# Ito's formula and Ito's isometry

Hi, I have a question about martingale with respect to Brownian motion process, and I am looking for a detailed explanation. Thank you.

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#### Solution Preview

Let be a probability space with a filtration .

Let be a standard one-dimensional - Brownian motion .

Let be a constant.

Prove that the process is and -martingale, where defined by:

Hint: You may use Ito's formula and Ito's isometry.

There are two ways to solve this problem. I'll demonstrate both approaches. The first one is to use "loose" definition of the martingale a s a drift ...

#### Solution Summary

Ito's formula and Ito's isometry are applied.

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