On the Young integral, truncated variation and rough pathsweb.sgh.waw.pl/~rlocho/UCT_talk.pdf ·...

On the Young integral, truncated variation and roughpaths

Rafa l M. Lochowski

Warsaw School of Economics and AIMS

UCT Department of Mathematics Seminar talk

R. Lochowski (WSE and AIMS) On the Young integral... UCT, April 2017 1 / 28

The classical Riemann-Stieltjes integral

We say that the Stieltjes integral∫ ba

f (x)dg(x)

exists in the Riemann sense with the value I , if the sum

n∑i=1

f (ξi ) {g (xi )− g (xi−1)} ,

where a = x0 < x1 < . . . < xn = b and ξi ∈ [xi−1, xi ] is as close to I as wewish, provided the mesh of the partition π = {x0, x1, . . . , xn} ,

mesh (π) := max1≤i≤n

(xi − xi−1) ,

is sufficiently small.


The classical Riemann-Stieltjes integral cont.There is no problem with the existence of the Riemann-Stieltjes integral ifthe total variation of the integrator

TV(g , [a, b]) := supn

supa≤x0

(Strong) p-variation

In his famous paper An inequality of the Holder type, connected withStieltjes integration (Acta Math. 67 (1): 251-282, 1936) LaurenceChisholm Young treated the case where both - the integrand andintegrator may have infinite total variations.

He was considering the functions of finite p and q-variations. Let us recallthe definition of a (strong) p-variation (p > 0): for any f : [a, b]→ R

Vp(f , [a, b]) := supn

supa≤x0

(Strong) p-variation as a rate-independent measure ofirregularity

Strong p-variation may be viewed as a measure of path irregularity.

If Vp(f , [a, b]) < +∞ for p ≥ 1 then Vq(f , [a, b]) < +∞ for all q > p.

This folows from inequality (∑

i |ai |q)1/q ≤ (

∑i |ai |p)

1/p . This is thereason to introduce the variation index:

Indvar (f , [a, b]) := inf {p ≥ 1 : Vp(f , [a, b]) < +∞} .

Another measure of path irregularity is Hölder exponent: we say that thefunction f : [a, b]→ R is Hölder continuous with the Hölder exponent α if

supa≤s

Examples of paths of a standard Brownian motion, whichhave almost surely variation index 2 and any Hölderexponent > 0.5

0 200 400 600 800 1000

−50

−40

−30

−20

−10

010

Index

y0


The Loeve-Young inequality

Young (together with Loeve) proved the following estimate: ifa = x0 < x1 < . . . < xn = b, ξi ∈ [xi−1, xi ] , and similarlya = x ,0 < x

,1 < . . . < x

,n = b, ξ

,i ∈[x ,i−1, x

,i

], and p > 0, q > 0 then∣∣∣∣∣

n∑i=1

f (ξi ) {g (xi )− g (xi−1)} −n∑

i=1

f(ξ,i) {

g(x ,i)− g

(x ,i−1

)}∣∣∣∣∣≤ 2

(1 + ζ

(1

p+

1

q

))(Vp(f , [a, b]))1/p (Vq(g , [a, b]))1/q . (1)

Here, ζ denotes the famous Riemann zeta function,

ζ(α) =∞∑k=1

1

kα.


The Young integralAs a result of the Loeve-Young estimate we get

Theorem (L. C. Young, 1936)

The Stieltjes integral∫ ba f (x)dg(x) exists in the Riemann sense whenever

f and g have no common discontinuities, and Vp(f , [a, b]) < +∞,Vq(g , [a, b]) < +∞ for some p > 0, q > 0 such that p−1 + q−1 > 1.

Moreover, for any ξ ∈ [a, b] one has the following estimate∣∣∣∣∫ ba

f (x)dg(x)− f (ξ) [g(b)− g(a)]∣∣∣∣

≤ 2(

1 + ζ

(1

p+

1

q

))(Vp(f , [a, b]))1/p (Vq(g , [a, b]))1/q .

Remark

Young also provided an example of f and g with finite strong 2-variationsand such that

∫ ba f (x)dg(x) does not exists in the Riemann sense, and

outlined a counter-example for other p and q such that p−1 + q−1 = 1.R. Lochowski (WSE and AIMS) On the Young integral... UCT, April 2017 8 / 28

(Strong) Φ-variation

In his later work General inequalities for Stieltjes integrals and theconvergence of Fourier series. (Math. Ann. 115: 581-612, 1938) Youngconsidered more general (strong) Φ-variations.

