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On pathwise stochastic integration

Rafa l Marcin Lochowski

Afican Institute for Mathematical Sciences,Warsaw School of Economics

UWC seminar

Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 1 / 29

The Riemann-Stieltjes integral

The Riemann-Stieltjes integral of a deterministic function f : [a, b]→ R(integrand) with respect to another deterministic function g : [a, b]→ R(integrator) is defined as the limit of sums

n∑i=1

f (si ) g (ti )− g (ti−1) ,

where si ∈ [ti−1; ti ] , π = a = t0 < t1 < ... < tn = b , as the mesh of thepartition π, mesh(π) := maxi=1,2,...,n |ti − ti−1| , goes to 0.

The Riemann-Stieltjes integral, denoted by

(RS)

∫[a;b]

f dg

may not exist for a given pair (f , g). The simplest situation when ithappens is when the total variation of g is infinite.


The Riemann-Stieltjes integral

The Riemann-Stieltjes integral of a deterministic function f : [a, b]→ R(integrand) with respect to another deterministic function g : [a, b]→ R(integrator) is defined as the limit of sums

n∑i=1

f (si ) g (ti )− g (ti−1) ,

where si ∈ [ti−1; ti ] , π = a = t0 < t1 < ... < tn = b , as the mesh of thepartition π, mesh(π) := maxi=1,2,...,n |ti − ti−1| , goes to 0.The Riemann-Stieltjes integral, denoted by

(RS)

∫[a;b]

f dg

may not exist for a given pair (f , g). The simplest situation when ithappens is when the total variation of g is infinite.


The total variation and the existence of the R-S integral

Recall that the total variation of a deterministic function g : [a, b]→ R isdefined as

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

supa≤t0<t1<...<tn≤b

n∑i=1

|g (ti )− g (ti−1)| ,

The finiteness of TV (g , [a; b]) together with the continuity of f guaranteethe existence of (RS)

∫[a;b] f dg .

However, still, for a bounded, Borel-measurable integrand f and finite(total) variation integrator g , (RS)

∫[a;b] f dg may not exist.

This may happen e.g. when the jumps of the function f coincide with thejumps of g (the invention of an appropriate example might be a good,instructive exercise for students).


The refinement of the R-S integral - the Lebesgue-Stieltjesintegral

A fine refinement of the R-S integral is the Lebesgue-Stieltjes integral.The construction is made by the introduction of the measure space([a; b],B ([a; b]) , µg ) , where B ([a; b]) denotes the σ-field of Borel subsetsof [a; b] and µg is a signed, σ-finite measure on [a; b].To define µg we consier the cadlag version of g (right-continuous with leftlimits) which we will also denote by g and for a ≤ c ≤ d ≤ b define

µg (c ; d ] := g(d)− g(c).

Now we extend the measure µg to all Borel subsets of [a; b](Caratheodory’s extension) and define the Lebesgue-Stieltjes integral asthe usual Lebesgue integral of f with respect to the measure µg :

(LS)

∫[a;b]

f dg :=

∫[a;b]

f dµg .


The Lebesgue-Stieltjes integral - properties

Now, finiteness of TV (g , [a; b]) together with boundedness andBorel-measurability of the integrand f guarantee the existence of(LS)

∫[a;b] f dg .

If both, f and g are cadlag and have finite total variation we haveimportant integration by parts formula

(LS)

∫(a;b]

f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)

−(LS)

∫(a;b]

g(t−)df (t)−∑

a<s≤b∆f (s)∆g(s).

where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).Notice, that using this formula one may propose a reasonable value for(LS)

∫(a;b] f (t−)dg(t) whenever f and g are cadlag, TV (f , [a; b]) < +∞

and g is bounded! (The same observation may be used to differentiatedistributions but it is another story...)




∫[a;b] f dg .


(LS)

∫(a;b]

f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)

−(LS)

∫(a;b]

g(t−)df (t)−∑


where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).

Notice, that using this formula one may propose a reasonable value for(LS)






∫[a;b] f dg .


(LS)

∫(a;b]

f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)

−(LS)

∫(a;b]

g(t−)df (t)−∑


where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).Notice, that using this formula one may propose a reasonable value for(LS)




The Lebesgue-Stieltjes integral still insufficient

(A standard) Brownian motion B is one of the simplest continuous-timeprocess with continuous trajectories, widely used in stochastic modellingand optimisation.Unfortunately, its paths have a.s. infinite total variation. When one wantsto integrate locally finite variation deterministic function or locally finitevariation stochastic process with respect to a Brownian trajectory Bt ,t ∈ [0;T ], one may use the relation (1) (this idea is due to Paley, Wienerand Zygmund).

