Ψ(x)). Inserting this in the above yields

dµ(x) = d(

ρ(Ψ(x)))

= − ρ(Ψ(x))

Ψ(x) − xdΨ(x) = − µ(x)

Ψ(x) − xdΨ(x)

Ψ(x)dµ(x) − xdµ(x) = −µ(x)dΨ(x)

Ψ(x)dµ(x) + µ(x)dΨ(x) = xdµ(x)

d(

Ψ(x)µ(x))

= xdµ(x)

Ψ(x)µ(x) =

∫

[x,+∞)

ydµ(y),

and we find the formula (5.3).

5.5 Some properties

The Azema-Yor solution is known to maximize the supremum in the followingsense. Let µ be a centered probability measure and R any stopping time withMR ∼ µ and (MR∧t) a uniformly integrable martingale. Then for any z > 0,P(SR ≥ z) ≤ P(STAY

≥ z), where St = supMs : s ≤ t is the supremum pro-cess. This was observed already by Azema and Yor [2], as a direct consequenceof work by Dubins and Gilat [24], which in turn relied on Blackwell and Dubins[7]. We will try to present here a unified and simplified version of the argumen-tation. The property was later argued in an elegant way by Perkins [65] and itis also clear from our proof in Section 5.1 (as stressed when discussing measureswith finite support, see page 23).

5It can also be verified for f an indicator function and then extended through the monotoneclass theorem.

5.5 Some properties 32

For simplicity assume that µ has a positive density, so that µ−1 is a welldefined continuous function. Let f(x) = 1[z,+∞)(x), then NF

t = (St − z)+ −(St −Mt)1[z,+∞)(St) is a martingale by (5.12). Applying the optimal stoppingtheorem for R ∧ n and passing to the limit (using the monotone convergencetheorem and the uniform integrability of (MR∧t)) we see that z P(SR ≥ z) =EMR1[z,+∞)(SR). It suffices now to note, that if we denote P(SR ≥ z) = pthen

zp = EBR1[z,+∞)(SR) ≤ EBR1[µ−1(p),+∞)(BR)

=

∫ +

µ−1(p)

∞xdµ(x) = pΨµ

(

µ−1(p))

, thus

z ≤ Ψµ

(

µ−1(p))

, which yields

p ≤ µ(

Ψ−1µ (z)

)

= P(

BTAY≥ Ψ−1

µ (z))

= P(STAY≥ z),

which we wanted to prove. Note that the passage between the last two lines isjustified by the fact that Ψ−1

µ is increasing and µ is decreasing.This optimality property characterizes the Azema-Yor solution. Further-

more, it characterizes some of its generalization - for example in the case ofBrownian motion with drift, see Peskir [68]. It also induces symmetrical optimal-ity on the reversed solution presented in Seciton 5.3. Namely, with R as above,we have P(sR ≥ z) ≤ P(sTrAY

≥ z), for any z ≥ 0, where st = − infu≤tMu.The second optimality property says that this solution is pointwise the small-

est one. More precisely, if for any stopping time S ≤ TAY and MS ∼ MTAY

then S = TAY a.s. Actually, it is a general phenomena that we will discuss inSection 8.

Finally, we want to present one more property of any solution to the Sko-rokhod embedding problem. The reason why we do it here is that it was shownusing precisely Azema-Yor solution and its reversed version presented in Section5.3.

Let (Bt) be Brownian motion. We say that a stopping time T is standard if(BT∧t) is a uniformly integrable martingale6. Further, we say that T is ultimate,whenever for a probability measure ν with νU(x) ≥ µU(x) for all x ∈ R, whereµ ∼ BT , there exists a stopping time S ≤ T , that embeds ν. The followingtheorem says that an ultimate solution to the Skorokhod problem exists only intrivial cases.

Theorem 5.4 (Meilijson [51], van der Vecht [86]). If T is a standard andultimate stopping time and µ ∼ BT then µ(a, b) = 1, for some a < 0 < b.

The proof consists in showing that such a solution would have to coincidewith both the standard Azema and Yor’s solution and its reversed version, whichin turn are shown to coincide only in trivial cases.

6Actually, this is not the original definition, but it was shown by Falkner [26] to be equiv-alent to the original one.

6 The H1-embedding 33

6 The H1-embedding

Given a solution T to the Skorokhod stopping problem we can ask about thebehavior of supremum and infimum of our process up to time T . This questionhas attracted a lot of attention. In particular some solutions were constructedprecisely in order to minimize the supremum and maximize the infimum andinversly.

Let us denote by (Xt) the real valued process we are considering and µ aprobability measure on the real line. The canonical problem is that of Hp-embedding:

Find necessary and sufficient conditions on (X,µ) for existence of a solutionT to the Skorokhod embedding problem such that X∗

T = sup|Xs| : s ≤ T is inLp.

We will restrain ourselves to the Brownian setup, even though most of theideas below work for real diffusions on natural scale. We will present a solutionof the Hp embedding by Perkins. He actually gives a solution to the Skorokhodembedding problem, which minimizes X∗

T . We then proceed to Jacka’s solution,which is a counterpart as it maximizes X∗

T .

6.1 A minimal H1-embedding

For X Brownian motion and a mean-zero probability measure µ, the problemwas considered by Walsh and Davis [19] but received a complete and accuratedescription in Perkins’ work in 1985 [65]. Using Doob’s inequalities one can seeeasily that for p > 1 an Hp-embedding exists if and only if

∫

|x|pdµ(x) <∞, andif p < 1 every embedding is in fact an Hp-embedding. For p = 1 the situationis more delicate, and it goes back to an analytical problem solved by Cereteli[11].

Theorem 6.1 (Cereteli-Davis-Perkins). Consider Brownian motion and acentered probability measure µ. The following are equivalent:

• There exists an H1-embedding of µ.

• The Skorokhod solution is an H1-embedding of µ.

• H(µ) =∫∞0

1λ |

∫

Rx1|x|≥λdµ(x)|dλ <∞.

Perkins associated with every centered probability measure two functions:γ− and γ+ (see the original work for explicit formulae). Then if

St = supu≤t

Bu and st = − infu≤t

Bu (6.1)

are respectively the supremum and infimum processes we can define:

ρ = inft > 0 : Bt /∈ (−γ+(St), γ−(st)).

6.2 A maximal H1 embedding 34

If the target distribution has an atom in 0, we just have to stop in 0 with theappropriate probability. This is done through an external randomization. Wetake an independent random variable U , uniformly distributed on [0, 1] and put

TP = ρ1U>µ(0),

where P stands for Perkins. TP is a stopping time that embeds µ and TP is anH1-embedding if there exists one. Moreover, Perkins defined his stopping timein order to control the law of the supremum and the infimum of the process andhis stopping time is optimal.

Theorem 6.2 (Perkins). (Bt∧TP) is a uniformly integrable martingale and

BTP∼ µ. TP is an H1-embedding if and only if H(µ) < ∞. Let T be any

embedding of µ. Then, for all λ > 0,

P(STP≥ λ) ≤ P(ST ≥ λ) (6.2)

P(sTP≥ λ) ≤ P(sT ≥ λ) (6.3)

P(B∗TP

≥ λ) ≤ P(B∗T ≥ λ) (6.4)

Actually this theorem is not hard to prove - the stopping time is constructedin such a way to satisfy the above. The hard part was to prove that TP embedsµ. Construction of TP allows Perkins to give explicit formulae for the threeprobabilities on the left side of the above display. Using Perkins’ framework wecould also define a different stopping time in such a way that it would maximizestochastically the supremum S. Of course what we end up with is the Azema-Yor stopping time (see Section 5.1 below) and this is yet another approach whichallows to recover their solution in a natural way.

The work of Perkins has been gradually generalized. Several authors exam-ined the law of supremum (and infimum) of a martingale with a fixed terminaldistribution. More generally the link between the Skorokhod embedding prob-lem and optimal stopping theory was studied. We develop this subject in Section10.

6.2 A maximal H1 embedding

We saw in Section 5 that the Azema-Yor construction maximizes stochasticallythe distribution of ST , where BT ∼ µ, fixed, in the class of stopping times suchthat (Mt∧T )t≥0 is a uniformly integrable martingale. Moreover, we saw alsothat its reversed version TrAY , minimizes the infiumum. How can we combinethese two? We tried to describe the idea in Section 3.14. It comes from Jackaand allowed him to construct a maximal H1 embedding. We will describe hisidea using the Chacon and Walsh methodology, which is quite different fromhis, yet allows very well, in our opinion, to see what is really being done.

Let M∗t = supu≤t |Mu|, where M is some continuous local martngale, M0 =

0 a.s. We want to find a stopping time T such that MT ∼ µ, (Mt∧T ) is auniformly integrable martingale and if R is another such stopping time, thenP(M∗

R ≥ z) ≤ P(M∗T ≥ z), for z ≥ 0. Naturally big values of M∗ come either

7 Root’s solution 35

from big values of S or s. We should therefore wait “as long as possible” todecide which of the two we choose, and then apply the appropriate extremalembedding. To see if we should count on big negative or positve values we willstop the martingale at some exit time form an interval [−k, k], which we takesymmetric. This corresponds to cutting the inital potential with a horizontalline. The biggest value of k which we can take is obtained by requirying thatthe horizontal line be tanget to the potential µU .

For sake of simplicity we supposed that the measur µ has a positive density.As usual, it will be easier to work with such a nice measure. We look thereforefor a point x∗ in which the derivative of µU becomes zero. Looking at (5.2) wesee that µ(x∗) = 1

2 . In other words, we chose the median of µ for x∗. The valueof k∗ is then given as the intersection of the tangent to µU in x∗ with the liney = −x, which is precisely k∗ = Ψµ(x∗).