Let V denotes the class of all functions Φ : [0,∞)→ [0,∞) which arestrictly increasing, continuous, unbounded, and 0 at 0. For any Φ ∈ V wedefine

VΦ(f , [a, b]) := supn

supa≤x0

The generalised Young integral

Theorem (L. C. Young, 1938)

If f and g have no common discontinuities, VΦ(f , [a, b]) < +∞,VΨ(g , [a, b]) < +∞ and Θ (φ, ψ) < +∞ then the Stieltjes integral∫ ba f (x)dg(x) exists in the Riemann sense.

Moreover, for any ξ ∈ [a, b] one has the following estimate∣∣∣∣∫ ba

f (x)dg(x)− f (ξ) [g(b)− g(a)]∣∣∣∣

≤ 20∞∑k=1

φ

(VΦ(f , [a, b])

k

)ψ

(VΨ(g , [a, b])

k

).

Remark

Leśniewicz and Orlicz showed in 1973 that the series condition Θ (φ, ψ)< +∞ is the best possible for convex functions Φ and Ψ satisfying the ∆2and ∇d growth conditions (d > 1) : Ψ(2u) ≤ CΨ(u), 2Ψ(u) ≤ Ψ(du)/d .


The proofs of Youngs therems

In all proofs of both Youngs therems found in the literature theLoeve-Young inequality is the most cruicial step.

The original proof of this inequality was elementary but prettycomplicated. Later, the proof was simplified. For example, in the book byTerry Lyons and Zhongmin Quian System Control and Rough Paths,Oxford Univ. Press 2002, or in lecture notes by Terry Lyons, MichaelCaruana and Thierry Levy Differential Equations Driven by Rough Paths,Springer 2007, it was based on the approximation of the Riemann sum

n∑i=1

f (ξi ) {g (xi )− g (xi−1)}

by a sum where one of the numbers x1 < . . . < xn−1 is excluded from thepartition. Iterating this procedure one finds that this sum is sufficientlyclose to the difference f (ξ) {g (b)− g (a)} for some ξ ∈ [a, b] .


Alternative approach

Yet another approach, which gives even better order of the error of theapproximation ∫ b

af (x)dg(x)− f (ξ) {g (b)− g (a)}

may be based on the concept of (uniform) approximation of irregularsignals f and g with given accuracy ε > 0 by their regularisations - signalswith finite total variations.Let us define

TVε(f , [a, b]) := supn

supa≤x0

The truncated variation

Theorem ( Lochowski, 2013 for cadlag functions; Ghomrasni and Lochowski for regulated functions, 2015)

If f [a, b]→ R is regulated then

TVε(f , [a, b]) = inf

{TV(f ε, [a, b]) : sup

a≤s≤b|f (s)− f ε(s)| ≤ ε

2

}. (2)

Remark

The bound

TVε(f , [a, b]) ≤ inf

{TV(f ε, [a, b]) : sup

a≤s≤b|f (s)− f ε(s)| ≤ ε

2

}

follows immediately from the estimate: if supa≤s≤b |f (s)− f ε(s)| ≤ ε2 then

max (|f (xi )− f (xi−1)| − ε, 0) ≤ |f ε (xi )− f ε (xi−1)| .


The truncated variation - remarksLet p ≥ 1, x ≥ 0 and ε > 0. From the inequality

εp−1 max (x − ε, 0) ≤ xp

we immediately get that whenever Vp(f , [a, b]) < +∞ then

supε>0

{εp−1TVε(f , [a, b])

}≤ Vp(f , [a, b])

or, equivalently,

TVε(f , [a, b]) ≤ Vp(f , [a, b])

εp−1for any ε > 0.

Thus, for any function with finite (strong) p-variation (p > 1) but infinitetotal variation (corresponding to p = 1) we get that its truncated variationTVε(f , [a, b]) tends to +∞ no faster than 1/εp−1.

Every such function may be uniformly approximated with the accuracyε > 0 by a function whose total variation is of order 1/εp−1.


A theorem on the existence of the Riemann-Stieljesintegral in terms of the truncated variations

Theorem

Let f , g : [a; b]→ R be two regulated functions which have no commonpoints of discontinuity. Let η0 ≥ η1 ≥ . . . and θ0 ≥ θ1 ≥ . . . be twosequences of non-negative numbers, such that ηk ↓ 0, θk ↓ 0 as k → +∞.Define η−1 := supa≤t≤b |f (t)− f (a)| and

S :=+∞∑k=0

2kηk−1 · TVθk (g , [a; b]) +∞∑k=0

2kθk · TVηk (f , [a; b]) .

