Stochastic integral

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integral

Introduction Ito integral References

Appendices

Stochastic integral

80-646-08 Stochastic Calculus I

Geneviève Gauthier

HEC Montréal

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integral

Introduction Riemann integral Ito integral References Appendices

Introduction

The Ito integral

The theories of stochastic integral and stochastic

di¤erential equations have initially been developed by

Kiyosi Ito around 1940 (one of the …rst important papers

was published in 1942).

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integral

Riemann integral I

Let f

: R

!

^R

be a real-valued function. Typically, when we speak of the integral of a function f , we refer to the Riemann integral, R

b

a

f ( t ) dt, which calculates the area

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

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integral

Riemann integral II

First, we must realize that not all functions are Riemann integrable.

Let’s explain a little further : in order to construct the

Riemann integral of the function f on the interval [ a, b ]

,

we divide such an interval into n subintervals of equal

length

^{b a}_n

and, for each of them, we determine the

greatest and the smallest value taken by the function f .

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integral

Riemann integral III

Thus, for every k 2 f ^{1, ...,} ⁿ g

^,

the smallest value taken by the function f on the interval a + ( k 1 )

^{b a}_n ,

a + k

^{b a}_n

is

f

⁽_kⁿ⁾

inf f ( t ) a + ( k 1 ) ^b ^a

n t a + k b a n and the greatest is

_

f

⁽_kⁿ⁾

sup f ( t ) a + ( k 1 ) ^b ^a

n t a + k b a

n

.

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integral

Riemann integral IV

The area under the curve t ! ^f ( t ) between the bounds a + ( k 1 )

^{b a}_n

and a + k

^{b a}_n

is therefore greater than the area

^{b a}_n

f

⁽_kⁿ⁾

of the rectangle of base

^{b a}_n

and height f

⁽_kⁿ⁾

, and smaller than the area

^{b a}_n

_

f

_k⁽ⁿ⁾

of the rectangle with the same base but with height

_

f

⁽_kⁿ⁾

. As a consequence, if we de…ne

I

_n

( f )

∑

n k=1

b a

n f

⁽_kⁿ⁾

and

_

I

n

( f )

∑

n k=1

b a n

_

f

⁽_kⁿ⁾

then

I

_n

( f )

Z b

a

f ( t ) dt

_

I

_n

( f )

.

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integral

Riemann integral V

Illustrations of I

_n

( f ) and

_

I

_n

( f )

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integral

Riemann integral VI

Riemann integrable functions are those for which the limits lim

n!∞

I

_n

( f ) and lim

n!∞

_

I

n

( f ) exist and are equal.

We then de…ne the Riemann integral of the function f as

Z b

a

f ( t ) dt lim

n!∞

I

_n

( f ) = lim

n!∞ _

I

n

( f )

.

It is possible to show that the integral R

b

a

f ( t ) dt exists

for any function f continuous on the interval [ a, b ]

.

Many

other functions are Riemann integrable.

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integral

Riemann integral VII

However, there exist functions that are not Riemann integrable.

For example, let’s consider the indicator function

I_Q:

[ 0, 1 ] ! f ^0, ¹ g which takes the value of 1 if the argument 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 de…ned for such a

function.

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integral

Riemann integral VIII

We will examine a very simple case : the Riemann integral for boxcar functions, i.e. the functions f

:

[ 0;

∞

) !

R

which admit the representation f ( t ) = c

I(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 ]

^.

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integral

Riemann integral IX

If f ( t ) = c

I(a,b]

( t ) then the Riemann integral of f is

Z t

0

f ( s ) ds =

Z t

0

c

I(a,b]

( s ) ds

=

8<

:

0 if t a

c ( t a ) if a < t b

c ( b a ) if t > b.

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integral

Introduction Ito integral

Basic process Moments Simple process Predictable process

In summary Generalization

References Appendices

Ito integral I

Let f ^W

^t ^:

^t ⁰ g be a standard Brownian motion built on a …ltered probability space (

_Ω,

F

^,F,P

)

,

F

= fF

^t ^:

^t ⁰ g ^.

