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Stochasticintegral

Introduction

Ito integral

References

Appendices Stochastic integral80-646-08

Stochastic Calculus I

Geneviève Gauthier

HEC Montréal

Stochasticintegral

IntroductionRiemann integral

Ito integral

References

Appendices

IntroductionThe Ito integral

The theories of stochastic integral and stochasticdi¤erential equations have initially been developed byKiyosi Ito around 1940 (one of the rst important paperswas published in 1942).

Stochasticintegral


Ito integral

References

Appendices

Riemann integral I

Let f : R ! R be a real-valued function. Typically, whenwe speak of the integral of a function f , we refer to theRiemann integral,

R ba f (t) dt, which calculates the area

under the curve t ! f (t) between the bounds a and b.

Stochasticintegral


Ito integral

References

Appendices

Riemann integral II

First, we must realize that not all functions are Riemannintegrable.

Lets explain a little further : in order to construct theRiemann integral of the function f on the interval [a, b] ,we divide such an interval into n subintervals of equallength

ban

and, for each of them, we determine the

greatest and the smallest value taken by the function f .

Stochasticintegral


Ito integral

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Appendices

Riemann integral IIIThus, for every k 2 f1, ..., ng , the smallest value taken bythe function f on the interval

a+ (k 1) ban , a+ k

ban

is

f (n)k inff (t)

a+ (k 1) b an t a+ k b an

and the greatest is

_f(n)

k supf (t)

a+ (k 1) b an t a+ k b an

.

Stochasticintegral


Ito integral

References

Appendices

Riemann integral IV

The area under the curve t ! f (t) between the boundsa+ (k 1) ban and a+ k ban is therefore greater than

the area ban f

(n)k of the rectangle of base ba

n and height

f (n)k , and smaller than the area ban

_f(n)

k of the rectangle

with the same base but with height_f(n)

k .

As a consequence, if we dene

I n (f ) n

∑k=1

b an

f (n)k and_I n (f )

n

∑k=1

b an

_f(n)

k

then

I n (f ) Z b

af (t) dt

_I n (f ) .

Stochasticintegral


Ito integral

References

Appendices

Riemann integral V

Illustrations of I n (f ) and_I n (f )

Stochasticintegral


Ito integral

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Appendices

Riemann integral VI

Riemann integrable functions are those for which the limitslimn!∞ I n (f ) and limn!∞

_I n (f ) exist and are equal.

We then dene the Riemann integral of the function f asZ b

af (t) dt lim

n!∞I n (f ) = lim

n!∞

_I n (f ) .

It is possible to show that the integralR ba f (t) dt exists

for any function f continuous on the interval [a, b] . Manyother functions are Riemann integrable.

Stochasticintegral


Ito integral

References

Appendices

Riemann integral VII

However, there exist functions that are not Riemannintegrable.

For example, lets consider the indicator functionIQ : [0, 1]! f0, 1g which takes the value of 1 if theargument is rational and 0 otherwise.

Then, for all natural number n, I n = 0 and_I n = 1, which

implies that the Riemann integral is not dened for such afunction.

Stochasticintegral


Ito integral

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Appendices

Riemann integral VIII

We will examine a very simple case : the Riemann integralfor boxcar functions, i.e. the functions f : [0;∞)! R

which admit the representation f (t) = cI(a,b] (t) , a < b,c 2 R where I(a,b] represents the indicator function

I(a,b] (t) =1 if t 2 (a, b]0 if t /2 (a, b] .

Stochasticintegral


Ito integral

References

Appendices

Riemann integral IX

If f (t) = cI(a,b] (t) then the Riemann integral of f isZ t

0f (s) ds =

Z t

0cI(a,b] (s) ds

=

8<:0 if t a

c (t a) if a < t bc (b a) if t > b.

Stochasticintegral

Introduction

Ito integralBasic processMomentsSimple processPredictableprocessIn summaryGeneralization

References

Appendices

Ito integral I

Let fWt : t 0g be a standard Brownian motion built ona ltered probability space (Ω,F ,F,P) ,F = fFt : t 0g.Technical condition. Since we will work with almost sureequalities 1, we require that the set of events which havezero probability to occur are included in the sigma-algebraF0, i.e. that the set

N = fA 2 F : P (A) = 0g F0.

