Hey,

I've to show the following convergence.

,

,

whereas

is a standard Brownian motion

is a normal distributed random variable with continous paths

is a random variable with values in the interval , depends on and has continous paths

is a constant

is a constant

I started by using the taylor expansion of the exponential function for

and also the taylor expansion for the logarithm:

But how can I approximate the remainder term?

Can I change the order of the expectation value and the limit?

I know that

The following equation holds due to the definition of the Itô integral. Am I right?

where the limit is taken in

Does anybody have an idea how to show this convergence?

Can anybody help me?

Thanks!