Technical condition . Since we will work with almost sure equalities

¹

, we require that the set of events which have zero probability to occur are included in the sigma-algebra F

⁰

, i.e. that the set

N = f ^A 2 F

^:^P

( A ) = 0 g F

⁰^.

That way, if X is F

^t

measurable and Y = X

P

almost

surely, then we know that Y is F

^t

measurable.

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integral

Ito integral II

Let f ^X

^t ^:

^t ⁰ g be a predictable stochastic process. It is tempting to de…ne the stochastic integral of X with respect to W , pathwise, by using a generalization of the Stieltjes integral:

8

^ω

2

^Ω^,

Z

X (

ω

) dW (

ω

)

.

That would be possible if the paths of Brownian motion W had bounded variation, i.e. if they could be expressed as the di¤erence of two non-decreasing functions.

But, since Brownian motion paths do not have bounded variation, it is not possible to use such an approach.

We must de…ne the stochastic integral with respect to the Brownian motion as a whole, i.e. as a stochastic process in itself, and not pathwise.

1X =Y P almost surely if the set of ωfor whichX is di¤erent from Y has a probability of zero, i.e.

Pf^ω2^Ω^:^X(ω)6=Y(ω)g=0.

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integral

Basic process I

Ito integral

De…nition

We call X a basic stochastic process if X admits the following representation:

X

t

(

ω

) = C (

ω

)

_I₍_a,b_]

( t )

where a < b 2

R

and C is a random variable, F

^a

^measurable and square-integrable, i.e. E

^P

C

²

<

_∞

.

Note that such a process is adapted to the …ltration

F

. Indeed,

Xt= 8<

:

0 if 0 t a C ifa<t b 0 ifb<t

which isF⁰ measurable thereforeF^t ^measurable which isF^a measurable thereforeF^t ^measurable which isF⁰ measurable thereforeF^t measurable.

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integral

Basic process II

Ito integral

In fact, X is more than

F

adapted, it is

F

predictable, but the notion of predictable process is more delicate to de…ne when we work with continuous-time processes.

Note, nevertheless, that adapted processes with continuous path are predictable.

Intuitively, if X

_t

represents the number of security shares

held at time t, then X

_t

(

ω

) = C (

ω

)

_I₍_a,b_]

( t ) means that

immediately after prices are announced at time a and

based on the information available at time a (C being

F

^a

measurable) we buy C (

ω

) security shares which we

hold until time b. At that time, we sell them all.

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integral

Basic process III

Ito integral

Theorem

F

adapted processes, the paths of which are left-continuous, are

F

predictable processes. In particular,

F

adapted

processes with continuous paths are

F

predictible. (cf. Revuz

and Yor)

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integral

Basic process IV

Ito integral

De…nition

The stochastic integral of X with respect to the Brownian motion is de…ned as

Z t

0

X

s

dW

s

(

ω

)

= C (

ω

) ( W

_t_^_b

(

ω

) W

_t_^_a

(

ω

))

=

8<

:

0 if 0 t a

C (

ω

) ( W

_t

(

ω

) W

_a

(

ω

)) if a < t b C (

ω

) ( W

_b

(

ω

) W

_a

(

ω

)) if b < t.

Note that, for all t, the integral R

t

0

X

_s

dW

_s

is a random variable.

Moreover,

n

R

t

0

X

s

dW

s :

t 0

o

is a stochastic process.

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integral

Basic process V

Ito integral

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

0 12 3 4 5 6 7 8 9 10

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

t

-5 -4 -3 -2 -10123456

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

t

Integrale_stochastique.xls

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integral

Basic process VI

Ito integral

Note that

Z t

0

X

_s

dW

_s

=

Z _∞

0

X

_sI(0,t]

( s ) dW

_s

.

Exercise. Verify it!