That way, if X is Ftmeasurable and Y = X Palmostsurely, then we know that Y is Ftmeasurable.

Stochasticintegral

Introduction


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Appendices

Ito integral IILet fXt : t 0g be a predictable stochastic process. It istempting to dene the stochastic integral of X withrespect to W , pathwise, by using a generalization of theStieltjes integral:

8ω 2 Ω,ZX (ω) dW (ω) .

That would be possible if the paths of Brownian motionW had bounded variation, i.e. if they could be expressedas the di¤erence of two non-decreasing functions.But, since Brownian motion paths do not have boundedvariation, it is not possible to use such an approach.We must dene the stochastic integral with respect to theBrownian motion as a whole, i.e. as a stochastic processin itself, and not pathwise.

1X = Y Palmost surely if the set of ω for which X is di¤erent fromY has a probability of zero, i.e.

P fω 2 Ω : X (ω) 6= Y (ω)g = 0.

Stochasticintegral

Introduction


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Basic process IIto integral

DenitionWe call X a basic stochastic process if X admits thefollowing representation:

Xt (ω) = C (ω) I(a,b] (t)

where a < b 2 R and C is a random variable, Fameasurableand square-integrable, i.e. EP

C 2< ∞.

Note that such a process is adapted to the ltration F. Indeed,

Xt =

8<: 0 if 0 t aC if a < t b0 if b < t

which is F0 measurable therefore Ft measurablewhich is Fa measurable therefore Ft measurablewhich is F0 measurable therefore Ft measurable.

Stochasticintegral

Introduction


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Basic process IIIto integral

In fact, X is more than Fadapted, it is Fpredictable,but the notion of predictable process is more delicate todene when we work with continuous-time processes.

Note, nevertheless, that adapted processes withcontinuous path are predictable.

Intuitively, if Xt represents the number of security sharesheld at time t, then Xt (ω) = C (ω) I(a,b] (t) means thatimmediately after prices are announced at time a andbased on the information available at time a (C beingFameasurable) we buy C (ω) security shares which wehold until time b. At that time, we sell them all.

Stochasticintegral

Introduction


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Appendices

Basic process IIIIto integral

TheoremFadapted processes, the paths of which are left-continuous,are Fpredictable processes. In particular, Fadaptedprocesses with continuous paths are Fpredictible. (cf. Revuzand Yor)

Stochasticintegral

Introduction


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Appendices

Basic process IVIto integral

DenitionThe stochastic integral of X with respect to the Brownianmotion is dened asZ t

0XsdWs

(ω)

= C (ω) (Wt^b (ω)Wt^a (ω))

=

8<:0 if 0 t a

C (ω) (Wt (ω)Wa (ω)) if a < t bC (ω) (Wb (ω)Wa (ω)) if b < t.

Note that, for all t, the integralR t0 Xs dWs is a random

variable.

Moreover,nR t

0 Xs dWs : t 0ois a stochastic process.

Stochasticintegral

Introduction


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Basic process VIto integral

Paths of a basic stochastic process, and its stochastic integralwith respect to the Brownian motion

0123456789

10

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1t

543210123456

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

t

Integrale_stochastique.xls

Stochasticintegral

Introduction


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Basic process VIIto integral

Note that Z t

0XsdWs =

Z ∞

0XsI(0,t ] (s) dWs .

Exercise. Verify it!

Stochasticintegral

Introduction


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Appendices

Basic process VIIIto integral

Interpretation. If, for example, the Brownian motion Wrepresents the variation of the share present value relativeto its initial value (Wt = St S0, where St is the sharepresent value at time t), then

R t0 XsdWs is the present

value of the prot earned by the investor.

This example is somewhat lame: since the Brownianmotion is not lower bounded, the example implies that it ispossible for the share price to take negative values. Oops!

Note on the graphs that, over the time period when theinvestor holds a constant number of security shares, thepresent value of his or her prot uctuates. That is onlycaused by share price uctuations.

