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as an example of calculating the Itoâ€™s integral. If the function W(t) is

deterministic, then the Riemann-Stieltjes integral IRÃ€S (t2 , t1 ) equals

IRÃ€S (t2 , t1 ) Â¼ 0:5[W(t2 )2 Ã€ W(t1 )2 ] (4:4:12)

However, when W(t) is the Wiener process, the Itoâ€™s integral II (t2 , t1 )

leads to a somewhat unexpected result

II (t2 , t1 ) Â¼ 0:5[W(t2 )2 Ã€ W(t1 )2 Ã€ (t2 Ã€ t1 )] (4:4:13)

This follows directly from equation (4.3.6). Obviously, the result

(4.4.13) can be derived directly from the definition of the Itoâ€™s integral

(see Exercise 1). Note that the mean of the Itoâ€™s integral (4.4.11)

equals zero

E[II (t2 , t1 )] Â¼ 0 (4:4:14)

39

Stochastic Processes

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

determined by the particular choice of a Â¼ 0 in (4.4.8). Stratonovich

has offered another definition of the stochastic integral by choosing

a Â¼ 0:5. In contrast to equation (4.4.9), the Stratonovichâ€™s integral is

defined as

X tiÃ€1 Ã¾ ti

Ã°

T

n

f(t)dW(t) Â¼ msÃ€lim [W(ti ) Ã€ W(tiÃ€1 )] (4:4:15)

f

2

n!1

iÂ¼1

0

For the integrand in (4.4.11), the Stratonovichâ€™s integral IS (t2 , t1 )

coincides with the Riemann-Stieltjes integral

IS (t2 , t1 ) Â¼ 0:5[W(t2 )2 Ã€ W(t1 )2 ] (4:4:16)

Both Itoâ€™s and Stratonovichâ€™s formulations can be transformed into

each other. In particular, the Itoâ€™s stochastic differential equation (4.3.1)

dyI (t) Â¼ mdt Ã¾ sdW(t) (4:4:17)

is equivalent to the Stratonovichâ€™s equation

@s

dyS (t) Â¼ m Ã€ 0:5s dt Ã¾ sdW(t) (4:4:18)

@y

The applications of stochastic calculus in finance are based almost

exclusively on the Itoâ€™s theory. Consider, for example, the integral

Ã°

t2

(4:4:19)

s(t)dW(t)

t1

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

is assumed, then the Itoâ€™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 Stratonovichâ€™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

40 Stochastic Processes

E[X(t Ã¾ 1)jX(t), X(t Ã€ 1), . . . ] Â¼ X(t) (4:5:1)

The equivalent definition is given by

E[X(t Ã¾ 1) Ã€ X(t)jX(t), X(t Ã€ 1), . . . ] Â¼ 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 Ã¾ 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 Ã¾ 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 Itoâ€™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

2tÃ¾Dt 3

Ã°

E4 s(z)dW(z)5 Â¼ 0 (4:5:3)

t

Therefore,

2tÃ¾Dt 3t

Ã° Ã°

E4 s(z)dW(z)5 Â¼ s(z)dW(z) (4:5:4)

0 0

41

Stochastic Processes

and the integral (4.4.19) satisfies the martingale definition. Note that

for the Brownian motion with drift (4.2.14)

2 3

Ã°

tÃ¾dt

E[y(t Ã¾ dt)] Â¼ E4y(t) Ã¾ dy5 Â¼ y(t) Ã¾ mdt (4:5:5)

t

Hence, the Brownian motion with drift is not a martingale. However,

the process

z(t) Â¼ y(t) Ã€ mt (4:5:6)

is a martingale since

E[z(t Ã¾ dt)] Â¼ z(t) (4:5:7)

This result follows also from the Doob-Meyer decomposition theorem,

which states that a continuous submartingale X(t) at 0 t 1 with

finite expectation E[X(t)] < 1 can be decomposed into a continuous

martingale and an increasing deterministic process.

4.6 REFERENCES FOR FURTHER READING

Theory and applications of the stochastic processes in natural

sciences are described in [1, 6]. A good introduction to the stochastic

calculus in finance is given in [2]. For a mathematically inclined

reader, the presentation of the stochastic theory with increasing

level of technical details can be found in [7, 8].

4.7 EXERCISES

1. Simulate daily price returns using the geometric Brownian

motion (4.3.7) for four years. Use equation (4.2.15) for approxi-

mating DW. Assume that S(0) Â¼ 10, m Â¼ 10%, s Â¼ 20% (m and

s are given per annum). Assume 250 working days per annum.

2. Prove that

Ã° Ã°

t2 t2

1 n

W(s)n dW(s) Â¼ [W(t2 )nÃ¾1 Ã€ W(t1 )nÃ¾1 ] Ã€ W(s)nÃ€1 ds

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