As suggested, we first stop at the exit time from [−k∗, k∗]. Obviously Brown-ian motion stops either in k∗, in which case it will be stopped inside [µU(k∗),∞),or in −k∗ in which case it will be stopped inside (−∞, µU(k∗)]. If we exit atthe top, we want to maximize the supremum, and so we will use the standardAzema-Yor: T1 = inft > T−k,k : St ≥ Ψµ(Mt). If we exit at the bottom, weneed to mimimize the infimum, so we will use the reversed Azema-Yor solution:T3 = inft > T−k,k : −st ≤ Θµ(Mt). We have then the following propositon,

Proposition 6.3 (Jacka [43]). In the setup above

P(

M∗R ≥ z) ≤ min

1, µ([ψ−1µ (z),+∞)) + µ((−∞,Θ−1

µ (z)])

. (6.5)

Furthermore, this inequlaity is optimal as equality is attained for a stopping timeTJ defined through

TJ = T1 ∧ T2 = inf

t ≥ T−k∗,k∗ : M∗t ≥ max(Ψµ(Mt),Θµ(Mt)

. (6.6)

If the measure µ has atoms the above Proposition is still true upon a correctextension of meaning of Ψ−1

µ ,Θ−1µ and k∗. Actually only an atom in the median

of µ poses a problem. If we write m = infx : µ(x) ≥ 12 for the meadian of

µ, then if µ(m) > 0 we have a family of tangents to µU in m. The valueΨµ(m) results from takieng the letf-most tangent (cf. Section 5.1) and not thehorizontal one. We have therefore k∗ > Ψµ(m). The difference is given bythe slope of the left-most tangent, (2µ(m) − 1), times the increment interval(Ψµ(m) −m). We obtian therefore: k∗ = Ψµ(m) + (2µ(m) − 1)(Ψµ(m) −m),which precisely the quantity given in Lemma 6 in Jacka [43]. Finally, one shouldtake left inverses of Ψµ and Θµ.

7 Root’s solution

All the solutions described so far have one thing in common - stopping times aredefined through some functionals of Brownian motion up to time T . Intuitively,if one searches for a stopping time with minimal variance it should have a

7.1 Solution 36

very simple history-dependence structure - preferably, it should just be the firsthitting time of some set. Of course this cannot be just a subset of our spaceR but rather of our time-space R+ ×R. Root [74] showed that indeed for anycentered probability measure µ on R with finite second moment, there exists aset Rµ, such that the first entry time into Rµ, embeds µ in Brownian motion(Bt). Actually what we consider is just a first hitting time for the time-spaceprocess of Brownian motion. These correspond to reduites of some potentialmeasures and we should not be surprised that to investigate Root’s solutionRost [75], [76], [77] needed a fair amount of potential theory.

7.1 Solution

The key notion is that of a barrier.

Definition 7.1. A subset R of [0,+∞] × [−∞,+∞] is a barrier if

1. R is closed,

2. (+∞, x) ∈ R for all x ∈ [−∞,+∞],

3. (t,±∞) ∈ R for all t ∈ [0,+∞],

4. if (t, x) ∈ R then (s, x) ∈ R whenever s > t.

Theorem 7.2 (Root [74]). For any probability measure µ on R with∫

x2dµ(x) =v <∞ and

∫

xdµ(x) = 0 there exists a barrier Rµ such that the stopping time

TR = inft ≥ 0 : (t, Bt) ∈ Rµ, (7.1)

solves the Skorokhod problem for µ, i.e. BTR∼ µ and ETR = v.

Remark on notation: We write TR for “Root’s stopping time” and sometimesTRµ

to stress the dependence on a particular barrier Rµ. Also, for a barrier R,we will write sometimes Bt ∈ R instead of (t, Bt) ∈ R.

The proof is carried out through a limit procedure. It is not hard to see thata barrier Rµ exists for a measure with finite support µ =

∑ni=1 αiδxi

. Rµ is justof the form

Rµ =

n⋃

i=1

[bi,+∞] × xi ∪(

[0,+∞] × [−∞, x1] ∪ [0,+∞] × [xn,+∞])

. (7.2)

The points (bi) are not given explicitly but only their existence is proven. Onethen needs some technical observations in order to pass to the limit.

Let H denote the closed half plane, H = [0,+∞] × [−∞,∞]. Root definesa metric r on C space of all closed subsets of H . Map H homeomorphicallyto a bounded rectangle F by (t, x) → ( t

1+t ,x

1+|x|). Let F have the ordinary

Euclidian metric ρ and denote by r the induced metric on H . For C,D ∈ C put

r(C,D) = maxsupx∈C

r(x,D), supy∈D

r(y, C). (7.3)

7.2 Some properties 37

Equipped with r, C is a separable, compact metric space and R, the subspaceof all barriers R is closed in C and hence compact. Furthermore, this metricallows us to deal with convergence in probability of stopping times.

Lemma 7.3 (Root [74], Loynes [50]). Let R be a barrier with correspondingstopping time TR. If P(TR < ∞) = 1, then for any ǫ > 0 there exists δ > 0such that if R1 ∈ R with corresponding stopping time TR1 and r(R,R1) < δthen P(|TR − TR1 | > ǫ) < ǫ. If P(TR = ∞) = 1 then for any K > 0 there existsδ > 0 such that r(R,R1) implies P(TR1 < K) < ǫ.

If a sequence of barriers converges, Rnr−−−−→

n→∞R with ETRn

< K <∞, then

ETR ≤ K and P(|TRn− TR| > ǫ)−−−−→

n→∞0.

With this lemma the theorem is proved taking a sequence µn ⇒ µ of mea-sures with finite supports.

7.2 Some properties

We want to start with some easier properties and then pass to the more impor-tant results of Rost. We keep the notation of the previous section.

Lemma 7.4 (Root [74]). Let µ be a centered probability measure with finitesecond moment and Rµ the corresponding barrier. Then

ET kRµ

<∞ ⇐⇒∫

|x|2kdµ(x) <∞.

This is a fairly easy result obtained using Hermite polynomials and associ-ated martingales (see Revuz-Yor [71] p.151).

Several further properties were described by Loynes [50], one of which isparticularly important for us. In order to state it, we need to introduce thenotion of a regular barrier. Let R ∈ R be a barrier. Define

xR+ = inf

y ≥ 0 : [0,+∞] × y ⊂ R

xR− = supy ≤ 0 : [0,+∞] × y ⊂ R, (7.4)

the first coordinates above and below zero respectively at which R is a horizontalline.

Definition 7.5. We say that a barrier R ∈ R is regular if [0,+∞]×[xR+,+∞] ⊂R and [0,+∞] × [−∞, xR−] ⊂ R.

We say that two barriers R1, R2 are equivalent R1 ∼ R2 if xR1− = xR2

− ,

xR1+ = xR2

+ and the two barriers coincide on [0,+∞] × [xR1− , xR1

+ ].

Thus a regular barrier is one that is composed of horizontal lines abovexR+ and below xR−. Clearly each barrier is equivalent with exactly one regularbarrier. Furthermore, for a barrier R we have that TR ≤ TxR

−,xR

+and so stopping

times corresponding to two equivalent barriers R1 ∼ R2 are equal TR1 = TR2 .This shows that we can limit ourselves to regular barriers.

7.2 Some properties 38

Proposition 7.6 (Loynes [50]). Let µ be a centered probability measure onR with

∫

x2dµ(x) = v < ∞ and (Bt) a Brownian motion. Then there existsexactly one regular barrier Rµ, such that BTRµ

∼ µ and ETRµ= v.

Loynes showed also that a wider class of probability measures µ can beembedded in Brownian motion through Root’s construction. However, we mayloose the uniqueness of barrier generating µ.

The fundamental property of Root’s construction is that it minimizes thevariance of the stopping time. This conjecture, made by Kiefer [48], was provedby Rost.

Theorem 7.7 (Rost [78]). Let µ be a centered probability measure on R with∫

x2dµ(x) = v < ∞, (Bt) a Brownian motion and TR Root’s stopping timeassociated with µ. Assume that ET 2

R <∞. Then, for any stopping time S suchthat BS ∼ µ and ES = v, we have: E T 2

R ≤ ES2 and ET 2R = ES2 if and only

if S = TR a.s.More generally, for any p < 1, E T p

R ≥ ESp and for any p > 1, E T pR ≤ ESp.

We will try to explain briefly, and on a intuitive level, how to arrive at theabove result. Actually, Rost considers a more general setup of a Markov process(that is assumed transient for technical reasons but the generalization is fairlysimple).

Let (Xt) be a Markov process (relative to its natural filtration (Ft)) on acompact metric space E with semigroup (Pt). Note U =

∫∞0 Ptdt its potential

kernel7. On the borel sets of E we are given two probability measures µ and νwith σ-finite potentials µU and νU . The measure ν is the initial law: X0 ∼ νand µ is the law we want to embed.

For an almost borel set A ⊂ E we note the first hitting time of A byTA := inft ≥ 0 : Xt ∈ A. It is a stopping time. Note that the measure νPTA

is just the law of XTA.

The reduite of a measure ρ is the smallest excessive measure which majorizesρ. It exists, is unique and can be written as νPTA

for a certain A. We have thefollowing characterization of balayage order (see Meyer [56], Rost [76]8).

The measure µ admits a decomposition µ = µ+µ∞, where µ∞U is the reduiteof (µ − ν)U and µ is of the form νPT , T a stopping time. Furthermore, thereexists a finely closed set A which carries µ∞ and for which µ∞ = (µ − ν)PTA

or (µ− ν)PTA= 0. In the special case µU ≤ νU we have µ∞ = 0 and µ = νPT

for some T .

We are looking for a stopping time with minimal variance. It is shown

actually that it satisfies a stronger condition of minimizing Eν

∫ T

T∧tf(Xs)ds for

Borel, positive functions f and any t > 0. This in turn is equivalent withminimizing νPT∧tU for all t > 0. We know that if µU ≤ νU then there exists a

7We point out that this is not the same U as in Section 3, which was defined on R.8Note that in our notation the roles of µ and ν are switched compared to the article of

Rost.