Stochasticintegral

Introduction


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Appendices

Basic process VIIIIto integral

TheoremLemma 1. If X is a basic stochastic process, thenZ t

0XsdWs : t 0

is a (Ω,F ,F,P)martingale.

Proof of Lemma 1. Recall that, if M is a martingale and τ astopping time, then the stopped stochastic processMτ = fMt^τ : t 0g is also a martingale.Since the Brownian motion is a martingale, and τa and τbwhere 8ω 2 Ω, τa (ω) = a and τb (ω) = b are stoppingtimes, then the stochastic processes W τa = fWt^a : t 0gand W τb = fWt^b : t 0g are both martingales.

Stochasticintegral

Introduction


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Appendices

Basic process IXIto integral

Now, lets verify condition (M1).

If t a, then

EP

Z t

0XudWu

= EP [jC (Wt^b Wt^a)j]

= EP [jC (Wt Wt )j]= 0.

Stochasticintegral

Introduction


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Appendices

Basic process XIto integral

By contrast, if t > a, then

EP

Z t

0XudWu

= EP [jC (Wt^b Wa)j] = EP [jC j jWt^b Waj]

r

EPhC 2 (Wt^b Wa)

2i(see next page)

=

rEPhC 2EP

h(Wt^b Wa)

2 jFaiiC being Fa mble.

=

rEPhC 2EP

h(Wt^b Wa)

2ii

Wt^b Wa being independent from Fa.

=q(t ^ b a)EP [C 2] < ∞

Stochasticintegral

Introduction


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Appendices

Basic process XIIto integral

Justication for the inequality: for any random variable Ysuch that E

Y 2< ∞, we have

0 Var [Y ] = EY 2 (E [Y ])2

which implies that

E [Y ] q

E [Y 2].

Stochasticintegral

Introduction


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Basic process XIIIto integral

With respect to condition (M2),

Z t

0XsdWs

=

8<: 0 if 0 t aC (Wt Wa) if a < t bC (Wb Wa) if b < t .

F0 measurable therefore Ft measurable,Ft measurable,Fb measurable therefore Ft measurable.

Stochasticintegral

Introduction


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Appendices

Basic process XIIIIto integral

Condition (M3) is also veried since 8s, t 2 R, 0 s < t,

EP

Z t

0XudWu jFs

= EP [C (Wt^b Wt^a) jFs ] .

Stochasticintegral

Introduction


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Basic process XIVIto integral

But, if s a, then Fs Fa. Since C is Fa measurable, then

EP

Z t

0XudWu jFs

= EP

hEP [C (Wt^b Wtâ) jFa ] jFs

i= EP

hCEP [(Wt^b Wtâ) jFa ] jFs

i= EP

264C (Wa^b Waâ)| z =WaWa=0

jFs

375since W τa and W τb are martingales,

= 0

=Z s

0XudWu .

Stochasticintegral

Introduction


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Appendices

Basic process XVIto integral

If s > a, then C is Fs measurable and, as a consequence,

EP

Z t

0XudWu jFs

= EP [C (Wt^b Wtâ) jFs ]

= CEP [(Wt^b Wtâ) jFs ]= C (Ws^b Wsâ)

=Z s

0XudWu .

The processRXsdWs is indeed a martingale.

Stochasticintegral

Introduction


References

Appendices

Moments IBasic process

We will need a few results later on, that we shall now startto develop.

Since the Ito integral for basic processes is a martingale,then

EP

Z t

0XsdWs

= EP

Z 0

0XsdWs

= 0.

The next result will allow us to calculate variances andcovariances.

Stochasticintegral

Introduction


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Appendices

Moments IIBasic process

TheoremLemma 2. If X and Y are basic processes, then for all t 0,

EP

Z t

0XsYsds

= EP

Z t

0XsdWs

Z t

0YsdWs

.

Stochasticintegral

Introduction


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Appendices

Moments IIIBasic process

Proof of Lemma 2. It will be su¢ cient to show that

EP

Z ∞

0XsYsds

= EP

Z ∞

0XsdWs

Z ∞

0YsdWs

(1)

sinceX = CI(a,b] and Y = eCI(ea,eb]

being basic processes,

XI(0,t ] = CI(a^t ,b^t ],

Y I(0,t ] = eCI(ea^t ,eb^t]et XY I(0,t ] = C eCI((a_ea)^t ,(b^eb)^t]

are also basic processes.