Unfortunately, this is still not sufficient to calculate (or at least to give areasonable meaning if the calculations were too hard) e.g.

(LS)

∫(a;b]

BtdBt .


The Lebesgue-Stieltjes integral still insufficient

(A standard) Brownian motion B is one of the simplest continuous-timeprocess with continuous trajectories, widely used in stochastic modellingand optimisation.Unfortunately, its paths have a.s. infinite total variation. When one wantsto integrate locally finite variation deterministic function or locally finitevariation stochastic process with respect to a Brownian trajectory Bt ,t ∈ [0;T ], one may use the relation (1) (this idea is due to Paley, Wienerand Zygmund).Unfortunately, this is still not sufficient to calculate (or at least to give areasonable meaning if the calculations were too hard) e.g.

(LS)

∫(a;b]

BtdBt .


Functions with finite and infinite total variation - examples

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.0

0.5

t

Figure : A typical path of a Brownian motion (green) and a function with finitetotal variation (blue)


A standard Brownian motion - an ”axiomatic” definition

A standard Brownian motion is a stochastic process Bt , t ≥ 0 defined onsome (rich enough) probability space (Ω,F ,P) with the followingproperties

B0 ≡ 0;

B has continuous paths, i.e. the probability P that for ω ∈ Ω, thepath

[0; +∞) 3 t 7→ Bt(ω) ∈ R

is continuous equals 1;

for any 0 ≤ s < t < u increments Bu − Bt and Bt − Bs areindependent;

for any 0 ≤ s < t the increment Bt − Bs has normal distribution withmean 0 and variance t − s, N (0, t − s).


Finiteness of the quadratic variation of a Brownian motion

Cruicial observation which is sometimes utilised to define a stochasticintegral with respect to the Brownian motion or even with respect to muchmore general family of processes - semimartingales - is the finiteness oftheir quadratic variation.But even with the quadratic variation we must be careful. When we define

V2 (B, [a; b]) := supn


n∑i=1

(Bti − Bti−1

)2,

then V2 (B, [a; b]) is a.s. infinite.

However, one may relatively easily prove that for any sequence of

partitions πk =a = tk0 < tk1 < ... < tkn(k) = b

, with mesh(πk) ↓ 0 as

k ↑ +∞ we have

n(k)∑i=1

(Btki− Btki−1

)2→P [B]b − [B]a := b − a,

where →P denotes the convergence in probability.


Finiteness of the quadratic variation of a Brownian motion

Cruicial observation which is sometimes utilised to define a stochasticintegral with respect to the Brownian motion or even with respect to muchmore general family of processes - semimartingales - is the finiteness oftheir quadratic variation.But even with the quadratic variation we must be careful. When we define

V2 (B, [a; b]) := supn


n∑i=1

(Bti − Bti−1

)2,

then V2 (B, [a; b]) is a.s. infinite.However, one may relatively easily prove that for any sequence of

partitions πk =a = tk0 < tk1 < ... < tkn(k) = b

, with mesh(πk) ↓ 0 as

k ↑ +∞ we have

n(k)∑i=1

(Btki− Btki−1

)2→P [B]b − [B]a := b − a,

where →P denotes the convergence in probability.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 9 / 29

ψ- variation of a Brownian motion

For x > 0 define

ψ(x) :=x2

ln(ln(1/x) ∨ 2)

and ψ(0) := 0 then, by the result of S. J. Taylor (Exact asymptoticestimates of Brownian path variation, Duke Math. J. 39), almost surely:

Vψ (B, [a; b]) := supn

sup0≤t0<t1<...<tn≤T

n∑i=1

ψ(∣∣Bti − Bti−1

∣∣) < +∞, (2)

moreover, ψ is a function with the greatest possible order at 0, for which(2) holds.

Remark

Moreover, when mesh(π(n)

)→ 0, then

limn→∞

supn


n∑i=1

ψ

(Bt

(n)i

− Bt

(n)i−1

)= T a.s.


A small detour - why the quadratic variation [B] of theBrownian motion on the interval [a; b] equals b − a?