7.3 The reversed barrier solution of Rost 39

stopping time T that embeds µ. The idea is to look at the space-time process.We consider a measure L on E × R defined through

L(A× B) =

∫

BLt(A)dt, where Lt = νU1t<0 + νPT∧tU1t≥0.

We take its reduite M (with respect to the semigroup of the space-time process)and show that it is of the form

M(A× B) =

∫

BMt(A)dt,

with M. decreasing, right continuous and Mt = νPTR∧tU for a certain stoppingtime with νPTR

= µ. It follows that TR minimizes νPT∧tU . The last stepconsists in showing that TR is actually Root’s stopping time for a certain barrierR. This is not surprising after all, as we saw that reduites are realized as hittingtimes of some sets and Root’s stopping time is just a hitting time for time-spaceprocess.

7.3 The reversed barrier solution of Rost

In this section we want to describe a solution proposed by Rost, which is acounterpart for Root’s solution as it maximizes the variance of the stoppingtime. We have not found it published anywhere and we cite it following thenotes by Meilijson [53]. The basic idea is to look at reversed barriers. Intuitivelywe just substitute the condition that if a point belongs to a barrier then all thepoints to the right of it also belong to the barrier, by a condition that all thepoints to the left of it also belong to the barrier.

Definition 7.8. A subset ρ of [0,+∞]× [−∞,+∞] is called a reversed barrierif

1. ρ is closed,

2. (0, x) ∈ ρ for all x ∈ [−∞,+∞],

3. (t,±∞) ∈ ρ for all t ∈ [0,+∞],

4. if (t, x) ∈ ρ then (s, x) ∈ ρ whenever s < t.

Theorem 7.9 (Rost). For any probability measure µ on R with , µ(0) = 0,∫

x2dµ(x) = v < ∞ and∫

xdµ(x) = 0 there exists a reversed barrier ρµ suchthat the stopping time

Tρ = inft ≥ 0 : (t, Bt) ∈ ρµ, (7.5)

solves the Skorokhod problem for µ, i.e. Bρ ∼ µ and ETρ = v. Furthermore,if S is a stopping time such that BS ∼ µ and ES = v then for any p < 1,ET p

ρ ≤ ESp and for any p > 1, ET pρ ≥ ESp.

7.4 Certain generalizations 40

Note that there is a certain monotony in the construction of a reversedbarrier, which is not found in the construction of a barrier. For a point x in thesupport of µ, say x > 0, we must have the following property: there exists sometx such that (tx, x) ∈ ρµ and (tx, y) /∈ ρµ for any y ∈ [0, x). In other words,the reversed barrier ρµ, on the level x, has to extend “more to the right” thananywhere below, between 0 and x. This, in a natural way, could lead us to revisethe notion of a reversed barrier and introduce regular reversed barriers, whichwould satisfy ∀x > 0 if (t, x) ∈ ρ then (t, y) ∈ ρ, ∀y ∈ [x,+∞) and similarlyfor x < 0. We could then obtain uniqueness results for Rost’s embedding in ananalogy to Proposition 7.6 above. We do not pursue these matters here.

7.4 Certain generalizations

The problem we want to solve is a certain generalization of the one solved byRost. In [78] H. Rost showed that among all stopping times which solve a givenSkorokhod embedding problem, the stopping time defined by Root has minimalsecond moment (which is equivalent to minimal variance). In the Skorokhodembedding problem, the distribution of BT is fixed (where (Bt) is a Brownianmotion). Here we only fix two moments of this distribution and want to findthe stopping time with the minimal variance. That is:

Problem. Let us fix 4 > p > 1 and a constant cp > 0. Among all stoppingtimes T such that EBT = 0, ET = EB2

T = 1 and E|BT |p = cp, find Tmin - theone with the minimal variance.

Unfortunately, we did not succeed in solving this problem. We know it has asolution and we will present a certain hypothesis about it.

Proposition 7.10. The stopping time Tmin exists.

Proof. Recall Definition 7.8 and (7.1). It is an immediate observation that thestopping time we are looking for will be a Root’s stopping time TR for somebarrierR. Indeed, if the µ ∼ BTmin

is the distribution of BTmin, then there exists

a barrier R such that the Root’s stopping time TR yields BTR∼ µ. Moreover,

thanks to Rost, we know that ET 2R ≤ ET 2

min, as Root’s stopping time minimizesthe variance, and so ET 2

R = ET 2min form the definition of Tmin. Therefore, using

Theorem 7.6 we conclude that TR = Tmin a.s.We can limit ourselves to Root’s stopping times. Let

Acp =

µ ∈M1(R) :

∫

xµ(dx) = 0,

∫

x2µ(dx) = 1,

∫

|x|pµ(dx) = cp

.

For every µ ∈ Acp there exists an unique regular barrier Rµ such that BTRµ∼ µ.

Let Rcp be the subset of barriers corresponding to the measures in Acp .Let H denote the closed half plane and recall the definition (7.3) of metric

r, under which C the space of all closed subsets of H , is a separable, compactmetric space and R the subspace of all barriers is closed in C and hence compact.

We can introduce an order in Rcp given by: C 4 D ⇔ ET 2C ≤ ET 2

D, forC,D ∈ Rcp . If we prove that there exists Rmin - a minimal element of Rcp

7.4 Certain generalizations 41

relative to 4 - then, we will have Tmin = TRmin. In other words

If there exists a Rmin ∈ Rcp such that

ET 2Rmin

= infR∈Rcp

ET 2R, then Tmin = TRmin

.

To complete the proof we need to show that Rcp is closed under r. Choose asequence of barriers Rn ∈ Rcp , such that ET 2

Rnց infR∈Rcp

ET 2R. We can sup-

pose that Rn converges (possibly choosing a subsequence) in r to some barrierR. To conclude that R = Rmin we need to know that R ∈ Rcp that is indeedwe just need to know that Rcp is a closed subset of R.

In order to prove that Rcp is a closed subset of R we essentially follow the

ideas used by Root in the proof of Theorem 7.2. Take any sequence Rnr→ R

with Rn ∈ Rcp . Then ETRn= 1 for all n. It follows from Lemma 7.3 that for

any ǫ > 0, P(|TR − TRn| > ǫ)−−−−→

n→∞0 and ETR < ∞. Choosing a subsequence,

we may assume that TRn→ TR a.s. Furthermore, we can always construct a

measure µ ∈ Acp with∫

x4dµ(x) <∞ and so E T 2Rµ

<∞, we may assume that

the sequence TRnis bounded in L2, and therefore uniformly integrable. We see

then, that the sequence of stopping times converges also in L1. In particularETR = 1. Finally, a simple calculation based on the martingales generated by

the Hermite polynomials shows that EB4T ≤ 1/(1 −

√

23 )2ET 2, which implies

that the sequence (BTRn) is bounded in L4. It converges almost surely, thanks

to the continuity of paths and therefore it converges in Lp. We see then, thatE|BTR

|p = cp. This implies that R ∈ Rcp and proves that Rcp is a closed set.We conclude that the stopping time Tmin exists and is equal a.s. to TR for acertain barrier R.

We do not know how to obtain Tmin explicitly. We do not even know if it isunique. Still, we want to give a simple observation, which motivates a conjecturebelow. We will let cp change now, so we will write Tmin = T

cpmin = TRcp

and

vcp := ET 2Rcp

.

Put cp = E|B1|p, then the minimal stopping time is trivial Tcpmin = 1 and it

is unique. It corresponds to the vertical barrier N = (x, t) : t ≥ 1. Take anycp > 0 and define an order in the set Rcp by: C ≺ D iff r(C,N) ≤ r(D,N).Let RN

min be the minimal element relative to this order. RNmin ∈ Rcp as Rcp is

closed. Denote the stopping time corresponding to this barrier by TNmin = TRN

min

The above observation suggested us the following conjecture.

Conjecture: The two minimal stopping times coincide a.s.: TNmin = Tmin.

This hypothesis yields a manageable description of the stopping time - solu-tion to our problem. This comes from the fact that it can be translated somehowinto information about the distribution of BT

cpmin

. However, we do not know how

to prove it, nor if it is true. We only have some idea that we wish to presenthere.

8 Shifted expectations and uniform integrability 42

Assume that 1 < p < 2. Consider f : [0, 1) → [1,∞), f(cp) = vcp . We willnow show that f is a continuous function. We start with the right-continuity.Fix cp < 1 and take a sequence cnp decreasing to cp. We have to show that

for each subsequence cnkp there exists a sub-subsequence c

nklp such that v

cnklp

converge to vcp . Take an arbitrary subsequence of cnp and denote it still cnp . Forsimplicity of notation put c := cp, v := v(cp), R = Rcp , cn = cnp and vn = vcnp .It is an easy observation that vn cannot converge to anything smaller than v assuch a situation would yield a stopping time with smaller variance than T

cpmin. So

it is enough to find a subsequence of barriers Rnkconverging to R, Rnk

∈ Rcnk,

such that the corresponding stopping times TRnkwill converge in L2.

Let µ be the distribution of BT cmin

. Define a family of probability measuresby

µs =s

2(δ−1 + δ1) + (1 − s)µ, for 0 < s < 1.

Observe that∫

xµs(dx) = 0,∫

x2µs(dx) = 1 and∫

|x|pµs(dx) = c + s(1 −c) > c. Corresponding to our sequence of (cn) we have a decreasing sequence(sn), sn ց 0,

∫

|x|pµsn(dx) = cn. Let Rn be the sequence of barriers suchthat BTRn

∼ µsn . Note that Rn ∈ Rcn . We need also to point out that∫

x4µs(dx) ≤∫

x4µ(dx) < ∞ (this implies that the family TRnis uniformly

bounded in L2). Now we can choose a converging subsequence Rnkand the

corresponding stopping times converge with ET 2Rnk

→ v.