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integral

Basic process VII

Ito integral

Interpretation. If, for example, the Brownian motion W represents the variation of the share present value relative to its initial value (W

t

= S

t

S

0

, where S

t

is the share present value at time t), then R

t

0

X

s

dW

s

is the present value of the pro…t earned by the investor.

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

Note on the graphs that, over the time period when the

investor holds a constant number of security shares, the

present value of his or her pro…t ‡uctuates. That is only

caused by share price ‡uctuations.

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integral

Basic process VIII

Ito integral

Theorem

Lemma 1. If X is a basic stochastic process, then

Z t

0

X

_s

dW

_s :

t 0 is a (

Ω,

F

^,^F^,^P

) martingale.

Proof of Lemma 1. Recall that, if M is a martingale and

τ

a stopping time, then the stopped stochastic process

M

^τ

= f ^M

^t^^τ :

t 0 g is also a martingale.

Since the Brownian motion is a martingale, and

τa

and

τb

where 8

^ω

2

Ω, τa

(

ω

) = a and

τb

(

ω

) = b are stopping

times, then the stochastic processes W

^τ^a

= f ^W

^t^^a :

t 0 g

and W

^τ^b

= f ^W

^t^^b :

t 0 g are both martingales.

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integral

Basic process IX

Ito integral

Now, let’s verify condition ( M1 )

.

If t a, then

E^P Z t

0

X

_u

dW

_u

=

E^P

[ j ^C ( W

_t_^_b

W

_t_^_a

) j ]

=

E^P

[ j ^C ( W

t

W

t

) j ]

= 0.

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integral

Basic process X

Ito integral

By contrast, if t > a, then

E^P

Z t

0

X

_u

dW

_u

=

E^P

[ j ^C ( W

_t_^_b

W

a

) j ] =

E^P

[ j ^C j j ^W

^t^^b

W

a

j ]

r

E^Ph

C

²

( W

_t_^_b

W

_a

)

²ⁱ

(see next page)

=

r

E^Ph

C

²E^Ph

( W

_t_^_b

W

a

)

²

jF

^aⁱⁱ

^C ^being F

^a

^mble.

=

r

E^Ph

C

²E^Ph

( W

_t_^_b

W

a

)

²ⁱⁱ

W

_t_^_b

W

_a

being independent from F

^a^.

=

q

( t ^ ^b ^a )

E^P

[ C

²

] <

_∞

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integral

Basic process XI

Ito integral

Justi…cation for the inequality: for any random variable Y such that

E

Y

²

<

∞,

we have

0 Var [ Y ] =

E

Y

²

(

E

[ Y ])

²

which implies that

E

[ Y ]

q

E

[ Y

²

]

.

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integral

Basic process XII

Ito integral

With respect to condition ( M 2 ) ,

Zt

0 XsdWs

= 8<

:

0 if 0 t a

C(Wt Wa) ifa<t b C(Wb Wa) ifb<t.

F⁰ measurable thereforeF^t measurable, F^t measurable,

Fb measurable thereforeF^t measurable.

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integral

Basic process XIII

Ito integral

Condition ( M 3 ) is also veri…ed since 8 ^s

^,

^t 2

^R

^{, 0} ^s < t,

E^P

Z t

0

X

u

dW

u

jF

^s

=

E^P

[ C ( W

_t_^_b

W

t^^a

) jF

^s

]

.

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integral

Basic process XIV

Ito integral

But, if s a, then F

^s

F

^a

^{. Since} ^C ^is F

^a

measurable, then

E^P

Z t

0

X

u

dW

u

jF

^s

=

E^Ph

E^P

[ C ( W

_t_^_b

W

t^^a

) jF

^a

] jF

^sⁱ

=

E^Ph

C

E^P

[( W

_t_^_b

W

_t_^_a

) jF

^a

] jF

^sⁱ

=

E^P 2

64

C ( W

_a_^_b

W

_a_^_a

)

| {z }

=Wa Wa=0

jF

^s 3 75

since W

^τ^a

and W

^τ^b

are martingales,

= 0

=

Z s

0

X

_u

dW

_u.

