stochastic integration i

7/24/2019 Stochastic Integration I

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Stochastic Integration I


Timothy Robin Teng

Stochastic Integration I 1 / 29


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Let (,F,P) be given.

Recall that ifX = {Xn} is a sequence of random variables, andC= {Cn} is a previsible process, then the martingale transform ofXbyC, denoted by C X, is such that

(C X)n =n1k=0

Ck+1(Xk+1 Xk)

The process{Cn}can be viewed as a gambling strategy, where Cnrepresents the stake at game n, and Xn+1 Xn represents netwinnings per unit stake at game n. Hence (C X)n represents totalwinnings up to time n.



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Applied to mathematical finance, ifC = {Cn} is a self-financingtrading strategy, and Xn represents the price of the asset at time n

(in general, Xn is a vector of prices), then (C X)n represents thetotal change in the value of the portfolio up to time n.



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We will carry the notion of martingale transform from discrete-timeinto continuous-time, which deals with the stochastic integral

CtdXt. However, asset prices are typically functions of one orseveral Brownian motions, which brings our focus to

f(t)dWt

where{Wt} is a Wiener process. The main goal of this section is todefine this integral.


S h i I i I


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The first thing to note is that stochastic integrals with respect toBrownian motion, if they exist, must be quite different from theclassical measure-theoretic integrals. This is because for almost everyWiener trajectory, it has infinite variation on any interval, and henceis not differentiable at any point.

Recall that ifg(t) is continuously differentiable and f R(g) i.e. fis Riemann-Stieltjes integrable with respect to gon [0,T], then wecan define

T

0

f(t)dg(t) := T

0

f(t)g(t)dt

where the right hand side is an ordinary Riemann integral with respectto time. This, however, will not work for the Brownian motion.


St h ti I t ti I


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Stochastic integration was introduced by Kiyoshi Ito in 1944, whodevised a way around the non-differentiability of the Brownian paths.He used a mix of ideas from the construction of integrals in theRiemann, Lebesgue settings.

The method involves defining the integral first for a simple class offunctions. Then show for every f in a class of integrands can beapproximated (in a sense to be defined) by simple processes s and

use this to define T

0 ftdWtas the limit of

T

0 tdWt.




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We will now show how to define the Ito integral.

I(f)T = T

0

ftdWt (1)

where 0 T and Wt is the 1-dimensional Brownian motion.




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IntegrandFor the above integral to make sense, the integrand f must meetsome basic measurability, adaptivity, and integrability conditions.

Let T 0,Bdenote the Borel -field on [0,+),{Ft}0tTbe thenatural Brownian filtration, i.e.,Ft= (Ws : 0 s t). Thefollowing describes the class of functions for which the integral will bedefined.




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Definition

LetHTdenote the class of functionsf : [0,) R

(t, )

ft()

such that

1 f isB F-measurable2 f isFt-adapted, i.e. ft isFt-measurable for each t.3 E

T

0 |ft|2 dt


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g

We then define the stochastic integral of the simple function t as

T0

tdWt=

n1i=0

aiWti+1 Wti

For tk

t

tk+1

t0

sdWs =k1i=0

aiWti+1 Wti

+ak[WtWtk]

=

ki=0

aiWti+1tWtit




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g

We can think of the relationship between the simple processt=

n1i=0 ai1[ti,ti+1)(t) and the Brownian motion Wt in the following

manner:

Regard Wtas the price per share of a stock at time t(We willreplace Wtby the more appropriate geometric Brownian motionlater). Think oft0, t1, ..., tn1 as the trading dates in the market, and

a0, a1, ..., an1 as the position taken in the stock at each trading dateand held to the next trading date. The gain from trading at time t,where tk


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g

Properties:

Let , HTbe simple processes and t T1) For the indicator function of an interval 1[a,b)(s), where0 a


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3) Zero mean property: E

t0sdWs

= 0

4) Isometry Property: Et

0sdWs

2

=Et

02sds

5) E[I()t |Fs] =I()s a.s. (0 s t). Hence{I()t}t[0,T] is acontinuous martingale.




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Remark: With (1) and (2), we have

b

a sdWs=

c

a sdWs+

b

c sdWs, where 0 a


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Ito Integral for General Integrands

In this section, we define the stochastic integral T0 f

tdW

t for

f HT, where is no longer assumed to be a simple process.

In order to defineT

0 ftdWt, we approximate fby simple processes.

In general, it is natural to approximate f byi

fti

1[ti,ti+1)(t)

where ti

[ti, ti+1). Afterwards, we define T0 ftdWtas the limitof

ifti

Wti+1 Wti

as n .