To prove the left-continuity of f we follow the same reasoning, we just haveto define the appropriate measures which will play the role of µs. Fix a constanta > 1 so that 1

a2−p < c. Define a measure ν = (1 − 1a2 )δ0 + 1

2a2 (δ−a + δa),

which plays the role of 12 (δ−1 + δ1) above. We defined the measure so as to

have:∫

xν(dx) = 0,∫

x2ν(dx) = 1 and∫

|x|pν(dx) = 1a2−p . Now, for 0 < s < 1,

we take ηs = sν + (1 − s)µ, where µ ∼ BT cmin

. It follows that∫

xηs(dx) = 0,∫

x2ηs(dx) = 1 and∫

|x|pηs(dx) = c+ s( 1a2−p − c) < c.

It is an easy observation that f(c) ≥ 1 and f(c) = 1 if and only if thevariance of the corresponding stopping time T c

min is 0, that is if and only ifT cmin = t a.s. for some t > 0. Then, since ET c

min = 1, we have T cmin = 1 a.s.

and so c = E|B1|p. In other words, f has one global minimum. We think thatf is decreasing on left and increasing on right of this minimum.

8 Shifted expectations and uniform integrabil-ity

In so far we have considered the case when µU ≤ νU , where ν is the initial lawand µ is the measure we want to embed. A natural question to ask is: whatabout other probability measures? Can they be embedded into a continuousmartingale (Mt) as well? The answer is yes. But we have to pay for this - thestopped martingale will no longer be uniformly integrable.

If (Mt∧T ) is uniformly integrable, T < ∞ a.s. we have that EM0 = EMT .Suppose both ν and µ have finite expectations but

∫

xdν(x) 6=∫

xdµ(x). This

8 Shifted expectations and uniform integrability 43

corresponds to the simplest situation when we do not have inequality betweenthe two potentials. Then, if T embeds µ, the martingale (Mt∧T ) cannot beuniformly integrable, as EM0 6= EMT . The same applies to the situation when∫

|x|dν(x) ≤ ∞ and∫

|x|dµ(x) = ∞.It is very easy however to point out a number of possible stopping rules

which embed µ, in the case of∫

|x|dµ(x) <∞. It suffices to note that our usualassumption, < M,M >∞= ∞ a.s., implies that Ta = inft ≥ 0 : Mt = a isfinite a.s. for any a ∈ R. We can therefore first wait till M hits m =

∫

xdµ(x)and then use any of the embeddings described so far applied to the shiftedprocess starting at m.

We have to see therefore two more properties. Firstly, that any probabilitymeasure with

∫

|x|dµ(x) = ∞ can be embedded in M and secondly that in caseof

∫

|x|dµ(x) < ∞ but µU νU we have to loose the uniform integrability.Resolution of our first problem is trivial and was proposed by Doob as noted onpage 6. Namely, for any probability measure µ on R the variable F−1

µ (Φ(B1))has the distribution µ (where Fµ is the distribution function of µ and Φ is thedistribution function of a standard normal variable). It suffices to stop theBrownian motion, when it is equal to this variable. Define T = inft ≥ 2 :Bt = F−1

µ (Φ(B1)). Then obviously BT = F−1µ (Φ(B1)) ∼ µ, but E T = ∞

(see also exercise VI.5.7 in Revuz-Yor [71]). A different solution is found in Coxand Hobson [17] who generalized the Perkins stopping rule in such a way thatit works with any distribution on R. The construction itself is rather technicaland we will not present it here. We refer to Section 3.19 and to the originalpaper for details.

The second question asked above is also simple9. Suppose that M0 ∼ ν andMT ∼ µ. Then for any x ∈ R the process (Mt∧T −x : t ≥ 0) is a martingale andtherefore (|Mt∧T −x| : t ≥ 0) is a submartingale. In particular the expectationsof the latter increase and therefore −νU(x) = E |M0−x| ≤ E |MT−x| = −µU(x)for any x ∈ R, or equivalently µU ≤ νU . We have thus proved the followingproposition, which can be essentially attributed to Meilijson [53]:

Proposition 8.1. Let (Mt : t ≥ 0) be a continuous martingale with < M,M >∞=∞ a.s. and M0 ∼ ν. For any probability measure µ on R there exists a stoppingtime in the natural filtration of M , which embeds µ, that is MT ∼ µ. Further-more, the stopping time can be taken so that (Mt∧T : t ≥ 0) is a uniformlyintegrable martingale if and only if µU ≤ νU .

Finally we want to present in this section a notion of minimal stopping time,which is in close relation with the uniform integrability questions.

Definition 8.2. A stopping time T is called minimal if for any stopping timeS ≤ T , BS ∼ BT implies S = T a.s.

Proposition 8.3 (Monroe [57]). For any stopping time S there exists a mini-mal stopping time T ≤ S with BS ∼ BT . A stopping time T such that EBT = 0is minimal if and only if (Bt∧T ) is a uniformly integrable martingale.

9Strangely enough we found it treated only in notes by Meilijson [53] sent to us by theauthor together with remarks on the preliminary version of this work.

9 Working with real diffusions 44

This result was first obtained by Monroe [57], who made an extensive use ofthe theory of barriers (see Section 7). It was then argued in a much simpler wayby Chacon and Ghoussoub [14]. They showed, using one dimensional potentialtheory, that a minimal stopping time is essentially the same as a standardstopping time (see Section 5.5). Therefore, one can deduce properties of minimalstopping times from these of standard stopping times, which in turn are easy toestablish (cf. Chacon [13]).

We can apply the above proposition to see that any solution to the Sko-rokhod embedding problem in a classical setup is minimal. Indeed if µ satisfies∫

R|x|dµ(x) < ∞ and we have that BT ∼ µ and (Bt∧T ) is uniformly integrable

then T is minimal.

9 Working with real diffusions

As a careful reader might have observed, the methods used here do not rely onany special properties of Brownian motion. We underlined this in Section 3.8.There is only one, crucial, property that was used: the way potential changesif we stop our process at the exit time of [a, b]. Namely, if we start Brownianmotion with initial distribution ν and stop it at Ta,b - the first exit time of [a, b],then the potential of µ ∼ BTa,b

is equal to νU outside of [a, b] and is linear on[a, b]. This corresponds to the fact that Brownian motion is a diffusion in itsnatural scale. In other words, if B0 = x a.s., a < x < b then

P(BTa,b= a) =

b− x

b− a. (9.1)

This is also the only fact used by Zaremba [90] in his proof. There is one morebasic assumption that we made. We had νU ≥ µU , where ν is the initial dis-tribution and µ is the distribution we want to embed. This was used in Section5.2 to work with non-trivial initial laws. It implies that the two distributionshave equal expectations (cf. Lemma 2.3 in [13]), which in turn implies that

(νU(x) − µU(x))|x|→∞−−−−→ 0 (cf. Lemma 2.2 in [13]).

Generalizing this framework to real diffusions with continuous, strictly in-creasing scale functions is easy. It was done for the recurrent case in the originalwork of Azema and Yor [3] and then for general case by Pedersen and Peskir [64].We present here a complete description following mainly a recent paper by Coxand Hobson [17]. We note that more general, not continuous processes, weretreated as well. In Section 3.20 we discussed the solution for the age process ofBrownian excursions developed by Ob loj and Yor [61].

Let (Xt)t≥0 be a regular (time-homogeneous) diffusion taking values in aninterval I ⊂ R, X0 = 0 a.s. Then, there exists a continuous, strictly increas-ing scale function s : I → R and Mt = s(Xt) is a local martingale on s(I).Furthermore, we can choose s(0) = 0.

We transform the embedding problem for X into an embedding problem forM . Let µ be a probability measure on I, which we want to embed in X . Define

10 Links with optimal stopping theory 45

a measure µs through

µs(B) = µ(s−1(B)), B ∈ B(s(I)),

that is if a random variable ξ ∼ µ then s(ξ) ∼ µs. It is easily seen that if wefind a stopping time T such that MT ∼ µs then XT ∼ µ and our problem willbe solved.

At first glance one would like to conclude applying our previous results to M .However some care is needed as s(I) might be bounded and our local martingaleM won’t be recurrent.

We assume that∫

|x|dµs(x) =∫

|s(u)|dµ(u) <∞ and writem =∫

xdµs(x) =∫

s(u)dµ(u). We have three distinct cases: s(I) = R, s(I) = (−∞, α) ors(I) = (α,∞) and s(I) = (α, β), which will now discuss.

1. s(I) = R. This is a trivial situation since the local martingale (Mt) isrecurrent and we can use the standard Azema-Yor construction to embedµs in M .

2. • s(I) = (−∞, α). In this case (Mt) is a local martingale that isbounded from above by α. Suppose T is a stopping time such thatMT ∼ µs and let γn by a localizing sequence for M . Then, by Fatoulemma,

(−m) = E(−MT ) = E( limn→∞

−Mt∧γn) ≤ lim inf

n→∞E(−MT∧γn

) = 0

and so m ≥ 0 is a necessary condition for existence of an embedding.It is also sufficient as Tm = inft ≥ 0 : Mt = m is a.s. finite form ≥ 0 and we can put T = Tm + TAY θTm

, where TAY is definedthrough (5.4). We could also use different embedding in place of TAY

of course.

• s(I) = (α,∞). A completely analogous reasoning to the one abovegives that m ≤ 0 is a necessary and sufficient condition for existenceof a solution to the embedding problem for µ.

3. s(I) = (α, β). In this case we have a bounded martingale and we can usethe dominated convergence theorem to see thatm = EMT = limEMT∧γn

=0. It is of course also a sufficient condition as we can use normal AY em-bedding.