Stochasticintegral

Introduction


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Appendices

Moments IVBasic process

Thus, if equation (1) is veried, we can use it to establish thethird equality in the following calculation:

EP

Z t

0XsYsds

= EP

Z ∞

0XsYsI(0,t ] (s) ds

= EP

Z ∞

0

XsI(0,t ] (s)

YsI(0,t ] (s)

ds

= EP

Z ∞

0XsI(0,t ] (s) dWs

Z ∞

0YsI(0,t ] (s) dWs

= EP

Z t

0XsdWs

Z t

0YsdWs

.

Stochasticintegral

Introduction


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Appendices

Moments VBasic process

So, lets establish Equation (1)

EP

Z ∞

0XsYsds

= EP

Z ∞

0XsdWs

Z ∞

0YsdWs

.

Three cases must be treated separately:

(i) (a, b] \ea,ebi = ?,

(ii) (a, b] =ea,ebi ,

(iii) (a, b] 6=ea,ebi and (a, b] \ ea,ebi 6= ?.

Stochasticintegral

Introduction


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Moments VIBasic process

Proof of (i).

Since (a, b] \ea,ebi = ?, we can assume, without loss of

generality, that a < b ea < eb. SinceXsYs = C eCI(a,b] (s) I(ea,eb] (s) = 0,

then EPR ∞0 XsYsds

= 0.

Stochasticintegral

Introduction


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Appendices

Moments VIIBasic process

Now, since C , eC and (Wb Wa) are Feameasurable,EP

Z ∞

0XsdWs

Z ∞

0YsdWs

= EP

hC (Wb Wa) eC Web Weai

= EPhEPhC (Wb Wa) eC Web Wea jFea ii

= EP

264C eC (Wb Wa) EPWeb Wea jFea | z =WeaWea=0

375= 0 = EP

Z ∞

0XsYsds

thus establishing Equation (1) in this particular case.

Stochasticintegral

Introduction


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Appendices

Moments VIIIBasic process

Proof of (ii). Lets now assume that (a, b] =ea,ebi.

Then,

EP

Z ∞

0XsYsds

= EP

Z ∞

0C eCI(a,b]dt

= EP

C eC Z ∞

0I(a,b]dt

= EP

hC eC (b a)i

= (b a)EPhC eCi .

Stochasticintegral

Introduction


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Appendices

Moments IXBasic process

On the other hand,

EP

Z ∞

0XsdWs

Z ∞

0YsdWs

= EP

hC (Wb Wa) eC (Wb Wa)

i= EP

hC eCEP

h(Wb Wa)

2 jFaiisince C and eC are Fa mbles

= EPhC eCEP

h(Wb Wa)

2ii

since Wb Wa is independent of Fa= EP

hC eC (b a)i since Wb Wa is N (0, b a)

= (b a)EPhC eC i = EP

Z ∞

0XsYsds

thus establishing Equation (1) for this other particular case.

Stochasticintegral

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Appendices

Moments XBasic process

Proof of (iii). We have that (a, b] 6=ea,ebi and

(a, b] \ea,ebi 6= ?.

a = a _ ea and b = b ^ eb. On the one hand,EP

Z ∞

0XsYsds

= EP

Z ∞

0CI(a,b]

eCI(ea,eb]dt

= EP

Z ∞

0C eCI(a,b]dt

= EP

C eC Z ∞

0I(a,b]dt

= EP

hC eC (b a)i

= (b a)EPhC eCi .

