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@F s2 @ 2 F

@F

Â¼ dy Ã¾ Ã¾ dt (4:3:4)

2 @ y2

@y @t

The Itoâ€™s expression (4.3.4) has an additional term in comparison with

the differential for a function with deterministic independent vari-

36 Stochastic Processes

s2 @ 2 F

ables. Namely, the term dt has stochastic nature. If y(t) is the

2 @ y2

Brownian motion (4.3.1), then

@F s2 @ 2 F

@F

dF(y, t) Â¼ [mdt Ã¾ sdW(t)] Ã¾ Ã¾ dt

2 @ y2

@y @t

@F @F s2 @ 2 F @F

Â¼m Ã¾ Ã¾ dt Ã¾ s dW(t) (4:3:5)

2 @ y2

@ y @t @y

Let us consider the function F Â¼ W2 as a simple example for

employing the Itoâ€™s lemma. In this case, m Â¼ 0, s Â¼ 1, and equation

(4.3.5) is reduced to

dF Â¼ dt Ã¾ 2WdW (4:3:6)

Finally, we specify the Itoâ€™s expression for the geometric Brownian

@F @ 2 F @F

motion F Â¼ exp [y(t)]. Since in this case, Â¼ 2 Â¼ F and Â¼ 0,

@y @y @t

then

s2

dF Â¼ m Ã¾ Fdt Ã¾ sFdW(t) (4:3:7)

2

Hence, if F is the geometric Brownian motion, its relative change, dF/F,

behaves as the arithmetic Brownian motion.

The Itoâ€™s lemma is a pillar of the option pricing theory. It will be

used for deriving the classical Black-Scholes equation in Section 9.4.

4.4 STOCHASTIC INTEGRAL

Now that the stochastic differential has been introduced, let us

discuss how to perform its integration. First, the Riemann-Stieltjes

integral should be defined. Consider a deterministic function f(t)

on the interval t 2 [0, T]. In order to calculate the Riemann integral

of f(t) over the interval [0, T], this interval is divided into n sub-intervals

t0 Â¼ 0 < t1 < . . . < tn Â¼ T and the following sum should be computed

X

n

Sn Â¼ f(ti )(ti Ã€ tiÃ€1 ) (4:4:1)

iÂ¼1

where ti 2 [tiÃ€1 , ti ]. The Riemann integral is the limit of Sn

37

Stochastic Processes

Ã°

T

f(t)dt Â¼ lim Sn , max (ti Ã€ tiÃ€1 ) ! 0 for all i: (4:4:2)

0

Note that the limit (4.4.2) exists only if the function f(t) is sufficiently

smooth. Another type of integral is the Stieltjes integral. Let us define

the differential of a function g(x)

dg Â¼ g(x Ã¾ dx) Ã€ g(x) (4:4:3)

Then the Stieltjes integral for the function g(t) on the interval

t 2 [0, T] is defined as

X

n

Sn Â¼ f(ti )[g(ti ) Ã€ g(tiÃ€1 )] (4:4:4)

iÂ¼1

where ti 2 [tiÃ€1 , ti ]

Ã°

T

f(t)dg(t) Â¼ lim Sn , where max (ti Ã€ tiÃ€1 ) ! 0 for all i: (4:4:5)

0

dg

If g(t) has a derivative, then dg % dt Â¼ g0 (t)dt, and the sum (4.4.4)

dt

can be written as

X

n

f(ti )g0 (ti )(ti Ã€ tiÃ€1 )

Sn Â¼ (4:4:6)

iÂ¼1

Similarity between the Riemann sum (4.4.1) and the Stieltjes sum

(4.4.6) leads to the definition of the Riemann-Stieltjes integral. The

Riemann-Stieltjes integral over the deterministic functions does not

depend on the particular choice of the point ti within the intervals

[tiÃ€1 , ti ]. However, if the function f(t) is random, the sum Sn does

depend on the choice of ti . Consider, for example, the sum (4.4.4) for

the case f(t) Â¼ g(t) Â¼ W(t) (where W(t) is the Wiener process). It

follows from (4.1.16) that

" #

X

n

E[Sn ] Â¼ E W(ti ){W(ti ) Ã€ W(tiÃ€1 )}

iÂ¼1

X X

n n

Â¼ [ min (ti , ti ) Ã€ min (ti , tiÃ€1 )] Â¼ (ti Ã€ tiÃ€1 ) (4:4:7)

iÂ¼1 iÂ¼1

38 Stochastic Processes

Let us set for all i

ti Â¼ ati Ã¾ (1 Ã€ a)tiÃ€1 0 1 (4:4:8)

a

Substitution of (4.4.8) into (4.4.7) leads to E[Sn ] Â¼ aT. Hence, the

sum (4.4.7) depends on the arbitrary parameter a and therefore can

have any value. Within the Itoâ€™s formalism, the value a Â¼ 0 is chosen,

so that ti Â¼ tiÃ€1 . The stochastic Itoâ€™s integral is defined as

Ã°

T

X

n

f(t)dW(t) Â¼ msÃ€lim f(tiÃ€1 )[W(ti ) Ã€ W(tiÃ€1 )] (4:4:9)

n!1

iÂ¼1

0

The notation ms-lim stands for the mean-square limit. It means that

the difference between the Ito integral in the left-hand side of (4.4.9)

and the sum in the right-hand side of (4.4.9) has variance that ap-

proaches zero as n increases to infinity. Thus, (4.4.9) is equivalent to

2T 32

Ã° Xn

lim E4 f(t)dW(t) Ã€ f(tiÃ€1 ){W(ti ) Ã€ W(tiÃ€1 )}5 Â¼ 0 (4:4:10)

n!1

iÃ€1

0

Let us consider the integral

Ã°

t2

I(t2 , t1 ) Â¼ W(t)dW(t) (4:4:11)

t1

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