To give a hint why the quadratic variation of the Brownian motion on theinterval [a; b] equals b − a let us recall the simplest construction of astandard Brownian motion (due to Donsker we know that this constructionworks, though the convergence is relatively slow).

Pick (very) large integer n;

set t = 0, 1n ,

2n , ...,

nn ,

n+1n , ...; dt = 1

n ;

set B0 = 0 and for t = 0, 1n ,

2n , ...,

nn ,

n+1n , ...

Bt+dt = Bt+1/n = Bt +

1√n

with probability 1/2;

− 1√n

with probability 1/2.

Now notice that (dBt)2 = dt (the simplest version of Ito’s formula) and

thus ∑t=1/n,2/n,...,T

(dBt)2 =

∑t=1/n,2/n,...,T

dt = [B]T = T .


Another small detour - Young’s integral

When the integrand has finite p-variation, Vp the integrator has finiteq-variation, Vq, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define

(RYS)

∫[a;b]

f dg .

where (RYS)∫

denotes some refinement of the Riemann-Stieltjes integral(the refinement is made to avoid problems with discontinuities) and wheng is continuous, it coincides with the Riemann-Stieltjes integral.

This is proved with the following Love-Young inequality:∣∣∣∣∣n∑

i=1

f (si ) g(ti )− g(ti−1) − f (si0) g(b)− g(a)

∣∣∣∣∣ ≤ ζ (p−1 + q−1),

for any partition π = a = t0 < t1 < ... < tn = b , si ∈ [ti−1; ti ] ,i , i0 ∈ 1, 2, . . . , n . Here

ζ(r) =∞∑k=1

k−r .


Another small detour - Young’s integral

When the integrand has finite p-variation, Vp the integrator has finiteq-variation, Vq, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define

(RYS)

∫[a;b]

f dg .

where (RYS)∫

denotes some refinement of the Riemann-Stieltjes integral(the refinement is made to avoid problems with discontinuities) and wheng is continuous, it coincides with the Riemann-Stieltjes integral.This is proved with the following Love-Young inequality:∣∣∣∣∣

n∑i=1

f (si ) g(ti )− g(ti−1) − f (si0) g(b)− g(a)

∣∣∣∣∣ ≤ ζ (p−1 + q−1),

for any partition π = a = t0 < t1 < ... < tn = b , si ∈ [ti−1; ti ] ,i , i0 ∈ 1, 2, . . . , n . Here

ζ(r) =∞∑k=1

k−r .


The Young integral is still insufficient

Since the ψ- variation of a Brownian motion is finite then any p- variationwith p > 2 of the Brownian motion is locally finite. (Just to make itprecise let us write a formula for p-variation:

Vp (B, [a; b]) := supn


n∑i=1

∣∣Bti − Bti−1

∣∣p .)

Thus, the Young integral may be used for the pathwise integration

(RYS)

∫[a;b]

XsdBs

whenever X is a stochastic process with locally finite q-variation, where1 ≤ q < 2.Unfortunately, this is still unsufficient for calcuation of the integral∫

[a;b] BsdBs .


The Young integral is still insufficient

Since the ψ- variation of a Brownian motion is finite then any p- variationwith p > 2 of the Brownian motion is locally finite. (Just to make itprecise let us write a formula for p-variation:

Vp (B, [a; b]) := supn


n∑i=1

∣∣Bti − Bti−1

∣∣p .) Thus, the Young integral may be used for the pathwise integration

(RYS)

∫[a;b]

XsdBs

whenever X is a stochastic process with locally finite q-variation, where1 ≤ q < 2.Unfortunately, this is still unsufficient for calcuation of the integral∫

[a;b] BsdBs .


Stochastic integral - the quadratic variation approach

For simplicity we will fix on integration with respect to the Brownianmotion B.We consider a family of stochastic processes X with caglad (leftcontinuous with right limits) paths [0; +∞) 3 t 7→ Xt ∈ R such that forevery process X ∈ X the following hold

1 for every u > t ≥ 0 the increment Bu − Bt is independent of thevalues of Xs , 0 ≤ s ≤ t;

2 ‖X‖H := E∫ +∞

0 X 2s d[B]s = E

∫ +∞0 X 2

s ds =∫ +∞

0 EX 2s ds < +∞.

It appears that H = (X , ‖ · ‖H) is a Hilbert space and the family of simpleprocesses of the form

K = K−1 · 10 ++∞∑i=0

Ki · 1(ti ,ti+1],

where 0 = t0 < t1 < t2 < . . . with ti ↑ +∞ as i ↑ +∞ and Bu − Bti ,u ≥ ti being independent from Ki is dense in H.


