4

DAVID DOUGLAS ENGEL

T

J f(t)dX(t) = X(I).

0

By linearity we can extend this definition to all step functions, that

is if f(t) = a. for tel. where I ,...,1 are disjoint intervals contained

in [0,T], then

T

f

f(t)dX(t) = £ a±X(I).

L n=l

0

Clearly each of these integrals is a Gauss function of o with mean zero.

Computing the variance we get

T k

f(t)dX(t)||2 = ||S a. X(I±)||2

n=l

0

k k

*i *—% n m n m

n=l m=l

k

n=l

= V a E(X(I )2) since X(I ) I I X(I ) n * m

*"* n n n ---

LJ

m

k

£ anm(In

^-r, n n

n= l

| f ( t ) l l

2

2

o

So the operation of taking the stochastic integral preserves L -mean.

2

Therefore, if f is any function in L ([0,T]), by taking a sequence of

2

step functions f such that f - f in L ([0,T] ), we can define

T n

T

n

f(t)dX(t) = L.I.M.

n - °°

fn(t)dX(t),

The important properties of X(t) used in defining this stochastic

integral are: