Let Wt be a Brownian motion and f a continuous function. Prove that if f (Wt)is a sub martingale, then f must be convex. Prove the maximum principle for harmonic functions. This says that if h is...

Let Wt be a Brownian motion and f a continuous function. Prove that if f (Wt)is a sub martingale, then f must be convex.

Prove the maximum principle for harmonic functions. This says that if h is harmonic in a bounded domain D, then

If W is a d-dimensional Brownian motion started at 0, find ET, where T is the first time W exits the ball of radius r centered at the origin.

Chapter 22