Lecture 23
Math 50051, Topics in Probability Theory and Stochastic Processes
Properties of the stochastic integral:
1) Linearity:
Zt
0
aX(s) + bY (s)dW (s) = aZt
0
X(s)dW (s) + bZt
0
Y(s)dW (s)
2) E[Rt
0X(s)dW (s)] = 0
3) Isometry:
E[Rt
0X(s)dW (s)2] = Rt
0E[Y(s)2]ds
4) For each t≥0, let Yt=Rt
0g(s)dW (s). Then Yis adapted to FW.
5) Martingale property:
If It=Rt
0g(s)dW (s), then Iis a FW-martingale.
Examples: Models that describe the dynamic behavior of asset prices contain invention terms that
represent unpredictable news. As a result, an integral of the form
Zt+∆t
t
σ(s)dW (s)
is a sum of unpredictable disturbances that affect asset prices during an interval of length ∆t. Now,
if each increment is unpredictable given the information set at time t, the sum of these increments
should also be unpredictable. This means that
E(Zt+∆t
t
σ(s)dW (s)|Ft) = 0
But this is equivalent to:
E(Zt+∆t
0
σ(s)dW (s)|Ft) = E(Zt
0
σ(s)dW (s)|Ft) = Zt
0
σ(s)dW (s)
which is exactly the martingale property of the stochastic integral. Hence, the existence of unpre-
dictable innovation terms in equations describing the dynamics of asset prices coincides well with
the martingale property of the Ito integral. Let’s look at particular case: σ(s) = σ, a constant.
Then:
Zt+∆t
t
σ(s)dW (s) = σ(Wt+∆t−Wt)
Exercises:
1) Show, using the properties of the si that:
E(I(X)I(Y)) = Zt
0
E(X(s)Y(s))ds
1
