The difference between the right-hand sides of (4.4.12) and (4.4.13) is

determined by the particular choice of a ¼ 0 in (4.4.8). Stratonovich

has offered another definition of the stochastic integral by choosing

a ¼ 0:5. In contrast to equation (4.4.9), the Stratonovich’s integral is

defined as

ð

T

0

f(t)dW(t) ¼ mslim

n!1

X

n

i¼1

f

t

i1

þ t

i

2

[W(t

i

) W(t

i1

)] (4:4:15)

For the integrand in (4.4.11), the Stratonovich’s integral I

S

(t

2

,t

1

)

coincides with the Riemann-Stieltjes integral

I

S

(t

2

,t

1

) ¼ 0:5[W(t

2

)

2

W(t

1

)

2

](4:4:16)

Both Ito’s and Stratonovich’s formulations can be transformed into

eachother.In particular, theIto’s stochastic differentialequation(4.3.1)

dy

I

(t) ¼ mdt þ sdW(t) (4:4:17)

is equivalent to the Stratonovich’s equation

dy

S

(t) ¼ m 0:5s

@s

@y

dt þ sdW(t) (4:4:18)

The applications of stochastic calculus in finance are based almost

exclusively on the Ito’s theory. Consider, for example, the integral

ð

t

2

t

1

s(t)dW(t) (4:4:19)

If no correlation between the function s(t) and the innovation dW(t)

is assumed, then the Ito’s approximation is a natural choice. In this

case, the function s(t) is said to be a nonanticipating function [1, 2].

However, if the innovations dW(t) are correlated (so-called non-white

noise), then the Stratonovich’s approximation appears to be an ad-

equate theory [1, 6].

4.5 MARTINGALES

The martingale methodology plays an important role in the

modern theory of finance [2, 7, 8]. Martingale is a stochastic process

X(t) that satisfies the following condition

Stochastic Processes 39

E[X(t þ 1)jX(t), X(t 1), ...] ¼ X(t) (4:5:1)

The equivalent definition is given by

E[X(t þ 1) X(t)jX(t), X(t 1), ...] ¼ 0(4:5:2)

Both these definitions are easily generalized for the continuum pre-

sentation where the time interval, dt, between two sequent moments

t þ 1 and t approaches zero (dt ! 0). The notion of martingale is

rooted in the gambling theory. It is closely associated with the notion

of fair game, in which none of the players has an advantage. The

condition (4.5.1) implies that the expectation of the gamer wealth at

time t þ 1 conditioned on the entire history of the game is equal to the

gamer wealth at time t. Similarly, equation (4.5.2) means that the

expectation to win at every round of the game being conditioned on

the history of the game equals zero. In other words, martingale has no

trend. A process that has positive trend is named submartingale.

A process with negative trend is called supermartingale.

The martingale hypothesis applied to the asset prices states that the

expectation of future price is simply the current price. This assumption

is closely related to the Efficient Market Hypothesis discussed in

Section 2.3. Generally, the asset prices are not martingales for they

incorporate risk premium. Indeed, there must be some reward offered

to investors for bearing the risks associated with keeping the assets. It

can be shown, however, that the prices with discounted risk premium

are martingales [3].

The important property of the Ito’s integral is that it is martingale.

Consider, for example, the integral (4.4.19) approximated with the

sum (4.4.9). Because the innovations dW(t) are unpredictable, it

follows from (4.4.14) that

E

ð

tþDt

t

s(z)dW(z)

2

4

3

5

¼ 0(4:5:3)

Therefore,

E

ð

tþDt

0

s(z)dW(z)

2

4

3

5

¼

ð

t

0

s(z)dW(z) (4:5:4)

40 Stochastic Processes

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