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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