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

for the Brownian motion with drift (4.2.14)

E[y(t þ dt)] ¼ E y(t) þ

ð

tþdt

t

dy

2

4

3

5

¼ y(t) þ mdt (4:5:5)

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

the process

z(t) ¼ y(t) mt(4:5:6)

is a martingale since

E[z(t þ dt)] ¼ 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 1with

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) ¼ 10, m ¼ 10%, s ¼ 20% (m and

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

2. Prove that

ð

t

2

t

1

W(s)

n

dW(s) ¼

1

n þ 1

[W(t

2

)

nþ1

W(t

1

)

nþ1

]

n

2

ð

t

2

t

1

W(s)

n1

ds

Hint: Calculate d(W

nþ1

) using the Ito’s lemma.

Stochastic Processes 41

3. Solve the Ornstein-Uhlenbeck equation that describes the mean-

reverting process in which the solution fluctuates around its

mean

dX ¼mXdt þ s dW, m > 0

Hint: introduce the variable Y ¼ X exp (mt).

*4. Derive the integral (4.4.13) directly from the definition of the

Ito’s integral (4.4.9).

42 Stochastic Processes

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