Stochastic integral - the quadratic variation approach cont.

Now, for any t > 0 and every simple process K we define∫ t

0KdB := K−1 · B0 +

n−1∑i=0

Ki ·(Bti+1 − Bti

)+ Kn · (Bt − Btn) , (3)

whenever tn ≤ t < tn+1.

By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate

E(∫ t

0KdB

)2

≤ E(∫ +∞

0KdB

)2

= ‖K‖2H. (4)

The equality (4) is called Ito’s isometry. Now, for any X ∈ X and ε > 0we find a simple process Kε such that ‖X − Kε‖H < ε and define∫ t

0XdB = lim

ε↓0

∫ t

0KεdB.

By (4) this limit exists and is unique.




0KdB := K−1 · B0 +

n−1∑i=0

Ki ·(Bti+1 − Bti

)+ Kn · (Bt − Btn) , (3)

whenever tn ≤ t < tn+1.By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate

E(∫ t

0KdB

)2

≤ E(∫ +∞

0KdB

)2

= ‖K‖2H. (4)


0XdB = lim

ε↓0

∫ t

0KεdB.

By (4) this limit exists and is unique.




0KdB := K−1 · B0 +

n−1∑i=0

Ki ·(Bti+1 − Bti

)+ Kn · (Bt − Btn) , (3)

whenever tn ≤ t < tn+1.By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate

E(∫ t

0KdB

)2

≤ E(∫ +∞

0KdB

)2

= ‖K‖2H. (4)


0XdB = lim

ε↓0

∫ t

0KεdB.

By (4) this limit exists and is unique.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29


The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.

A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered

E [Mt −Ms | Fs ] = 0.

Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.



The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered

E [Mt −Ms | Fs ] = 0.

Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.

A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.




E [Mt −Ms | Fs ] = 0.

Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.

A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.




E [Mt −Ms | Fs ] = 0.

Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.


Semimartingales

The family of semimartingales is rich enough to encompass majority ofprocesses used in stochastic modelling (except maybe fractional Brownianmotions).Moreover, the deep result of Bichteler and Dellacherie states that thisfamily is in some sense the broadest possible family of good integrators.Namely, when we define for a given integrator M and every simple processthe integral

∫ t0 KdM with the formula analogous to the formula (3), then

the transformation

K 7→∫ t

0KdM

is continuous if and only if M is a semimartingale.

To deal with continuity we need to define topologies: the space of simpleprocesses is endowed with uniform convergence in (t, ω) topology and thespace of integrals is topologized by convergence in probability.


Semimartingales

The family of semimartingales is rich enough to encompass majority ofprocesses used in stochastic modelling (except maybe fractional Brownianmotions).Moreover, the deep result of Bichteler and Dellacherie states that thisfamily is in some sense the broadest possible family of good integrators.Namely, when we define for a given integrator M and every simple processthe integral

∫ t0 KdM with the formula analogous to the formula (3), then

the transformation

K 7→∫ t

0KdM

is continuous if and only if M is a semimartingale.To deal with continuity we need to define topologies: the space of simpleprocesses is endowed with uniform convergence in (t, ω) topology and thespace of integrals is topologized by convergence in probability.


Few properties of a stochastic integral

The stochastic integral is now defined for any semimartingale M and any(predictable) caglad process X . For example

∫ t0 BtdBt = 1

2B2t − 1

2 t.We have important

Theorem (counterpart of the Lebesgue dominated convergence)

For any caglad (predictable) processes X 1, X 2, . . . , such that X i ≤ |X | fori = 1, 2, . . . , where X is some caglad (predictable) process andlimi↑+∞ X i → 0 pointwise then we have

sup0≤s≤T

∣∣∣∣∫ s

0X idM

∣∣∣∣→P 0.

For any p > 0 we also have important Burkholder-Davis-Gundy’s inequality

E sup0≤s≤T

∣∣∣∣∫ s

0XdM

∣∣∣∣p ≤ Cp · E∣∣∣∣∫ T

0X 2d[M]

∣∣∣∣p/2

.