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integral

Basic process XV

Ito integral

If s > a, then C is F

^s

measurable and, as a consequence,

E^P

Z _t

0

X

_u

dW

_u

jF

^s

=

E^P

[ C ( W

_t_^_b

W

_t_^_a

) jF

^s

]

= C

E^P

[( W

_t_^_b

W

_t_^_a

) jF

^s

]

= C ( W

_s_^_b

W

s^^a

)

=

Z _s

0

X

_u

dW

_u.

The process R

X

_s

dW

_s

is indeed a martingale.

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integral

Moments I

Basic process

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

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

E^P Z t

0

X

s

dW

s

=

E^P Z 0

0

X

s

dW

s

= 0.

The next result will allow us to calculate variances and

covariances.

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integral

Moments II

Basic process

Theorem

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

E^P

Z _t

0

X

_s

Y

_s

ds =

E^P Z _t

0

X

_s

dW

_s Z _t

0

Y

_s

dW

_s .

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integral

Moments III

Basic process

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

E^P

Z _∞

0

X

_s

Y

_s

ds =

E^P Z _∞

0

X

_s

dW

_s

Z _∞

0

Y

_s

dW

_s

(1) since

X = C

I(a,b]

and Y = C

eI

(

ea,eb

] being basic processes,

X

I(0,t]

= C

I(a^^t^,b^^t],

Y

I(0,t]

= C

^eI

(

ea^^t^,e^b^^t

]

et XY

I(0,t]

= C C

eI

(

⁽^a_ea)^^t,

(

^b^^e^b

)

^^t

]

are also basic processes.

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integral

Moments IV

Basic process

Thus, if equation (1) is veri…ed, we can use it to establish the third equality in the following calculation:

E^P Z t

0

X

s

Y

s

ds

=

E^P Z _∞

0

X

_s

Y

_sI(0,t]

( s ) ds

=

E^P Z _∞

0

X

_sI(0,t]

( s ) Y

_sI(0,t]

( s ) ds

=

E^P Z _∞

0

X

_sI(0,t]

( s ) dW

_s

Z _∞

0

Y

_sI(0,t]

( s ) dW

_s

=

E^P Z t

0

X

s

dW

s

Z t

0

Y

s

dW

s .

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integral

Moments V

Basic process

So, let’s establish Equation (1)

E^P

Z _∞

0

X

_s

Y

_s

ds =

E^P Z _∞

0

X

_s

dW

_s

Z _∞

0

Y

_s

dW

_s .

Three cases must be treated separately:

( i ) ( a, b ] \

e

a,

e

b

i

= ?

^,

( ii ) ( a, b ] =

_e

a,

e

b

i

,

( iii ) ( a, b ] 6 =

_e

a,

e

b

i

and ( a, b ] \

e

a,

e

b

i

6 = ?

^.

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integral

Moments VI

Basic process

Proof of ( i ) . Since ( a, b ] \

e

a,

e

b

i

= ? , we can assume, without loss of generality, that a < b

e

a <

^e

b. Since

X

s

Y

s

= C C

eI(a,b]

( s )

_I

(

ea,eb

] ( ^s ) = 0, then E

^P

R

_∞

0

X

_s

Y

_s

ds = 0.

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integral

Moments VII

Basic process

Now, since C , C

e

and ( W

_b

W

_a

) are F

ea

measurable,

E^P

Z _∞

0

X

_s

dW

_s

Z _∞

0

Y

_s

dW

_s

=

E^Ph

C ( W

b

W

a

) C W

e _e_b

W

_e_a i

=

E^Ph E^Ph

C ( W

_b

W

_a

) C W

^e _e_b

W

_e_a

jF

ea

ii

=

E^P 2

64

C C

e

( W

_b

W

_a

)

E^P

W

_e_b

W

_e_a

jF

ea

| {z }

=W_ea W_ea=0

3 75

= 0 =

E^P Z _∞

0

X

_s

Y

_s

ds

thus establishing Equation (1) in this particular case.

