The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 235 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 235

T

E ∫( h – gn )^2 td → 0

S

■ Step 3. It can be demonstrated that any function ft( ω , ) ∈ Φ , not nec-

essarily bounded or continuous, can be approximated by a sequence of

bounded functions hn ( tω , ) ∈ Φ in the sense that

T

E ∫( f – hn )^2 td → 0

S

We now have all the building blocks to complete the definition of

Itô stochastic integrals. In fact, by virtue of the above three-step

approximation procedure, given any function ft( ω , ) ∈ Φ , we can

choose a sequence of elementary functions φn ( tω , ) ∈ Φ such that the

following property holds:

T

E ∫( f – φ )

2

n td →^0

S

Hence we can define the Itô stochastic integral as follows:

T T

If[]()w = ∫ ft( ω , ) dBt() ω = lim φ ( ntω , ) td

n → ∞∫

S S

The limit exists as

T

∫φ n( t ω , ) dBt()ω^

S

forms a Cauchy sequence by the Itô isometry, which holds for every

bounded elementary function.

Let’s now summarize the definition of the Itô stochastic integral:

Given any function ft( ω , ) ∈ Φ , we define the Itô stochastic integral

by