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However, unlike the Riemann-Stieltjes integral, the choice ofti

makes a difference. The following two choices have turned out to bethe most useful ones:

(1) ti =ti(the left endpoint of [ti, ti+1)). This leads to the Ito

integral, denoted byT

0 ftdWt.

(2) ti = 12(ti+ti+1) (the midpoint). This leads to the Stratonovich

integral, denoted by

T

0 ft dWt.




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To define the Ito integral

I(f)T = T

0ftdWt

we have to find a sequence {(n)}n0 of simple processes such that asn

, these processes converge to f in the following sense

Theorem

Letf HT. Then there exists a sequence{(n)}n0 of simpleprocesses such that

E

T

0

ft (n)t 2 dt 0 asn




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Outline of Proof:

Such a sequence is found in several steps

Step 1: Let g HTbe bounded and t gt() is continuous for all (sample paths ofgare continuous). Then there exists simple

processes (n)

HT such thatE

T0

gt (n)t

2

dt

0 as n

Here, (n)t () =

igti() 1[ti,ti+1)(t)




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Step 2: Let h HTbe bounded. Then there exists boundedprocesses g

(n)

HT such that t g(n)

t () is continuous for all and n, and

E T

0 ht g(n)t

2

dt 0 as n

Here g(n)t () =

t0n(s t)hs() ds, where n is a nonnegative

continuous function on Rsuch that

(1) n(x) = 0 for x 1

n and x 0;(2)

n(x)dx= 1




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Step 3: Let f HT. Then there exists a sequence{h(n)} HT suchthat h

(n)

is bounded for each n and

E

T0

ft h(n)t

2

dt

0 as n

Here

h(n)t () =

n ifft()< nft() if n ft() nn ifft()>n

For full details of the proof, refer to Bernt Oksendal (pp. 26-28)




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We are now ready to complete the definition of the Ito integral

DefinitionLet f

HT. Then the integral (from 0 to T) is defined by T

0

ftdWt = limn

T0

(n)t dWt (limit in L

2(P))

I(f)T = limn

I((n))T

i.e.EI(f)T I((n))T

2 0 as n where

{n}

is a sequence of simple processes such that

E

T0

ft (n)t

2

dt

0 as n




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This integral inherits the properties of Ito integrals of simple

processes.

Properties

Let Tbe a positive constant and f HT. For 0 t T, letI(f)t := t0 fsdWsbe as defined above. Then(1) I

1[a,b)f

T

=T

0 1[a,b)(s)fsdWs=

ba fsdWs

(2) Continuity: As a function of the upper limit of integration t, thepaths ofI(f)tare continuous.

(3) Adaptivity: For each t, I(f)t isFt-measurable.




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(4) Linearity: I(f + g)t=I(f)t+ I(g)t for all f, g HT and, R(5) Zero Mean Property: E

t0 fsdWs

= 0

(6) Martingale:{I(f)t}t[0,T] is a continuous martingale.

(7) Ito Isometry: E

t0 fsdWs

2=E

t0 f2sds

(8) Quadratic Variation: [I(f), I(f)] (t) = t0 f2sds




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We have defined the Ito integral for f HT. The integral will alsobe defined if we replace the third condition E

T

0 f2tdt


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Proposition

IfX is a continuous adapted process, then the Ito Integral T

0 XtdWt

exists. In particular,T

0 f (Wt) dWt, wheref is continuous functionR is well defined.



Ill


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Illustration:The integral

10 eWtdWtis well defined since the integrand is of the

form f (Wt), where f (x) =ex is continuous on R.

Since

E

10

e2Wtdt

=

10

E

e2Wt

dt

=

1

0

e2tdt=12

e2 1


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The integral

1

0 eW

2t dWtis well defined since the integrand is of the

form f (Wt), where f (x) =ex2 is continuous on R.

However, E1

0 e2W

2t dt

= , and hence we cannot claim that this

Ito integral has finite moments

Note that Wt N(0, t)

Ee2W

2t

=

e2x2

1

2tex2

2t

dx=

fort 14 . Moreover, using some martingale inequalities, it can shownthat the expectation of the given integral does not exist




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References

Shreve, Steven. Stochastic Calculus for Finance II:

Continuous-Time Models. Springer Science + Business Media,LLC, 2004.

Etheridge, Alison. A Course in Financial Calculus. CambridgeUniversity Press, 2002.

Klebaner, Fima. An Introduction to Stochastic Calculus withApplications, 2nd Edition. Imperial College Press, 2005.

Oksendal, Bernt. Stochastic Differential Equations: AnIntroduction with Applications. Springer-Verlag Berlin, 2003.

Damien Lamberton and Bernard Lapeyre. Introduction toStochastic Calculus Applied to Finance. Chapman and Hall,1996


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