The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 233 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 233

Let’s first define the set Φ(S,T) of functions Φ(S,T) ≡ {f(t,ω): [(0,∞) ×

Ω → R]} with the following properties:

■ Each f is jointly B × ℑ measurable.

■ Each f(t,ω) is adapted to ℑt.

■ Ef^2 ( t ).^3

S

T

∫ ω , td ∞ <

This is the set of paths for which we define the Itô integral.

Consider the time interval [S,T] and, for each integer n, partition

the interval [S,T] in subintervals: St≡ 0 < t 1 < ...ti < ...tn < ...tN ≡ T in

this way:

k 2 – n if Sk≤ 2 – n ≤ T

n 

tk = tk = S if k 2 – n < S



T if k 2

• n

  
  

  T

  
  

  
  

This rule provides a family of partitions of the interval [S,T] which can

be arbitrarily refined.

Consider the elementary functions φ(t,ω) ∈ Φ which we write as

φ(t ω , ) = ∑εi ()ω [ Iti+ 1 – ti )

i

As φ(t,ω) ∈ Φ, εi(ω) are ℑti measurable random variables.

We can now define the stochastic integral, in the sense of Itô, of ele-

mentary functions φ(t,ω) as

T

W = ∫φ(t ω , )dBt() ω = ∑εi ()ω[Bi + 1 () ω – Bi ()ω]

S i ≥ 0

where B is a Brownian motion. Note that the εi(ω) and the increments

Bj () ω – Bi () ω are independent for j > i. The key aspect of this definition

that was not included in the informal outline is the condition that the

εi(ω) are ℑti measurable.

For bounded elementary functions φ(t,ω) ∈ Φ the Itô isometry holds

(^3) This condition can be weakened.