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integral

Moments VIII

Basic process

Proof of ( ii ) . Let’s now assume that ( a, b ] =

_e

a,

e

b

i

. Then,

E^P Z _∞

0

X

_s

Y

_s

ds =

E^P Z _∞

0

C C

eI(a,b]

dt

=

E^P

C C

e Z _∞

0 I(a,b]

dt

=

E^P h

C C

e

( b a )

ⁱ

= ( b a )

E^Ph

C C

ei

.

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integral

Moments IX

Basic process

On the other hand,

E^P Z_∞

0 XsdWs Z _∞

0 YsdWs

= E^Ph

C(W_b Wa) Ce(W_b Wa)ⁱ

= E^Ph CCeE^Ph

(W_b W_a)²jFa

ii sinceC andCe areFa mbles

= E^Ph CCeE^Ph

(Wb Wa)²ⁱⁱ

sinceW_b Wa is independent ofF^a

= E^Ph

CCe(b a)ⁱ sinceW_b Wa isN(0,b a)

= (b a)E^Ph CCei

=E^P Z _∞

0 XsYsds

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

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integral

Moments X

Basic process

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

_e

a,

e

b

i

and ( a, b ] \

e

a,

e

b

i

6 = ? ^.

a = a _

e

a and b = b ^

^e

b. On the one hand,

E^P

Z _∞

0

X

_s

Y

_s

ds =

E^P Z _∞

0

C

I(a,b]

C

eI

(

ea,eb

] ^dt

=

E^P Z _∞

0

C C

eI(a ,b ]

dt

=

E^P

C C

e Z _∞

0 I(a ,b ]

dt

=

E^Ph

C C

e

( b a )

ⁱ

= ( b a )

E^Ph

C C

ei

.

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integral

Moments XI

Basic process

In what follows, in case there were intervals of the form (

α,β

] where

α β,

then we de…ne (

α,β

] = ?

^.

On the other hand, since

Z _∞

0 XsdWs

= C(W_b Wa)

= C(W_b W_b +W_b Wa +Wa Wa)

= C(W_b W_b ) +C(W_b Wa ) +C(Wa Wa)

= Z _∞

0 CI_(b _,b]dWs+ Z_∞

0 CI_(a_,b _]dWs+ Z_∞

0 CI_(a,a_]dWs, (2)

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integral

Moments XII

Basic process

Z _∞

0 YsdWs

= C We _b_e W_e_a

= C We _b_e W_b +W_b Wa +Wa W_e_a

= C We _b_e W_b +Ce(W_b Wa ) +Ce(Wa W_e_a)

= Z _∞

0

CIe (^b ^,e^b]^dW^s+ Z_∞

0

CIe _(a_,b _]dWs+ Z_∞

0

CeI_(e_a,a _]dWs (3)

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integral

Moments XIII

Basic process

and

(a,a ]\(_ea,a ] = ?^, (a,a ]\(a ,b ] =?^, (a,a ]\ ^b ^,^e^bⁱ=?^, (a ,b ]\(_ea,a ] = ?^, (a ,b ]\ ^b ^,^e^bⁱ=?^,

(b ,b]\(_ea,a ] = ?^, (b ,b]\(a ,b ] =?^, (b ,b]\ ^b ^,^e^bⁱ=?^,

we can use the results obtained in points ( i ) and ( ii ) in order

to complete the proof: using the expressions from lines (2) and

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integral

Moments XIV

Basic process

(3),

E^P Z _∞

0

X

_s

dW

_s

Z _∞

0

Y

_s

dW

_s

=

E^P Z _∞

0

C

I(a,b ]

dW

s

Z _∞

0

C

eI(a ,b ]

dW

s

= ( b a )

E^Ph

C C

ei

=

E^P Z _∞

0

X

_s

Y

_s

ds

.

The proof of Lemma 2 is now complete.