Stochasticintegral

Introduction


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Appendices

Moments XIBasic process

In what follows, in case there were intervals of the form (α, β]where α β, then we dene (α, β] = ?. On the other hand,since Z ∞

0XsdWs

= C (Wb Wa)

= C (Wb Wb +Wb Wa +Wa Wa)

= C (Wb Wb ) + C (Wb Wa ) + C (Wa Wa)

=Z ∞

0CI(b,b]dWs +

Z ∞

0CI(a,b ]dWs +

Z ∞

0CI(a,a ]dWs , (2)

Stochasticintegral

Introduction


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Appendices

Moments XIIBasic process

Z ∞

0YsdWs

= eC Web Wea= eC Web Wb +Wb Wa +Wa Wea= eC Web Wb

+ eC (Wb Wa ) + eC (Wa Wea)

=Z ∞

0eCI(b,eb]dWs +

Z ∞

0eCI(a,b ]dWs +

Z ∞

0eCI(ea,a ]dWs (3)

Stochasticintegral

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Appendices

Moments XIIIBasic process

and

(a, a] \ (ea, a] = ?, (a, a] \ (a, b] = ?, (a, a] \b, ebi = ?,

(a, b] \ (ea, a] = ?, (a, b] \b, ebi = ?,

(b, b] \ (ea, a] = ?, (b, b] \ (a, b] = ?, (b, b] \b, ebi = ?,

we can use the results obtained in points (i) and (ii) in orderto complete the proof: using the expressions from lines (2) and

Stochasticintegral

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Moments XIVBasic process

(3),

EP

Z ∞

0XsdWs

Z ∞

0YsdWs

= EP

Z ∞

0

CI(a,b]dWs

Z ∞

0eCI(a,b]dWs

= (b a)EP

hC eCi

= EP

Z ∞

0XsYsds

.

The proof of Lemma 2 is now complete.

Stochasticintegral

Introduction


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Appendices

Simple process IIto integral

DenitionWe call X a simple stochastic process if X is a nite sum ofbasic processes :

Xt (ω) =n

∑i=1Ci (ω) I(ai ,bi ] (t) .

Since such a process is a sum of Fpredictable processes,then it is itself predictable with respect to the ltration F.

Stochasticintegral

Introduction


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Appendices

Simple process IIIto integral

DenitionThe stochastic integral of X with respect to the Brownianmotion is dened as the sum of the stochastic integrals of thebasic processes which constitute X :Z t

0XsdWs =

n

∑i=1

Z t

0CiI(ai ,bi ] (s) dWs .

Stochasticintegral

Introduction


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Appendices

Simple process IIIIto integral

Paths of a simple stochastic process and its stochastic integralwith respect to the Brownian motion

4202468

1012

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

t

64202468

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

t

The simple process is of the form C1I( 110 , 610 ]+ C2I( 12 , 34 ]

Stochasticintegral

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Simple process IVIto integral

Integrale_stochastique.xls

Stochasticintegral

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Simple process VIto integral

First, we must ensure that such a denition contains noinconsistencies, i.e. if the simple stochastic process Xadmits at least two representations in terms of basicstochastic processes, say ∑n

i=1 CiI(ai ,bi ] and ∑mj=1

eCjI(eaj ,ebj ],then the integral is dened in a unique manner:

n

∑i=1

Z t

0CiI(ai ,bi ] (s) dWs =

m

∑j=1

Z t

0eCjI(eaj ,ebj ] (s) dWs .

Exercise. Prove that the denition of the stochasticintegral for simple processes does not depend on therepresentation chosen.

Stochasticintegral

Introduction


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Appendices

Simple process VIIto integral

TheoremLemma 3. If X is a simple stochastic process, thennR t

0 XsdWs : t 0ois a (Ω,F ,F,P)martingale.

Proof of Lemma 3. The stochastic integralZ t

0XsdWs =

n

∑i=1

Z t

0CiI(ai ,bi ] (s) dWs

of the simple process X = ∑ni=1 CiI(ai ,bi ] is the sum of the

stochastic integrals of the basic processes it is made up of.Since a nite sum of martingales is also a martingale, theresults comes from the fact that stochastic integrals of basicprocesses are martingales (Lemma 1).

Stochasticintegral

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Simple process VIIIto integral

The next result is rather technical and the proof can be foundin the Appendix. It will useful to us later on.

TheoremLemma 4. If X is a simple process, then for all t 0,

EP

Z t

0X 2s ds

= EP

"Z t

0XsdWs

2#.

