The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 234 Wednesday, February 4, 2004 12:50 PM

234 The Mathematics of Financial Modeling and Investment Management



2 T



T

E ∫ φ( tω , ) dBt()ω = E ∫φ( t ω , )^2 td

 

S S

The demonstration of the Itô isometry rests on the fact that

 0if ij≠

E[ε iε j( Bti – Bti)( Bt – Btj)] = 

• 1 j + (^1)  E ()ε^2 i if i = j
  This completes the definition of the stochastic integral for elementary
  functions.
  We have now completed the introduction of Brownian motions and
  defined the Itô integral for elementary functions. Let’s next introduce
  the approximation procedure that allows to define the stochastic inte-
  gral for any φ (t,ω ). We will develop the approximation procedure in the
  following three additional steps that we will state without demonstra-
  tion:
  ■ Step 1. Any function g(t,ω ) ∈ Φ that is bounded and such that all its
  time paths φ (·,ω ) are continuous functions of time can be approximated
  by

φn ( t ω , ) = ∑ gt(iω , ) It[i + 1 – ti )

i

in the sense that:

T

[( g – φ )

2

E∫ n td ] → 0 , n → ∞ , ∀ω

S

where the intervals are those of the partition defined above. Note that

φ ( n t ω , ) ∈ Φ given that gt( ω , ) ∈ Φ.

■ Step 2. We release the condition of time-path continuity of the

φ ( n t ω , ). It can be demonstrated that any function ht( ω , ) ∈ Φ which

is bounded but not necessarily continuous can be approximated by

functions gn( tω , ) ∈ Φ which are bounded and continuous in the sense

that