If S < +∞ and ξ ∈ [a, b] then the Riemann-Stieltjes integral∫ ba f dg

exists and one has the following estimate∣∣∣∣∫ ba

f dg − f (ξ) [g (b)− g (a)]∣∣∣∣ ≤ 2S .


Corollary - an estimate of the Loeve-Young type

Corollary

Let f , g : [a; b]→ R be two functions with no common points ofdiscontinuity. If

supε>0

εp−1TVε(f , [a, b]) < +∞ and supδ>0

δq−1TVδ(g , [a, b]) < +∞,

where p > 1, q > 1, p−1 + q−1 > 1, then the Riemann Stieltjes∫ ba f dg

exists. Moreover, there exist a constant Cp,q, depending on p and q only,such that for any ξ ∈ [a, b]∣∣∣∣∫ b

af dg − f (ξ) [g (b)− g (a)]

∣∣∣∣≤ Cp,q ‖f ‖p−p/qp−TV,[a;b] ‖f ‖

1+p/q−posc,[a;b] ‖g‖q−TV,[a;b] ,

where ‖f ‖p−TV,[a;b] :=(supε>0 ε

p−1TVε(f , [a, b]))1/p

.


Young’s theorem in Banach spaces

It is possible to state an analog of the Young theorem and theLoeve-Young inequality for integrals driven by signals attainig their valuesin some Banach space E .

In such a setting the integral ∫ ba

f (x)dg(x)

makes sense if g : [a, b]→ E and f : [a, b]→ L(E ,V ), where V is anotherBanach space and L(E ,V ) denotes the space of (continuous) linearmapings from E to V , F : E → V , with the usual norm

‖F‖L(E ,V ) = supu∈E ,‖u‖E=1

Fu.

To prove the existence of this integral using truncated variation techniquesone needs to prove an analog of relation (2) in higher dimensional spaces.


The truncated variation in higher dimensions - properties

If f : [a, b]→ E attains its values in a metric space E with the metric dthen we may define

TVε(f , [a, b]) := supn

supa≤x0

A theorem on the truncated variation in higher dimensions

Theorem (Lochowski 2016)

For any regulated f : [a, b]→ E there exists a step functionf c : [a, b]→ E such that supt∈[a,b] d (f (t) , f c (t)) ≤ c/2 and for anyλ > 1, TV(f c , [a, b]) ≤ λ · TV(λ−1)c/(2λ)(f , [a, b]) . Thus the followingestimates hold

TVc(f , [a, b]) ≤ infg∈B(f ,c/2)

TV(g , [a, b]) ≤ infλ>1

λ · TV(λ−1)c/(2λ)(f , [a, b]) .

Moreover, if E is a vector normed space with the norm ‖·‖E then thereexists f c,lin : [a, b]→ E such that f c,lin is piecewise linear, jumps of f c,linoccur only at the points where the jumps of f occur,supt∈[a,b]

∥∥f (t)− f c,lin (t)∥∥E≤ c and TV

(f c,lin, [a, b]

)= TV(f c , [a, b]) .


A theorem on the existence of the Riemann-Stieltjesintegral in higher dimensions

Theorem

Let f : [a, b]→ L (E ,V ) and g : [a, b]→ E be two regulated functionswhich have no common points of discontinuity. Let η0 ≥ η1 ≥ . . . andθ0 ≥ θ1 ≥ . . . be two sequences of positive numbers, such that ηk ↓ 0,θk ↓ 0 as k → +∞. Define η−1 := 12 supa≤t≤b ‖f (t)− f (a)‖L(E ,V ) and

S := 4+∞∑k=0

3kηk−1 · TVθk/4(g , [a, b]) + 4∞∑k=0

3kθk · TVηk/4(f , [a, b]) .

If S < +∞ then the Rieman-Stieltjes integral∫ ba f dg exists and for any

ξ ∈ [a, b] one has the following estimate∥∥∥∥∫ ba

f dg − f (ξ) [g (b)− g (a)]∥∥∥∥V

≤ 2S . (3)


An estimate of the Loeve-Young type in Banach spaces

Corollary

Let f : [a, b]→ L (E ,V ) , g : [a, b]→ E be two functions with no commonpoints of discontinuity. If

supε>0

εp−1TVε(f , [a, b]) < +∞ and supδ>0

δq−1TVδ(g , [a, b]) < +∞,

where p > 1, q > 1, p−1 + q−1 > 1, then the Riemann Stieltjes∫ ba f dg

exists. Moreover, there exist a constant Dp,q, depending on p and q only,such that for any ξ ∈ [a, b]∣∣∣∣∫ b

af dg − f (ξ) [g (b)− g (a)]

∣∣∣∣≤ Dp,q ‖f ‖p−p/qp−TV,[a;b] ‖f ‖

1+p/q−posc,[a;b] ‖g‖q−TV,[a;b] ,

where ‖f ‖p−TV,[a;b] :=(supε>0 ε

p−1TVε(f , [a, b]))1/p

.