Stochasticintegral

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Appendices

Simple process VIIIIto integral

That lemma is, besides, very useful to calculate the variance ofa stochastic integral. Indeed,

VarP

Z t

0XsdWs

= EP

"Z t

0XsdWs

2#

0BB@EP

Z t

0XsdWs

| z

=0

1CCA2

= EP

Z t

0X 2s ds

=Z t

0EPX 2sds.

It is possible to extend this calculation to other processes Xand toestablish in a similar manner a method to calculate thecovariance between two stochastic integrals (see the Appendix).

Stochasticintegral

Introduction


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Appendices

Predictable processes IIto integral

We would like to extend the class of processes for whichthe stochastic integral with respect to the Brownianmotion can be dened. We will choose the class ofFpredictable processes for which there exists a sequenceof simple processes approaching them.

TheoremFadapted processes, the paths of which are left-continuousare Fpredictable processes. In particular, Fadaptedprocesses with continuous paths are Fpredictible.

Let X be an Fpredictable process for which there existsa sequence

nX (n) : n 2 N

oof simple processes

converging to X when n increases to innity.

Stochasticintegral

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Predictable processes IIIto integral

You cant speak of convergence without speaking ofdistance. Indeed, how does one measure whether asequence of stochastic processes approaches some otherprocess? What is the distance between two stochasticprocesses? To answer such a question, we must dene thespace of stochastic processes on which we work as well asthe norm we put on that space.

Recall that kkA : A ! [0,∞) is a norm on the space A if

(i) kX kA = 0, X = 0;(ii) 8X 2 A and 8a 2 R, kaX kA = jaj kX kA ;(iii)

8X ,Y 2 A, kX + Y kA kX kA + kY kA(triangle inequality).

Stochasticintegral

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Predictable processes IIIIto integral

Let

A =X

X is aF predictable processsuch that EP

R ∞0 X

2t dt< ∞

.

Exercise, optional and rather di¢ cult! The functionkkA : A ! [0,∞) dened as

kXkA =s

EP

Z ∞

0X 2t dt

is a norm on the space A.

Stochasticintegral

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Predictable processes IVIto integral

It is said that the sequencenX (n) : n 2 N

o A

converges to X 2 A when n increases to innity if andonly if

limn!∞

X (n) X A

= limn!∞

sEP

Z ∞

0

X (n)t Xt

2dt

= 0.

Stochasticintegral

Introduction


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Appendices

Predictable processes VIto integral

It is tempting to dene the stochastic integral of X withrespect to the Brownian motion as the limit of thestochastic integrals of the simple processes, i.e.Z t

0XsdWs = lim

n!∞

Z t

0X (n)s dWs .

Stochasticintegral

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Predictable processes VIIto integral

The latter equation raises two issues, one of which beingwhether such a limit exists.

For example, lets assume we work on the space of strictlypositive numbers A fx 2 R jx > 0g and we study thesequence

1n : n 2 N

. That sequence does not converge

in the space A since limn!∞1n = 0 /2 A. Of course, that

sequence converges in space R.

The question is: does limn!∞R t0 X

(n)s dWs exist in the

space in which we work?The second issue is that this equation implies that, thegreater n , the closer

R t0 X

(n)s dWs is to

R t0 XsdWs . But,

what is the distance between two stochastic integrals?

Stochasticintegral

Introduction


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Predictable processes VIIIto integral

Recall that, in Lemma 3, we have established that, for alln, the integral

RX (n)s dWs of a simple process is a

martingale.

Moreover, it is possible to establish that

E

"ZX (n)s dWs

2#< ∞.

Stochasticintegral

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Predictable processes VIIIIto integral

As a consequence, the sequence used to approachRXs

dWs is contained in the space

M =

(M

M is a martingale

such thatqsupt0 EP [M2

t ] < ∞

).

Exercise, optional and laborious! The functionkkM : M! [0,∞) dened as

kMkM =rsupt0

EP [M2t ]

is a norm onM.

Stochasticintegral

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Predictable processes IXIto integral

Thus, we say that the sequencenM (n) : n 2 N

oof

martingales belonging to the normed space (M, kkM)converges to M if and only if M 2 M and

limn!∞

M (n) M M

= limn!∞

ssupt0

EP

M (n)t Mt

2= 0.