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integral

Simple process I

Ito integral

De…nition

We call X a simple stochastic process if X is a …nite sum of basic processes :

X

_t

(

ω

) =

∑

n i=1

C

_i

(

ω

)

I(ai,bi]

( t )

.

Since such a process is a sum of

F

predictable processes,

then it is itself predictable with respect to the …ltration

F

.

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integral

Simple process II

Ito integral

De…nition

The stochastic integral of X with respect to the Brownian motion is de…ned as the sum of the stochastic integrals of the basic processes which constitute X

:

Z _t

0

X

_s

dW

_s

=

∑

n i=1

Z _t

0

C

_iI(ai,bi]

( s ) dW

_s.

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integral

Simple process III

Ito integral

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

-4 -2 0 2 4 6 8 10 12

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

t

-6 -4 -2 0 2 4 6 8

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 C

₁I

(

10¹,₁₀⁶

] + ^C

²^I

(

¹2,³₄

]

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integral

Simple process IV

Ito integral

Integrale_stochastique.xls

(47)

integral

Simple process V

Ito integral

First, we must ensure that such a de…nition contains no inconsistencies, i.e. if the simple stochastic process X admits at least two representations in terms of basic stochastic processes, say

∑ⁿi=1

C

_iI(ai,bi]

and

∑^mj=1

C

e_jI

(

ea_j,eb_j

] ^, then the integral is de…ned in a unique manner:

∑

n i=1

Z _t

0

C

_iI(ai,bi]

( s ) dW

_s

=

∑

m j=1

Z _t

0

e

C

_jI

(

eaj,ebj

] ( ^s ) dW

_s.

Exercise. Prove that the de…nition of the stochastic

integral for simple processes does not depend on the

representation chosen.

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integral

Simple process VI

Ito integral

Theorem

Lemma 3. If X is a simple stochastic process, then

n

R

t

0

X

_s

dW

_s :

t 0

o

is a (

_Ω,

F

^,F,P

) martingale.

Proof of Lemma 3. The stochastic integral

Z t

0

X

_s

dW

_s

=

∑

n i=1

Z t

0

C

_iI(ai,bi]

( s ) dW

_s

of the simple process X =

_∑ⁿ_i₌₁

C

_iI(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, the

results comes from the fact that stochastic integrals of basic

processes are martingales (Lemma 1).

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integral

Simple process VII

Ito integral

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

Theorem

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

E^P

Z _t

0

X

_s²

ds =

E^P

" _Z

t 0

X

_s

dW

_s

2# .

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integral

Simple process VIII

Ito integral

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

Var^P Z t

0

X

_s

dW

_s

=

E^P

" _Z

t 0

X

_s

dW

_s

2# 0

BB

@E^P Z t

0

X

_s

dW

_s

| {z }

=0

1 CC A

2

=

E^P Z _t

0

X

_s²

ds =

Z _t

0 E^P

X

_s²

ds.

It is possible to extend this calculation to other processes X

and toestablish in a similar manner a method to calculate the

covariance between two stochastic integrals (see the Appendix).

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integral

Predictable processes I

Ito integral

We would like to extend the class of processes for which the stochastic integral with respect to the Brownian motion can be de…ned. We will choose the class of

F

predictable processes for which there exists a sequence of simple processes approaching them.

Theorem

F

adapted processes, the paths of which are left-continuous are

F

predictable processes. In particular,

F

adapted processes with continuous paths are

F

predictible.

Let X be an

F

predictable process for which there exists a sequence

n

X

⁽ⁿ⁾ :

n 2

^N^o

of simple processes

converging to X when n increases to in…nity.

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integral

Predictable processes II

Ito integral

You can’t speak of convergence without speaking of distance. Indeed, how does one measure whether a

sequence of stochastic processes ”approaches” some other process? What is the distance between two stochastic processes? To answer such a question, we must de…ne the space of stochastic processes on which we work as well as the norm we put on that space.