We want to point out that the simplifying assumption we made in the firstplace (namely

∫

|s(u)|dµ(u) <∞) is not necessary. One can see that in the firstcase a solution always exists and in two last cases the assumption is actually anecessary condition. We refer to Cox and Hobson [17] for more details.

10 Links with optimal stopping theory

There are two main ways to establish a link between an embedding problemand an optimal stopping problem. Firstly, one can prove that a solution to

10.1 The maximality principle revisited 46

the optimal stopping problem is given by a certain solution to the embeddingproblem (for some law). Secondly, one can look for such an optimal stoppingproblem that its solution solves a given embedding problem. The former isdiscussed in Section 10.1 and the latter in Section 10.2.

Closely connected with these subjects are also the works which followedand generalized the work of Perkins in studying the interplay of laws of thesupremum, the infimum and the terminal law of a martingale. Main literaturein this stream is discussed in Section 10.3.

In this section we only want to give a brief overview of these topics but withno claim at describing that vast domain in a complete way. In particular we donot treat optimal stopping theory itself. For a good and brief account of thebasic ideas as well as new developments in the latter see chapter 1 in [62].

10.1 The maximality principle revisited

Let φ be a non-negative, increasing, continuous function and c a continuous,positive function and consider the following optimal stopping problem of maxi-mizing

Vτ = E[

φ(Sτ ) −∫ τ

0

c(Bs)ds]

, (10.1)

over all integrable stopping times τ such that

E[

φ(Sτ ) +

∫ τ

0

c(Bs)ds]

<∞. (10.2)

Suppose, in the first moment, that φ(x) = x and c(x) = c > 0 is constant.In this formulation the problem was solved by Dubins and Schwarz [25] in anarticle on Doob-like inequalities. The optimal stopping time is just the Azema-Yor embedding for a shifted (to be centered) exponential distribution (withparameter 2c).

Keeping φ(x) = x let c(x) be a non-negative, continuous function. Thesetup was treated by Peskir in a series of articles ([66], [67], [69]). We want tomaximize (10.1) over all stopping times τ satisfying

E[

∫ τ

0

c(Bs)ds]

<∞. (10.3)

If there are no stopping times satisfying the above equation we can understandoptimality in a weaker sense.

Theorem 10.1 (Peskir [66]). The problem (10.1), for φ(x) = x and c(x) anon-negative, continuous function, has an optimal solution with finite payoff ifand only if there exists a maximal solution g∗ of

g′(s) =1

2c(g(s))(s− g(s))(10.4)

which stays strictly below the diagonal in R2. The Azema-Yor stopping timeτ∗ = inft ≥ 0 : Bt ≤ g∗(St) is then optimal and satisfies (10.3) wheneverthere exists a stopping time which satisfies (10.3).

10.1 The maximality principle revisited 47

Actually the above theorem was proven in a setup of any real, time-homogeneousdiffusion to which we will come back later, as it allows us to recover the solutionfor general φ as a corollary. The characterization of existence of a solution to(10.1) through existence of a solution to the differential equation (10.4) is calledthe maximality principle.

Let now φ be any non-negative, non-decreasing, right-continuous functionsuch that φ(Bt)− ct is a.s. negative on some (t0,∞) (with t0 random) and keepc constant. This optimal stopping problem was solved by Meilijson. Define

H(x) = supτ

E[

φ(x + Sτ ) − cτ]

.

Theorem 10.2 (Meilijson [54]). Suppose that ESc = E suptφ(Bt) − ct <∞. Then H is absolutely continuous and is the minimal solution of the differ-ential equation

H(x) − 1

4c(H ′(x))2 = φ(x) . (10.5)

If φ is constant on [x0,∞) then H is the unique solution of (10.5) that equals φon [x0,∞). The optimal stopping time τ∗ which yields H(0) is the Azema-Yor

stopping time given by τ∗ = inft ≥ 0 : Bt ≤ St − H′(St)2c .

Let us examine in more detail this result of Meilijson in order to try tocompare it with the results of Peskir. The Azema-Yor stopping time is defined

as τ∗ = inft ≥ 0 : Bt ≤ g(St) with g(x) = x − H′(x)2c . Let us determine the

differential equation satisfied by g. Note that H is by definition non-decreasing,so we have H ′(x) =

√

4c(H(x) − φ(x)). Differentiating this once again weobtain

H ′′(x) =2c(H ′(x) − φ′(x))√

4c(H(x) − φ(x))=

2c(H ′(x) − φ′(x))

H ′(x).

Therefore

g′(x) = 1 − H ′′(x)

2c=

φ′(x)

H ′(x)=

φ′(x)

2c(x− g(x)). (10.6)

We recognize immediately the equation (10.4) only there φ′(s) = 1 and c wasa function and not a constant. This motivated our investigation of the generalproblem 10.1, which is given below. It is given without proof, as it requires afair amount of new notation and more involved ideas. The interested reader isreferred to [60]. The following theorem is obtained from the theorem 3.1 in [66].

Theorem 10.3 (Peskir). The problem (10.1) for continuously derivable func-tion φ and c with countable number of points of discontinuity, has an optimalsolution with finite payoff if and only if there exists a maximal solution g∗ of

g′(s) =φ′(s)

2c(

g(s))(

s− g(s))) (10.7)

10.2 The optimal Skorokhod embedding problem 48

which stays strictly below the diagonal in R2, i.e. g∗(s) < s. In this case thepayoff is given by

V∗ = φ(0) − 2

∫ φ−1(0)

φ−1(gY∗(0))

uc(u)du, (10.8)

where gY∗ is a function given explicitly in [60]. The stopping time τ∗ = inft ≥0 : Bt ≤ g∗(St) is then optimal whenever it satisfies (10.2), otherwise it is“approximately” optimal10.Furthermore if there exists a solution ρ of the optimal stopping problem (10.1)then P(τ∗ ≤ ρ) = 1 and τ∗ satisfies (10.2).If there is no maximal solution of (10.7), which stays strictly below the diagonalin R2, then V∗ = ∞ and there is no optimal stopping time.

We recall the function φ was supposed strictly increasing and with a con-tinuous derivative. The assumptions made by Meilijson [54] are weaker. All heassumes is that φ is non-decreasing and right-continuous. In particular he alsotreats the case of piece-wise constant φ. The reasoning in this case is based onan iteration procedure. A similar argument can be used here to describe thestopping time and the payoff. We refer again to [60] for details.

10.2 The optimal Skorokhod embedding problem

Consider now the converse problem. That is, given a centered probability mea-sure µ describe all pairs of functions (φ, c) such that the optimal stopping timeτ∗, which solves (10.1) exists and embeds µ, that is Bτµ ∼ µ.

This may be seen as the “optimal Skorokhod embedding problem” as wenot only specify a method to obtain an embedding for µ but also construct anoptimal stopping problem, of which this embedding is a solution.

Again, consider first two special cases. First, let φ(x) = x. From the The-orem 10.1 we know that τ∗ is the Azema-Yor stopping time and the functiong∗ is just the inverse of barycentre function of some measure µ. The problemtherefore consists in identifying the dependence between µ and c. Suppose forsimplicity that µ has a strictly positive density f and that it satisfies L logL-integrability condition.

Theorem 10.4 (Peskir [67]). In the above setup there exists a unique function

c(x) =1

2

f(x)

µ(x)(10.9)

such that the optimal solution τ∗ of (10.1) embeds µ. This optimal solution isthen the Azema-Yor stopping time given by 5.4.

Now let c(x) = c be constant. Then we have

10In the sense of Peskir [66]

10.3 Investigating the law of the maximum 49

Theorem 10.5 (Meilijson [54]). In the setup above, there exists a uniquefunction φ defined through (10.5) with H ′(x) = 2c(x−Ψ−1

µ (x)), where Ψµ is thebarycentre function given in (5.3), such that the optimal solution τ∗ of (10.1)embeds µ. This optimal solution is then the Azema-Yor stopping time given by(5.4).

To solve the general problem we want to identify all the pairs (φ, c) suchthat the optimal stopping time τ∗ which solves (10.1) embeds µ in B, Bτµ ∼ µ.This is actually quite simple. We know that for τ∗ = inft ≥ 0 : Bt ≤ g∗(St)we have Bτ∗ ∼ µ if and only if g∗(s) = Ψ−1

µ (s). This is a direct consequenceof the Azema-Yor embedding (cf. Section 5). Let us investigate the differentialequation satisfied by Ψ−1

µ . For simplicity assume that µ has a strictly positivedensity f . It is easy to see that

(

Ψ−1µ (s)

)′=

µ(Ψ−1µ (s))

f(Ψ−1µ (s))(s − Ψ−1

µ (s)). (10.10)

Comparing this with the differential equation for g∗ (10.7) we see that we needto have

φ′(s)

2c(Ψ−1µ (s))

=µ(Ψ−1

µ (s))

f(Ψ−1µ (s))

. (10.11)

Equivalently, we have

φ′(Ψµ(u))

2c(u)=µ(u)

f(u), for u ∈ supp(µ). (10.12)

Indeed, if we take φ(s) = s so that φ′(s) = 1 we obtain a half of the hazardfunction, which is the result of Theorem 10.4. In general we have the followingtheorem

Theorem 10.6. Let µ be a probability measure on R with a strictly positive den-sity f , which satisfies L logL-integrability condition. Then the optimal stoppingtime τ∗, which solves (10.1) embeds µ in B, i.e. Bτ∗ ∼ µ if and only if (φ, c)satisfy (10.12) for all s ≥ 0. The stopping time τ∗ is then just the Azema-Yorstopping time given by (5.4).

10.3 Investigating the law of the maximum

Several authors investigated the laws of maximum and minimum of a stoppedprocess and their dependence on the terminal and initial laws. We will not dis-cuss here all the results as there are too many of them and they frequently varyconsiderably in methods and formulation. We give instead a quite complete,we hope, list of main references together with a short description and we leavefurther research to the reader.