Stochasticintegral

Introduction


References

Appendices

Predictable processes XIto integral

The idea behind the construction ofR t0 Xs dWs is that, on

the one hand, we have the space (A, kkA) of theprocesses for which we construct the stochastic integraland, on the other hand, we have the space (M, kkM)containing the stochastic integrals of the processes in therst space.

But, we have chosen the norms in such a way that, if theprocess X is a simple process, thenkXkA =

R Xs dWs M (Ref.: Lemma 5 in the

Appendix).

Stochasticintegral

Introduction


References

Appendices

Predictable processes XIIto integral

This implies that, if the sequence of simple processesnX (n) : n 2 N

ois a Cauchy sequence in the rst space,

i.e. X (n) X (m) Am,n!∞! 0,

then the sequencenR t

0 X(n)s dWs : n 2 N

oof stochastic

integrals is a Cauchy sequence in the second space: Z t

0X (n)s dWs

ZX (m)s dWs

M

m,n!∞! 0.

The sequencenM(n) : n 2 N

ois a Cauchy sequence on

the normed space (M, kkM) if

limn,m!∞

M(n) M(m) M= 0.

Stochasticintegral

Introduction


References

Appendices

Predictable processes XIIIto integral

IfnM(n) : n 2 N

ois a Cauchy sequence, we cannot, in

general, state that such a sequence converges, since we arenot certain that the limit-point belongs to the setM.

Stochasticintegral

Introduction


References

Appendices

Predictable processes XIIIIto integral

The only thing left to verify is that the limit-point of thesequence of stochastic integrals

nR t0 X

(n)s dWs : n 2 N

ois

indeed an element ofM (ref.: Lemma 6 in the Appendix).

We will then be able to dene the stochastic integral ofthe predictable process X = limn!∞ X (n) aslimn!∞

R t0 X

(n)s dWs .

Stochasticintegral

Introduction


References

Appendices

In summaryPredictible process

We have dened the stochastic integral with respect tothe (Ω,F ,F,P)Brownian motion for all XFpredictable processes satisfying the condition

EP

Z ∞

0X 2t dt

< ∞.

For each of these processes, we have established that thefamily of stochastic integrals

nR t0 XsdWs : t 0

ois a

(Ω,F ,F,P)martingale.

Stochasticintegral

Introduction


References

Appendices

Generalization IPredictible process

It is possible to extend the class of processes for which thestochastic integral can be dened.

It involves local martingales as well as semi-martingales.

We refer to the book by Richard Durrett those who wouldlike to know more.

Lets however mention that it is possible, for suchprocesses, that the family of stochastic integralsnR t

0 XsdWs : t 0ois no longer a martingale.

Stochasticintegral

Introduction


References

Appendices

Generalization IIPredictible process

Here is a result allowing us to verify whether the stochasticintegral with which we work is a martingale :

TheoremLemma. If X is a Fpredictible process such thatEPhR t0 X

2s dsi< ∞ then

R s0 XsdWs : 0 s t

is a

(Ω,F ,F,P)martingale (ref. Revuz and Yor, Corollary 1.25,page 124).

Stochasticintegral

Introduction


References

Appendices

Generalization IIIPredictible process

It is possible to dene the stochastic integral with respectto other stochastic processes than the Brownian motion byusing the same construction as the one we have justprovided.

Stochasticintegral

Introduction

Ito integral

References

Appendices

References I

DURRETT, Richard (1996). Stochastic Calculus, APractical Introduction, CRC Press, New York.

KARATZAS, Ioannis and SHREVE, Steven E. (1988).Brownian Motion and Stochastic Calculus,Springer-Verlag, New York.

KARLIN, Samuel and TAYLOR, Howard M. (1975). AFirst Course in Stochastic Processes, second edition,Academic Press, New York.

LAMBERTON, Damien and LAPEYRE, Bernard (1991).Introduction au calcul stochastique appliqué à la nance,Éllipses, Paris.

REVUZ, Daniel and YOR, Marc (1991). ContinuousMartingale and Brownian Motion, Springer-Verlag, NewYork.

Stochasticintegral

Introduction

Ito integral

References

AppendicesLemma 4Lemma 5Lemma 6Comments

Lemma 4 I

Here is a second result similar to Lemma 2, but for simpleprocesses.