Recall thatk k_A^:A ![0,∞)is a norm on the spaceA^if (i) k^Xk_A=0,^X =0;

(ii) 8^X 2 Aând8â2^R, kâXk_A=jâj k^Xk_A^; (iii) 8^X^,^Y 2 A^, k^X+Yk_A k^Xk_A+k^Yk_A

(triangle inequality).

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integral

Predictable processes III

Ito integral

Let

A = X X is a

F

predictable process such that

E^P

R

_∞

0

X

_t²

dt <

∞ .

Exercise, optional and rather di¢ cult! The function k k

_A ^:

A ! [ 0,

∞

) de…ned as

k ^X k

_A

=

s

E^P Z _∞

0

X

_t²

dt

is a norm on the space A ^.

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integral

Predictable processes IV

Ito integral

It is said that the sequence

n

X

⁽ⁿ⁾:

n 2

^N^o

A converges to X 2 A ^when ⁿ increases to in…nity if and only if

n

lim

!∞

X

⁽ⁿ⁾

X

A

= lim

n!∞

s E^P

Z _∞

0

X

_t⁽ⁿ⁾

X

t 2

dt

= 0.

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integral

Predictable processes V

Ito integral

It is tempting to de…ne the stochastic integral of X with respect to the Brownian motion as the limit of the stochastic integrals of the simple processes, i.e.

Z t

0

X

s

dW

s

= lim

n!∞

Z t

0

X

s⁽ⁿ⁾

dW

s.

Stochastic integral

Stochastic integral

Geneviève Gauthier

Introduction

The theories of stochastic integral and stochastic

di¤erential equations have initially been developed by

Kiyosi Ito around 1940 (one of the …rst important papers

was published in 1942).

Riemann integral I

Let f

!

be a real-valued function. Typically, when we speak of the integral of a function f , we refer to the Riemann integral, R

f ( t ) dt, which calculates the area

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

Riemann integral II

First, we must realize that not all functions are Riemann integrable.

Let’s explain a little further : in order to construct the

Riemann integral of the function f on the interval [ a, b ]

we divide such an interval into n subintervals of equal

length

and, for each of them, we determine the

greatest and the smallest value taken by the function f .

Riemann integral III

Thus, for every k 2 f 1, ..., n g

the smallest value taken by the function f on the interval a + ( k 1 )

a + k

is

f

inf f ( t ) a + ( k 1 ) b a

n t a + k b a n and the greatest is

f

sup f ( t ) a + ( k 1 ) b a

n t a + k b a

n

Riemann integral IV

The area under the curve t ! f ( t ) between the bounds a + ( k 1 )

and a + k

is therefore greater than the area

f

of the rectangle of base

and height f

, and smaller than the area

f

of the rectangle with the same base but with height

f

. As a consequence, if we de…ne

I

( f )

∑

b a

n f

and

I

( f )

∑

b a n

f

then

I

( f )

f ( t ) dt

I

( f )

Riemann integral V

Illustrations of I

( f ) and

I

( f )

Riemann integral VI

Riemann integrable functions are those for which the limits lim

I

( f ) and lim

I

( f ) exist and are equal.

We then de…ne the Riemann integral of the function f as

f ( t ) dt lim

I

( f ) = lim

I

( f )

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

Thus, for every k 2 f ^{1, ...,} ⁿ g

inf f ( t ) a + ( k 1 ) ^b ^a

sup f ( t ) a + ( k 1 ) ^b ^a

The area under the curve t ! ^f ( t ) between the bounds a + ( k 1 )

[ 0, 1 ] ! f ^0, ¹ g which takes the value of 1 if the argument is rational and 0 otherwise.

^where

( t ) = ^{1 if} ^t 2 ( a, b ]

Let f ^W

^t ⁰ g be a standard Brownian motion built on a …ltered probability space (

^t ⁰ g ^.

N = f ^A 2 F

Let f ^X

^t ⁰ g be a predictable stochastic process. It is tempting to de…ne the stochastic integral of X with respect to W , pathwise, by using a generalization of the Stieltjes integral: