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 ¼mXdt 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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