• Dubins - Schwarz [25] - this article focuses more on Doob-like inequal-ities but some useful descriptions and calculations for Brownian motionstopping times are provided. Formulae for constant gap stopping times areestablished (see above).

10.3 Investigating the law of the maximum 50

• Kertz and Rosler [46] - authors build a martingale with assigned ter-minal and supremum distributions. More precisely, consider a proba-bility measure µ on R with

∫

|x|dµ(x) < ∞ and any other measure νwith µ < ν < µ∗, where µ∗ is Hardy and Littlewood maximal probabilitymeasure associated with µ. Authors show that there exists a martingale(Xt)0≤t≤1 such that X1 ∼ µ and S1 ∼ ν. It’s worth mentioning that theconverse problem, to prove that if ν ∼ S1 then µ < ν < µ∗, was solvedmuch earlier in Dubins and Gilat [24].

• Jacka [44] - again this work is concerned with Doob-like inequalities. Thecase of reflected Brownian motion and its supremum process is treated. Akind of maximality principle in this special case is establish although it isnot visible at first reading.

• Rogers [73] - a characterization of possible joint laws of the maximumand terminal value of a uniformly integrable martingale (or convergentcontinuous local martingale vanishing at zero) is developed. A number ofprevious results from the previous three articles on the list can be deduced.

• Hobson [39] - for fixed initial and terminal laws µ0 and µ1 respectively,the author shows that there exists an upper bound, with respect to stochas-tic ordering, on the law of S1, where S is the supremum process of amartingale (Xt)0≤t≤1 with X0 ∼ µ0 and X1 ∼ µ1. The bound is cal-culated explicitely and the martingale such that the bound is attained isconstructed. The methods used are closely connected with those of Chaconand Walsh and therefore with our presentation of the Azema-Yor solution(Section 5).

• Pedersen [62] - this Ph.D. thesis contains several chapters dealing withoptional stopping and the Skorokhod problem. In particular Doob’s Maxi-mal Inequality for Bessel processes is discussed.

• Peskir [66], [67] - the works discussed above.

• Brown, Hobson and Rogers [10] - this paper is very similar in spirit to[39]. Authors investigate an upper bound on the law of sup0≤t≤2Mt withgiven M0 ≡ 0, M1 ∼ µ1 and M2 ∼ µ2, where (Mt)0≤t≤2 is a martingale.A martingale for which this upper bound is attained is constructed usingexcursion theory.

• Hobson and Pedersen [40] - in this paper the authors find a lower boundon the law of the supremum of a continuous martingale with fixed initialand terminal distribution. An explicit construction, involving the Sko-rokhod embedding, is given. The results are applied to the robust hedgingof a forward start digital option.

• Cox and Hobson [17] - see Section 3.19.

11 Some applications 51

11 Some applications

The Skorokhod embedding problem is quite remarkable as it exists in the lit-erature for over 40 years by now, but it still comes back in various forms, itposes new challenges and is used for new applications. A thorough account ofapplications of this problem would require at least as much space as we havetaken so far to describe the problem itself and its solutions. With some regret,this is a project we shall postpone for the moment. What we will do instead isto try to sketch the main historical applications and give relevant references.

11.1 Embedding processes

So far we have considered the problem of embedding, by means of a stoppingtime, a given distribution in a given process. Naturally, once this was done,various applications were being considered. Primarily a question of embeddinga whole processes into, say, Brownian motion was of great interest. This ledfinally to the famous paper of Monroe [59], where he showed how all localsemimartingales are in fact time-changed Brownian motions. We will not presenthere a full account of the domain but rather try to sketch the important steps.

The very basic question consists in representing (in law) a martingale (Xn)as (BTn

), where (Bt) is a Brownian motion and (Tn) is an increasing family ofstopping times. This question actually motivated both Skorokhod and Dubinsand is in fact solved by any solution to the Skorokhod embedding problem.Indeed, let (Xn) be a martingale, X0 = 0 a.s, and Tµ be some solution to theSkorokhod embedding problem. The idea is then to build consecutive stoppingtimes Tn to embed in the new starting Brownian motion (BTn−1+s −BTn−1)s≥0

the conditional distribution of Xn+1 −Xn given X1 = BT1 , . . . Xn−1 = BTn−1 .Formal notation is easier if we use the Wiener space. Let ω(t) be the coor-

dinate process in the space of real, continuous functions, zero at zero, equippedwith the Wiener measure, so that (ω(t))t≥0 is a Brownian motion. Fix an em-bedding method, which associates to a measure µ a stopping time Tµ whichembeds it into ω: ω(Tµ(ω)) ∼ µ and ω(Tµ(ω)∧ t) is uniformly integrable. Writeµ0 for the law of X1 and µn(x1, . . . , xn) for the regular conditional distributionof Xn+1 −Xn given Xk = xk, 1 ≤ k ≤ n. Let T1 = Tµ0 and

Tn(ω) = Tn−1(ω) + Tµn(b1,...,bn)(ω′n), (11.1)

where bi = ω(Ti(ω)) and ω′n(s) = ω(s + Tn−1(ω)) − ω(Tn−1(ω)). It is then

straightforward to see the following proposition holds.

Proposition 11.1 (Dubins [23]). Let (Xn) be a real valued martingale withX0 = 0 a.s. and define an increasing family of stopping times (Tn) through(11.1). Then ω(Tn(ω))n≥ has the same distribution as (Xn)n≥1. Furthermoreif EX2

n = vn <∞ for all n ≥ 1 then ETn = vn for all n ≥ 1.

The next step consisted in extending this result to right-continuous mar-tingales with continuous time parameter. It was done by Monroe [57] in the

11.2 Invariance principles 52

same paper, where he established the properties of minimal stopping times (seeSection 8). Finally, as mentioned above, in a subsequent paper Monroe [59] es-tablished that any local semimartingale can be embedded in Brownian motion.Monroe used randomized stopping times and they were obtained as a limit, sothat no explicit formulae were given. Some years later Scott and Huggins [80],developed an embedding for reverse martingales and established for them, as anapplication, a functional law of iterated logarithm.

11.2 Invariance principles

In Skorokhod’s original work [81] the embedding problem was introduced asa tool to prove some invariance principles (and their approximation rates) forrandom walks. This was continued afterwards and extended.

We want to start however with a simple reasoning found in the notes byMeilijson [53] which shows how to obtain the central limit theorem for i.i.d.variables from the weak law of large numbers. Let µ be a centered probabilitymeasure with finite second moment

∫

Rx2dµ(x) = v < ∞ and let T be some

solution that embeds µ in B, ET = v. Write T1 = T and let T2 be the stoppingtime T for the new Brownian motion βt = Bt+T1 − BT1 . Apply this again toobtain T3 etc. We have thus a sequence i.i.d. of random variables T1, T2, . . . and

Sn = B∑

ni=1 Ti

is a sum of n i.i.d. variables with law µ. Define the processB(n)t =

1√nBnt, which is again a Brownian motion, thanks to the scaling property.

The pairs (B,B(n)) converge in law to a pair of two independent Brownianmotions (B, β). The stopping times Ti are measurable with respect to theBrownian motion B and by the strong law of large numbers 1

n

∑ni=1 Ti−−−−→n→∞

v

a.s. This implies that B(n)1n

∑

ni=1 Ti

converges in law to βv ∼ N (0, v), which ends

the argument as B(n)1n

∑

ni=1 Ti

= Sn√n

. This reasoning was generalized into

Proposition 11.2 (Holewijn and Meilijson [41]). Let X = (Xi : i ≥ 1) be astationary and ergodic process such that EX1 = 0 and E(Xn|X1, . . . , Xn−1) = 0a.s., n = 2, 3, . . . and (Bt : t ≥ 0) a Brownian motion. Then there exists asequence of standard (randomized) stopping times 0 ≤ T1 ≤ T2 ≤ . . . , such that

• (BT1 , BT2 , . . . ) is distributed as X,

• The process of pairs ((BT1 , T1), (BT2 −BT1 , T2−T1), . . . ) is stationary andergodic,

• ET1 = V ar(X1).

We now turn to invariance principles in their pure form. The general problemcan be summarized in the following way: construct a sequence of independentrandom vectors Xn, with given distributions, and a sequence of independentGaussian vectors Yn, each Yn having the same covariance structure as Xn, soas to minimize

∆(X,Y ) := max1≤k≤n

∣

∣

∣

k∑

i=1

Xi −k∑

i=1

Yi

∣

∣

∣. (11.2)

11.3 A list of some other applications 53

The rate of approximation for the invariance principle expresses how small is ∆as n→ ∞.

The classical case deals with real i.i.d. variables. Assume Xn are i.i.d.,EX1 = 0, EX2

1 = 1 and EX41 = b < ∞. It is fairly easy to see how the

Skorokhod embedding can be used to evaluate ∆(X,Y ). It suffices to apply

Proposition 11.1 to the martingale Mk =∑k

i=1Xk. Since the martingale in-crements are i.i.d. so are the consecutive stopping differences: (Tk+1 − Tk),k ≥ 0, where T0 ≡ 0. The Gaussian variables are given in a natural way byYk = Bk − Bk−1, k ≥ 1. The differences in (11.2) reduce to |BTk

− Bk|. In hispaper Strassen [82], proved that

Z = lim supn→∞

|BTn−Bn|

(n log logn)1/4(log n)1/2,

is of order O(1). Strassen used the original Skorokhod’s randomized embed-ding, but the result do not depend on this particular construction of stoppingtimes. This subject was further developed by Kiefer [47], who proved that therate ((logn)2n log log n)1/4 could not be improved under any additional momentconditions on the distribution ofX1. The case of independent, but not i.i.d. vari-ables was also treated. The Skorokhod embedding method was regarded as thebest one in proving the rates of approximation in invariance principles for sometime. However, in the middle of 70-ties, a new method, known as the Komlos-Major-Tusnady construction, was developed (see [18] and [49]) that proved farbetter than the Skorokhod embedding and provided an exhaustive solution tominimizing (11.2). The reason why Skorokhod type embedding could not beoptimal is quite clear. It was created as a mean to embed a single distributionin Brownian motion and to embed a sequence one just iterates the procedure,while the KMT construction was created from the beginning as an embeddingof random walk in Brownian motion.

