The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 223 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 223

cumulates the products of the elementary functions φ(t,ω) and of the

increments of the Brownian motion Bt(ω).

It can be demonstrated that the following property, called Itô

isometry, holds for Itô stochastic integrals defined for bounded ele-

mentary functions as above:



2 T



T

E ∫ φ(t, ω)dBt()ω = E ∫φ(t, ω)^2 td

S  S

The Itô isometry will play a fundamental role in Step 3.

■ Step 3. The third step consists in using the Itô isometry to show that

each function g which is square-integrable (plus other conditions that

will be made precise in the next section) can be approximated by a

sequence of elementary functions φn(t,ω) in the sense that

T

E ∫[g – φ (t, ω)]

2

n td →^0

S

If g is bounded and has a continuous time-path, the functions φn(t,ω)

can be defined as follows:

φ (n t, ω)= ∑gt(i , ω)It[i + 1 ,ti )

i

where I is the indicator function. We can now use the Itô isometry to

define the stochastic integral of a generic function f(t,ω) as follows:

T T

∫ft(, ω)dBt()ω = lim φ (nt, ω)dBt()

n → ∞∫

ω

S S

The Itô isometry insures that the Cauchy condition is satisfied

and that the above sequence thus converges.

In outlining the above definition, we omitted an important point

that will be dealt with in the next section: The definition of the stochas-

tic integral in the sense of Itô requires that the elementary functions be

without anticipation—that is, they depend only on the past history of