Problem with the definition of integrals driven by moreirregular paths

All the results mentioned so far are insufficient to define for example theintegral ∫ T

0BtdBt ,

where Bt is a standard Brownian motion - a stochastic process widely usedin modelling the evolution of stock prices, stock indices etc.

It is well known that Indvar (B, [0,T ]) = 2, which makes it impossible toapply Young’s theorem in this case.

A real breakthrough in the understanding of this difficulcy was provided bythe rough paths theory, developed mainly by Terry Lyons in early 90s ofthe last century.


Multiplicative functionalsAssume that V is a Banach space with norm ‖ · ‖ and we are given asequence of tensor products V⊗k = V ⊗ V ⊗ . . .⊗ V (k copies of V )together with norms ‖ · ‖k on V⊗k satisfying ‖ξ ⊗ η‖k+l ≤ ‖ξ‖k‖η‖l forξ ∈ V⊗k and η ∈ V⊗l . For each n = 1, 2 . . . we build the following(truncated) tensor algebra T (n)V :

T (n)V =n∑

k=0

⊕V⊗k , V⊗0 = R

with multiplication (tensor product) being the usual multiplication aspolynomials, except that the higher-order (than degree n) terms areomitted.

Let now ∆ := {(s, t) : 0 ≤ s ≤ t ≤ T} . X : ∆→ T (n)V ,Xs,t =

(X

(0)s,t ,X

(1)s,t , . . . ,X

(n)s,t

)is called a multiplicative functional if

X(0)s,t = 1 and

Xs,t ⊗ Xt,u = Xs,u, for any (s, t), (t, u) ∈ ∆. (4)


Abstract iterated integrals

Equality (4) is called Chen’s identity and it corresponds to the additiveproperty of increments of iterated integrals over concatenated intervals.For example, if x : [0,T ]→ R is a continuous path with finite totalvariation and we define

x(1)s,t =

∫ ts

dxv and x(2)s,t =

∫s≤v1

Controls and Lyons Extension TheoremA control ω is a continuous, super-additive function on the simplex ∆with values in [0; +∞) such that ω(s, s) = 0. The superadditivity of ωmeans that

ω(s, t) + ω(t, u) ≤ ω(s, u) for any 0 ≤ s ≤ t ≤ u ≤ T .

One says that a map X : ∆→ T (n)V possesses finite p-variation(p ≥ 1) if

‖X (i)s,t ‖i ≤ ω(s, t)i/p for i = 1, 2, . . . , n; (s, t) ∈ ∆.

Theorem (Lyons Extension theorem)

If X : ∆→ T (n)V is a multiplicative functional with finite p-variation andn ≥ bpc then all abstract iterated integrals of order greater than n may becalculated with the integrals of order 1, 2, . . . , n which means simply thatfor any m > n the functional X may be uniquely extended to amultiplicative functional X̃ : ∆→ T (m)V with finite p-variation, whichagrees with X on first n components.


Ambiguity of iterated integrals driven by signals withinfinite p-variation

The simplest non-trivial case of Lyons Extension Theorem may be viewedas an abstract formulation of Young’s theorem for p = q ∈ (1, 2), since forsuch p and q, 1/p + 1/q = 2/p > 1 and thus, Young’s theorem guarantees

the existence of x(2)s,t =

∫s≤v1

References

L.C. Young, 1936, An inequality of the Holder type, connected withStieltjes integration, Acta Math. 67 (1): 251-282.

L.C. Young, 1938,General inequalities for Stieltjes integrals and theconvergence of Fourier series, Math. Ann. 115: 581-612.

Terry Lyons and Zhongmin Quian, System Control and Rough Paths,Oxford Univ. Press 2002.

Terry Lyons, Michael Caruana and Thierry Levy, Differential EquationsDriven by Rough Paths, Springer 2007.R. Lochowski, 2015, A new theorem on the existence of theRiemann-Stieltjes integral and an improved version of the Love-Younginequality, J. Ineq. Appl. 378: 1–17.

R. Lochowski, 2015, Riemann-Stieltjes integrals driven by irregular signalsin Banach spaces and rate-independent characteristics of their irregularity,ArXiv:1602.02269.


Thank you for your attention!


The Young integral

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