Wong-Zakai’s pathwise approach to the stochastic integral

Since many years mathematicians tried to define the stochastic integral ina pathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:

(A) for all s ∈ [0;T ], Bns → Bs pointwise as n ↑ +∞, where Bn,

n = 1, 2, . . . , are continuous and have locally bounded variation;

(B) (A) and there exists such a bounded process Z that for all s ∈ [0;T ],|Bn

s | ≤ Zs ;

and stated the following approximation theorem.

Theorem (Wong-Zakai (1965))

Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ

∂x and let Bn satisfy

(B), then for the Lebesgue-Stieltjes integrals∫ T

0 ψ (t,Bnt )dBn

t , a.s.,

limn→∞

∫ T

0ψ (t,Bn

t )dBnt =

∫ T

0ψ (t,Bt)dBt +

1

2

∫ T

0

∂ψ

∂x(t,Bt)dt.


Wong-Zakai’s pathwise approach to the stochastic integral

Since many years mathematicians tried to define the stochastic integral ina pathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:

(A) for all s ∈ [0;T ], Bns → Bs pointwise as n ↑ +∞, where Bn,

n = 1, 2, . . . , are continuous and have locally bounded variation;

(B) (A) and there exists such a bounded process Z that for all s ∈ [0;T ],|Bn

s | ≤ Zs ;

and stated the following approximation theorem.

Theorem (Wong-Zakai (1965))

Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ

∂x and let Bn satisfy

(B), then for the Lebesgue-Stieltjes integrals∫ T

0 ψ (t,Bnt )dBn

t , a.s.,

limn→∞

∫ T

0ψ (t,Bn

t )dBnt =

∫ T

0ψ (t,Bt)dBt +

1

2

∫ T

0

∂ψ

∂x(t,Bt) dt.


Wong-Zakai’s pathwise approach to the stochasticintegral, cont.

The correction term in Wong-Zakai’s theorem,∫ T

0∂ψ∂x (t,Bt)dt, is simply

the quadratic covariation of the processes Bt and ψ (t,Bt) .For two semimartingales X and Y the quadratic covariation, [X ,Y ], isthe (unique) limit in the probability of the sums

n(k)∑i=1

(Xtki− Xtki−1

)·(Ytki− Ytki−1

),

where the sequence of partitions πk =a = tk0 < tk1 < ... < tkn(k) = b

is

such that mesh(πk) ↓ 0 as k ↑ +∞.

For two cadlag semimartingales X and Y the integral

(S)

∫ T

0Y−dX :=

∫ T

0Y−dX +

1

2[X ,Y ]

is called the Stratonovich integral.


Wong-Zakai’s pathwise approach to the stochasticintegral, cont.

The correction term in Wong-Zakai’s theorem,∫ T

0∂ψ∂x (t,Bt)dt, is simply

the quadratic covariation of the processes Bt and ψ (t,Bt) .For two semimartingales X and Y the quadratic covariation, [X ,Y ], isthe (unique) limit in the probability of the sums

n(k)∑i=1

(Xtki− Xtki−1

)·(Ytki− Ytki−1

),

where the sequence of partitions πk =a = tk0 < tk1 < ... < tkn(k) = b

is

such that mesh(πk) ↓ 0 as k ↑ +∞.For two cadlag semimartingales X and Y the integral

(S)

∫ T

0Y−dX :=

∫ T

0Y−dX +

1

2[X ,Y ]

is called the Stratonovich integral.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 20 / 29

Disadvantages of the Wong-Zakai construction

The Wong-Zakai construction works only for very limited family ofintegrators: Brownian motions and processes of the formXt =

∫ t0 µsds +

∫ t0 σsdBs and integrands must be functions of the

integrators.

The generalisation for any pair of semimartingale integrator andintegrand is impossible.

It is relatively easy to give an example of two sequences ofcontinuous, with locally finite variation, bounded (and adapted to thenatural Brownian filtration) processes Bn and Bn such that Bn ⇒ B,Bn ⇒ B uniformly but ∫ 1

0BndBn

diverges (Lochowski (2013)).


