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