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
¼ y(t) þ mdt (4:5:5)
Hence, the Brownian motion with drift is not a martingale. However,
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 . For a mathematically inclined
reader, the presentation of the stochastic theory with increasing
level of technical details can be found in [7, 8].
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
n þ 1
Hint: Calculate d(W
) using the Ito’s lemma.
Stochastic Processes 41