A modification of the Wong-Zakai construction

In Lochowski (2013) for any cadlag process X the following construction isconsidered. For any c > 0 we there exists a process X c such that

(i) X c has locally finite total variation;

(ii) X c has cadlag paths;

(iii) for every T ≥ 0 there exists such KT < +∞ that for every t ∈ [0;T ] ,

|Xt − X ct | ≤ KT c ;

(iv) for every T ≥ 0 there exists such LT < +∞ that for every t ∈ [0;T ] ,

|∆X ct | ≤ LT |∆Xt | ,

where ∆X ct = X c

t − X ct−, ∆Xt = Xt − Xt−;

(v) the process X c is adapted to the natural filtration of X .


A modification of the Wong-Zakai construction, cont.

Further, in Lochowski (2013) it was shown that if processes X and Y arecadlag semimartingales then the sequence of pathwise Lebesgue-Stieltjesintegrals ∫ T

0Y−dX

c →Pc↓0

∫ T

0Y−dX + [X ,Y ]contT .∫ T

0 Y−dX denotes here the (semimartingale) stochastic integral and[X ,Y ]cont denotes here the continuous part of [X ,Y ], i.e.

[X ,Y ]contT = [X ,Y ]T −∑

0<s≤T∆Xs∆Ys .

Moreover, when c(n) > 0 and∑+∞

n=1 c(n)2 < +∞ then the convergence of∫ T0 Y−dX

c(n) to∫ T

0 Y−dX + [X ,Y ]contT holds almost surely.


Drawbacks of the construction presented

Unfortunately, the construction presented does not work for any cagladintegrand Y .It is possible to construct a continuous, bounded (and adapted to thenatural Brownian filtration) process Y and a sequence Bc(n), n = 1, 2, . . . ,satisfying all conditions (i)-(v) for X = B such that the integral∫ 1

0Y dBc(n)

diverges (cf. Lochowski (2013)).


Bichteler’s construction

The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation

limn→∞

sup0≤s≤T

∣∣∣∣∣Y0X0 +∞∑i=1

Yτni−1∧s

(Xτni ∧s − Xτni−1∧s

)−∫ s

0Y−dX

∣∣∣∣∣ = 0 a.s.,

where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,

τn = inft > τni−1 :

∣∣∣Yt − Yτni−1

∣∣∣ ≥ 2−n.

Remark

In fact, given c(n) > 0,∑∞

n=1 c2 (n) < +∞, Bichteler’s construction works

for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that

τn0 = 0 and for i = 1, 2, . . . , τni = inft > τni−1 :

∣∣∣Yt − Yτni−1

∣∣∣ ≥ c (n).


Norvaisa’s integral for functions with finite quadraticvariation

In a long (171 pages!) paper, Norvaisa develops (among other results) atheory of an integral for deterministic functions with finite λ - quadraticvariation.

A function f : [a; b]→ R has finite λ - quadratic variation if the sums

n(k)∑i=1

(f (tki )− f (tki−1)

)2

converge for any nested partitions i.e. πk ⊂ πk+1 with tk0 = a, tkn(k) = b

and mesh(πk)→ 0 as k ↑ +∞.

Sample paths of a Brownian motion have this property with probability Pequal 1 - a result due to Paul Levy.


Norvaisa’s integral for functions with finite quadraticvariation, cont.

Further, he defines Left-Cauchy λ integral of g with respect to f ,(LC )

∫ ba gdf , as a limit of the sums

n(k)∑i=1

g(tki−1)(f (tki )− f (tki−1)

)and proves its existence for the integrands of the form g(t) = ψ(f (t)),where ψ is a C 1 function.

He manages to prove a version of Ito’s formula for this integral. But stillthis is far from the theory of an integral for any (locally bounded)deterministic integrands and finite quadratic variation integrators.

The reason for this might be the fact that the analogous construction forsemimartingales does not require only finiteness of the quadratic variationof the integrator but also a centering property of the increments of a localmartingale part.


Some references

Bichteler, K., (1981) Stochastic integration and Lp− theory ofsemimartingales. Ann. Probab., 9(1):49–89.

Karandikar, R. L., (1995) On pathwise stochastic integration. Stoch.Process. Appl., 57(1):11–18.

Lochowski, R., M., (2013) Pathwise stochatic integration with finitevariation processes uniformly approximating cadlag processes,submitted.

Norvaisa, R., (2008) Quadratic Variation, p-variation and Integrationwith Applications to Stock Price Modelling, arXiv.

Wong, E. and Zakai, M., (1965) On the convergence of ordinaryintegrals to stochastic integrals. Ann. Math. Statist., 36:1560–1564.


Thank you!


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