Exercise 1. The Hermite polynomial of order is defined as
- Compute .
- Show that if is a Brownian motion, then the process is a martingale.
- Show that
Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if is a bounded harmonic function, then is constant.
Exercise 3. Show that if is a local martingale of a Brownian filtration , then there is a unique progressively measurable process such that for every , and
Exercise 4 [Skew-product decomposition]
Let be a complex Brownian motion started at .
- Show that for ,
- Show that there exists a complex Brownian motion such that
- Show that the process is independent from the Brownian motion .
- We denote which can be interpreted as a winding number around 0 of the complex Brownian motion paths. For , we consider the stopping time
- Compute for every , the distribution of the random variable
- Prove Spitzer theorem: In distribution, we have the following convergence
where is a Cauchy random variable with parameter 1 that is a random variable with density .