Still, the Skorokhod embedding techniques can be of some use in this field.One possible example involves extending the results known for Brownian motion(as the invariance principles withXn uniform empirical process and Yn Brownianbridges) to the setup of arbitrary continuous time martingales (see for exampleHaeusler and Mason [34]). Also, Skorokhod embedding might be used alongwith the KMT construction (see Zhang and Hsu [91]).

11.3 A list of some other applications

Finally, we want just to point out several more fields, where Skorokhod embed-ding is used for various purposes. The reader is referred to the original articlesfor all details.

• Financial mathematics : some examples of applications include movingbetween discrete time and continuous time models (Guo [33]) and classicalBlack-Scholes model and other schemes (Walsh and Walsh [87]).

• Processes in random environment: random environment version of theSkorokhod embedding serves to embed a random walk in a random en-

11.3 A list of some other applications 54

vironment into a diffusion process in the random environment (see Yu[42]).

• Study of 12 -stable distribution: see Donati-Martin, Ghomrasni and Yor

[22].

• Functional CLT: Courbot uses Skorokhod embedding to study the Pro-horov distance between the distributions of Brownian motion and a squareintegrable cadlag martingale [16].

Acknowledgment. The author would like to thank Professor Marc Yor for intro-

ducing him to the domain and for his constant help and support both with finding

sources and understanding them.

REFERENCES 55

References

[1] J. Azema and P. A. Meyer. Une nouvelle representation du type de Sko-rokhod. In Seminaire de Probabilites, VIII (Univ. Strasbourg, annee univer-sitaire 1972-1973), pages 1–10. Lecture Notes in Math., Vol. 381. Springer,Berlin, 1974.

[2] J. Azema and M. Yor. Le probleme de Skorokhod: complements a “Unesolution simple au probleme de Skorokhod”. In Seminaire de Probabilites,XIII (Univ. Strasbourg, Strasbourg, 1977/78), volume 721 of Lecture Notesin Math., pages 625–633. Springer, Berlin, 1979.

[3] J. Azema and M. Yor. Une solution simple au probleme de Skorokhod. InSeminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78),volume 721 of Lecture Notes in Math., pages 90–115. Springer, Berlin, 1979.

[4] R. F. Bass. Skorokhod imbedding via stochastic integrals. In Seminar onprobability, XVII, volume 986 of Lecture Notes in Math., pages 221–224.Springer, Berlin, 1983.

[5] J. R. Baxter and R. V. Chacon. Potentials of stopped distributions. IllinoisJ. Math., 18:649–656, 1974.

[6] J. Bertoin and Y. Le Jan. Representation of measures by balayage from aregular recurrent point. Ann. Probab., 20(1):538–548, 1992.

[7] D. Blackwell and L. E. Dubins. A converse to the dominated convergencetheorem. Illinois J. Math., 7:508–514, 1963.

[8] L. Breiman. Probability. Addison-Wesley Publishing Company, Reading,Mass., 1968.

[9] J. Bretagnolle. Une remarque sur le probleme de Skorokhod. In Seminairede Probabilites, VIII (Univ. Strasbourg, annee universitaire 1972-1973),pages 11–19. Lecture Notes in Math., Vol. 381. Springer, Berlin, 1974.

[10] H. Brown, D. Hobson, and L. C. G. Rogers. The maximum maximum ofa martingale constrained by an intermediate law. Probab. Theory RelatedFields, 119(4):558–578, 2001.

[11] O. D. Cereteli. A metric characterization of the set of functions whoseconjugate functions are integrable. Sakharth. SSR Mecn. Akad. Moambe,81(2):281–283, 1976.

[12] P. Chacon. The filling scheme and barrier stopping times. PhD thesis,Univ. Washington, 1985.

[13] R. V. Chacon. Potential processes. Trans. Amer. Math. Soc., 226:39–58,1977.

REFERENCES 56

[14] R. V. Chacon and N. A. Ghoussoub. Embeddings in Brownian motion.Ann. Inst. H. Poincare Sect. B (N.S.), 15(3):287–292, 1979.

[15] R. V. Chacon and J. B. Walsh. One-dimensional potential embedding. InSeminaire de Probabilites, X (Premiere partie, Univ. Strasbourg, Stras-bourg, annee universitaire 1974/1975), pages 19–23. Lecture Notes inMath., Vol. 511. Springer, Berlin, 1976.

[16] B. Courbot. Rates of convergence in the functional CLT for martingales.C. R. Acad. Sci. Paris Ser. I Math., 328(6):509–513, 1999.

[17] A. Cox and D. Hobson. An optimal Skorokhod embedding for diffusions.Stochastic Process. Appl., 111(1):17–39, 2004.

[18] M. Csorgo and P. Revesz. A new method to prove Strassen type laws ofinvariance principle. I, II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete,31:255–269, 1974/75.

[19] B. Davis. Hardy spaces and rearrangements. Trans. Amer. Math. Soc.,261(1):211–233, 1980.

[20] B. De Meyer, B. Roynette, P. Vallois, and M. Yor. Sur l’independance d’untemps d’arret T et de la position BT d’un mouvement brownien (Bu, u ≥0). C. R. Acad. Sci. Paris Ser. I Math., 333(11):1017–1022, 2001.

[21] B. de Meyer, B. Roynette, P. Vallois, and M. Yor. On independent timesand positions for Brownian motions. Rev. Mat. Iberoamericana, 18(3):541–586, 2002.

[22] C. Donati-Martin, R. Ghomrasni, and M. Yor. Affine random equationsand the stable (12 ) distribution. Studia Sci. Math. Hungar., 36(3-4):387–405, 2000.

[23] L. E. Dubins. On a theorem of Skorohod. Ann. Math. Statist., 39:2094–2097, 1968.

[24] L. E. Dubins and D. Gilat. On the distribution of maxima of martingales.Proc. Amer. Math. Soc., 68(3):337–338, 1978.

[25] L. E. Dubins and G. Schwarz. A sharp inequality for sub-martingales andstopping-times. Asterisque, (157-158):129–145, 1988. Colloque Paul Levysur les Processus Stochastiques (Palaiseau, 1987).

[26] N. Falkner. On Skorohod embedding in n-dimensional Brownian motion bymeans of natural stopping times. In Seminar on Probability, XIV (Paris,1978/1979) (French), volume 784 of Lecture Notes in Math., pages 357–391.Springer, Berlin, 1980.

[27] N. Falkner and P. J. Fitzsimmons. Stopping distributions for right pro-cesses. Probab. Theory Related Fields, 89(3):301–318, 1991.

REFERENCES 57

[28] P. J. Fitzsimmons. Penetration times and Skorohod stopping. In Seminairede Probabilites, XXII, volume 1321 of Lecture Notes in Math., pages 166–174. Springer, Berlin, 1988.

[29] P. J. Fitzsimmons. Corrections to: “Penetration times and Skorohod stop-ping” [in seminaire de probabilites, xxii, 166–174, Lecture Notes in Math.,1321, Springer, Berlin, 1988; MR 89k:60109]. In Seminaire de Probabilites,XXIII, volume 1372 of Lecture Notes in Math., page 583. Springer, Berlin,1989.

[30] P. J. Fitzsimmons. Skorokhod embedding by randomized hitting times. InSeminar on Stochastic Processes, 1990 (Vancouver, BC, 1990), volume 24of Progr. Probab., pages 183–191. Birkhauser Boston, Boston, MA, 1991.

[31] T. Fujita. Cartain martingales of simple symmetric random walk and theirapplications. Private Communcation, 2004.

[32] P. Grandits and N. Falkner. Embedding in Brownian motion with driftand the Azema-Yor construction. Stochastic Process. Appl., 85(2):249–254,2000.

[33] X. Guo. Information and option pricings. Quant. Finance, 1(1):38–44,2001.

[34] E. Haeusler and D. M. Mason. Weighted approximations to continuous timemartingales with applications. Scand. J. Statist., 26(2):281–295, 1999.

[35] W. Hall. On the Skorokhod embedding theorem. Technical Report 33,Stanford Univ., Dept. of Stat., 1968.

[36] W. J. Hall. Embedding submartingales in Wiener processes with drift, withapplications to sequential analysis. J. Appl. Probability, 6:612–632, 1969.

[37] B. Hambly, G. Kersting, and A. Kyprianou. Law of the iterated logarithmfor oscillating random walks conditioned to stay non-negative. StochasticProcess. Appl., 108:327–343, 2002.

[38] D. Heath. Skorokhod stopping via potential theory. In Seminaire de Proba-bilites, VIII (Univ. Strasbourg, annee universitaire 1972-1973), pages 150–154. Lecture Notes in Math., Vol. 381. Springer, Berlin, 1974.

[39] D. G. Hobson. The maximum maximum of a martingale. In Seminaire deProbabilites, XXXII, volume 1686 of Lecture Notes in Math., pages 250–263. Springer, Berlin, 1998.

[40] D. G. Hobson and J. L. Pedersen. The minimum maximum of a continuousmartingale with given initial and terminal laws. Ann. Probab., 30(2):978–999, 2002.

REFERENCES 58

[41] P. J. Holewijn and I. Meilijson. Note on the central limit theorem for sta-tionary processes. In Seminar on probability, XVII, volume 986 of LectureNotes in Math., pages 240–242. Springer, Berlin, 1983.