TheoremLemma 4. If X is a simple process, then for all t 0,

EP

Z t

0X 2s ds

= EP

"Z t

0Xs dWs

2#.

Stochasticintegral

Introduction

Ito integral

References


Lemma 4 IIProof of Lemma 4. Let X = ∑n

i=1 CiI(ai ,bi ] be a simpleprocess.

EP

"Z t

0XsdWs

2#

= EP

24 n

∑i=1

Z t


!235=

n

∑i=1

n

∑j=1

EP

Z t


Z t

0CjI(aj ,bj ] (s) dWs

=n

∑i=1

n

∑j=1

EP

Z t

0CiI(ai ,bi ] (s) CjI(aj ,bj ] (s) ds

by Lemma 2.

= EP

"Z t

0

n

∑i=1

n

∑j=1

CiI(ai ,bi ] (s) CjI(aj ,bj ] (s) ds

#

= EP

24Z t

0

n

∑i=1

CiI(ai ,bi ] (s)

!2ds

35 = EP

Z t

0X 2s ds

.

Stochasticintegral

Introduction

Ito integral

References


Lemma 5 I

Theorem

Lemma 5. For any simple process Y , kY kA = R YsdWs

M .

Proof of Lemma 5. Note that 8ω 2 Ω, the functiont !

R t0 Y

2s (ω) ds as well as the function t !EP

hR t0 Y

2s ds

i

Stochasticintegral

Introduction

Ito integral

References


Lemma 5 II

are non-decreasing. As a consequence,

kY k2A = EP

Z ∞

0Y 2s ds

= EP

limt!∞

Z t

0Y 2s ds

= lim

t!∞EP

Z t

0Y 2s ds

by monotone convergence theorem.

= supt0

EP

Z t

0Y 2s ds

= sup

t0EP

"Z t

0YsdWs

2#by Lemma 4.

=

Z YsdWs

2Mby the very denition of kkM .

Stochasticintegral

Introduction

Ito integral

References


Lemma 5 III

From the latter result, it follows that the sequencenRX (n)s dWs : n 2 N

oof stochastic integrals of simple

processes with respect to the Brownian motion is a Cauchysequence since

limn,m!∞

Z X (n)s dWs ZX (m)s dWs

M

= limn,m!∞

X (n) X (m) A

= 0

where the rst equality is a direct consequence from Lemma 5,X (n) and X (m) being simple stochastic processes, and thesecond equality comes from the fact that the sequencenX (n) : n 2 N

ois a Cauchy sequence since it converges to X .

Stochasticintegral

Introduction

Ito integral

References


Lemma 6 I

TheoremLemma 6. The normed space (M, kkM) is completea(Durrett, 1996, Theorem 4.6, page 57).

aA normed space is said to be complete if any Cauchy sequenceconverges, i.e. the limit-point of any Cauchy sequence belongs to thespace.

SincenR

X (n)s dWs : n 2 Nois a Cauchy sequence on the

complete space (M, kkM) then such a sequence converges,thus establishing the existence of what we have calledRXsdWs , and its limit-point belongs to (M, kkM), which

implies that the stochastic integral of X with respect to W ,RXsdWs , is a martingale, even for some adapted processes

which are not simple processes.

Stochasticintegral

Introduction

Ito integral

References


Comments I

It is possible to prove that the denition of the stochasticintegral, for the predictable stochastic processes X 2 Awhich own a sequence of simple processes approachingthem, does not depend on the sequence chosen, i.e. ifnX (n) : n 2 N

oand

neX (n) : n 2 Noare such two

sequences of simple processes, thatlimn!∞

X (n) X A= 0 and limn!∞

eX (n) X A= 0,

thenlimn!∞

ZX (n)s dWs = lim

n!∞

Z eX (n)dWs .

Stochasticintegral

Introduction

Ito integral

References


Comments II

Last comment: for any predictable stochastic processX 2 (A, kkA), there exists such a sequencenX (n) : n 2 N

oof simple processes that

limn!∞

X (n) X A= 0.

(Durrett, 1996, Lemma 4.5, page 57)

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