[42] Y. Hu. The logarithmic average of Sinai’s walk in random environment.Period. Math. Hungar., 41(1-2):175–185, 2000. Endre Csaki 65.

[43] S. D. Jacka. Doob’s inequalities revisited: a maximal H1-embedding.Stochastic Process. Appl., 29(2):281–290, 1988.

[44] S. D. Jacka. Optimal stopping and best constants for Doob-like inequalities.I. The case p = 1. Ann. Probab., 19(4):1798–1821, 1991.

[45] T. Jeulin and M. Yor. Sur les distributions de certaines fonctionnelles dumouvement brownien. In Seminar on Probability, XV (Univ. Strasbourg,Strasbourg, 1979/1980) (French), volume 850 of Lecture Notes in Math.,pages 210–226. Springer, Berlin, 1981.

[46] R. P. Kertz and U. Rosler. Martingales with given maxima and terminaldistributions. Israel J. Math., 69(2):173–192, 1990.

[47] J. Kiefer. On the deviations in the Skorokhod-Strassen approximationscheme. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13:321–332,1969.

[48] J. Kiefer. Skorohod embedding of multivariate rv’s, and the sample df. Z.Wahrscheinlichkeitstheorie und Verw. Gebiete, 24:1–35, 1972.

[49] J. Komlos, P. Major, and G. Tusnady. An approximation of partial sumsof independent RV’s and the sample DF. I. Z. Wahrscheinlichkeitstheorieund Verw. Gebiete, 32:111–131, 1975.

[50] R. M. Loynes. Stopping times on Brownian motion: Some propertiesof Root’s construction. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete,16:211–218, 1970.

[51] I. Meilijson. There exists no ultimate solution to Skorokhod’s problem. InSeminar on Probability, XVI, volume 920 of Lecture Notes in Math., pages392–399. Springer, Berlin, 1982.

[52] I. Meilijson. On the Azema-Yor stopping time. In Seminar on probability,XVII, volume 986 of Lecture Notes in Math., pages 225–226. Springer,Berlin, 1983.

[53] I. Meilijson. Skorokhod’s problem - embeddings in Brownian Motion. Notesfrom the 3rd winter school of probability in Chile, 1983.

[54] I. Meilijson. The time to a given drawdown in Brownian motion. In Seminaron probability, XXXVII, volume 1832 of Lecture Notes in Math., pages 94–108. Springer, Berlin, 2003.

REFERENCES 59

[55] P. A. Meyer. Sur un article de Dubins. In Seminaire de Probabilites, V(Univ. Strasbourg, annee universitaire 1969–1970), pages 170–176. LectureNotes in Math., Vol. 191. Springer, Berlin, 1971.

[56] P. A. Meyer. Le schema de remplissage en temps continu, d’apres H. Rost.In Seminaire de Probabilites, VI (Univ. Strasbourg, annee universitaire1970–1971), pages 130–150. Lecture Notes in Math., Vol. 258. Springer,Berlin, 1972.

[57] I. Monroe. On embedding right continuous martingales in Brownian mo-tion. Ann. Math. Statist., 43:1293–1311, 1972.

[58] I. Monroe. Using additive functionals to embed preassigned distributions insymmetric stable processes. Trans. Amer. Math. Soc., 163:131–146, 1972.

[59] I. Monroe. Processes that can be embedded in Brownian motion. Ann.Probability, 6(1):42–56, 1978.

[60] J. Ob loj. A note on the maximality principle. In preparation, 2004.

[61] J. Ob loj and M. Yor. An explicit Skorokhod embedding for the age ofBrownian excursions and Azema martingale. Stochastic Process. Appl.,110(1):83–110, 2004.

[62] J. L. Pedersen. Some results on optimal stopping and Skorokhod Embeddingwith Applications. PhD thesis, University of Aarhus, 2000.

[63] J. L. Pedersen and G. Peskir. Computing the expectation of the Azema-Yor stopping times. Ann. Inst. H. Poincare Probab. Statist., 34(2):265–276,1998.

[64] J. L. Pedersen and G. Peskir. The Azema-Yor embedding in non-singulardiffusions. Stochastic Process. Appl., 96(2):305–312, 2001.

[65] E. Perkins. The Cereteli-Davis solution to the H1-embedding problem andan optimal embedding in Brownian motion. In Seminar on stochastic pro-cesses, 1985 (Gainesville, Fla., 1985), volume 12 of Progr. Probab. Statist.,pages 172–223. Birkhauser Boston, Boston, MA, 1986.

[66] G. Peskir. Optimal stopping of the maximum process: the maximalityprinciple. Ann. Probab., 26(4):1614–1640, 1998.

[67] G. Peskir. Designing options given the risk: the optimal Skorokhod-embedding problem. Stochastic Process. Appl., 81(1):25–38, 1999.

[68] G. Peskir. The Azema-Yor embedding in Brownian motion with drift. InHigh dimensional probability, II (Seattle, WA, 1999), volume 47 of Progr.Probab., pages 207–221. Birkhauser Boston, Boston, MA, 2000.

[69] G. Peskir. Maximum process problems in optimal control theory. TechnicalReport 423, Dept. Theoret. Statist. Aarhus, 2001.

REFERENCES 60

[70] M. Pierre. Le probleme de Skorokhod: une remarque sur la demonstrationd’Azema-Yor. In Seminar on Probability, XIV (Paris, 1978/1979)(French), volume 784 of Lecture Notes in Math., pages 392–396. Springer,Berlin, 1980.

[71] D. Revuz and M. Yor. Continuous martingales and Brownian motion, vol-ume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamen-tal Principles of Mathematical Sciences]. Springer-Verlag, Berlin, thirdedition, 1999.

[72] L. C. G. Rogers. Williams’ characterisation of the Brownian excursion law:proof and applications. In Seminar on Probability, XV (Univ. Strasbourg,Strasbourg, 1979/1980) (French), volume 850 of Lecture Notes in Math.,pages 227–250. Springer, Berlin, 1981.

[73] L. C. G. Rogers. The joint law of the maximum and terminal value of amartingale. Probab. Theory Related Fields, 95(4):451–466, 1993.

[74] D. H. Root. The existence of certain stopping times on Brownian motion.Ann. Math. Statist., 40:715–718, 1969.

[75] H. Rost. Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichemPotential. Manuscripta Math., 3:321–329, 1970.

[76] H. Rost. The stopping distributions of a Markov Process. Invent. Math.,14:1–16, 1971.

[77] H. Rost. Skorokhod’s theorem for general Markov processes. In Trans-actions of the Sixth Prague Conference on Information Theory, StatisticalDecision Functions, Random Processes (Tech. Univ. Prague, Prague, 1971;dedicated to the memory of Antonın Spacek), pages 755–764. Academia,Prague, 1973.

[78] H. Rost. Skorokhod stopping times of minimal variance. In Seminairede Probabilites, X (Premiere partie, Univ. Strasbourg, Strasbourg, anneeuniversitaire 1974/1975), pages 194–208. Lecture Notes in Math., Vol. 511.Springer, Berlin, 1976.

[79] B. Roynette, P. Vallois, and M. Yor. A solution to Skorokhod’s embeddingfor linear Brownian motion and its local time. Studia Sci. Math. Hungar.,39(1-2):97–127, 2002.

[80] D. J. Scott and R. M. Huggins. On the embedding of processes in Brownianmotion and the law of the iterated logarithm for reverse martingales. Bull.Austral. Math. Soc., 27(3):443–459, 1983.

[81] A. V. Skorokhod. Studies in the theory of random processes. Translatedfrom the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co.,Inc., Reading, Mass., 1965.

REFERENCES 61

[82] V. Strassen. Almost sure behavior of sums of independent random variablesand martingales. In Proc. Fifth Berkeley Sympos. Math. Statist. and Prob-ability (Berkeley, Calif., 1965/66), pages Vol. II: Contributions to Proba-bility Theory, Part 1, pp. 315–343. Univ. California Press, Berkeley, Calif.,1967.

[83] D. W. Stroock. Markov processes from K. Ito’s perspective, volume 155 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ,2003.

[84] P. Vallois. Le probleme de Skorokhod sur R: une approche avec le tempslocal. In Seminar on probability, XVII, volume 986 of Lecture Notes inMath., pages 227–239. Springer, Berlin, 1983.

[85] P. Vallois. Sur la loi du maximum et du temps local d’une martingalecontinue uniformement integrable. Proc. London Math. Soc. (3), 69(2):399–427, 1994.

[86] D. P. van der Vecht. Ultimateness and the Azema-Yor stopping time. InSeminaire de Probabilites, XX, 1984/85, volume 1204 of Lecture Notes inMath., pages 375–378. Springer, Berlin, 1986.

[87] J. B. Walsh and O. D. Walsh. Embedding and the convergence of thebinomial and trinomial tree schemes. In Numerical methods and stochastics(Toronto, ON, 1999), volume 34 of Fields Inst. Commun., pages 101–121.Amer. Math. Soc., Providence, RI, 2002.

[88] M. P. Yershov. On stochastic equations. In Proceedings of the SecondJapan-USSR Symposium on Probability Theory (Kyoto, 1972), pages 527–530. Lecture Notes in Math., Vol. 330, Berlin, 1973. Springer.

[89] M. Yor. Some aspects of Brownian motion. Part II. Lectures in Mathemat-ics ETH Zurich. Birkhauser Verlag, Basel, 1997. Some recent martingaleproblems.

[90] P. Zaremba. Skorokhod problem—elementary proof of the Azema-Yor for-mula. Probab. Math. Statist., 6(1):11–17, 1985.

[91] H. Zhang and G.-H. Hsu. A new proof of the invariance principle for renewalprocesses. Sankhya Ser. A, 56(2):375–378, 1994.