MA5161. Take home exam

Exercise 1. The Hermite polynomial of order n is defined as
H_n (x)=(-1)^n e^{\frac{x^2}{2}} \frac{d^n}{dx^n} e^{-\frac{x^2}{2}}.

  • Compute H_0, H_1,H_2,H_3.
  • Show that if (B_t)_{t \ge 0} is a Brownian motion, then the process \left(t^{n/2}H_n (\frac{B_t}{\sqrt{t}})\right)_{t \ge 0} is a martingale.
  • Show that
    t^{n/2}H_n (\frac{B_t}{\sqrt{t}})=n! \int_0^t \int_0^{t_1} ... \int_0^{t_{n-1}} dB_{s_1}...dB_{s_n}.


Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if f:\mathbb{R}^n \to \mathbb{R} is a bounded harmonic function, then f is constant.

Exercise 3. Show that if (M_t)_{t \ge 0} is a local martingale of a Brownian filtration (\mathcal{F}_t)_{t \ge 0}, then there is a unique progressively measurable process (u_t)_{t \ge 0} such that for every t \ge 0, \mathbb{P} \left(\int_0^t u_s^2 ds < +\infty \right)=1 and M_t=\mathbb{E} (M_0)+ \int_0^{t} u_s dB_s.
Exercise 4 [Skew-product decomposition]
Let (B_t)_{t \ge 0} be a complex Brownian motion started at z \neq 0.

  1. Show that for t \ge 0,
    B_t=z \exp\left( \int_0^t \frac{dB_s}{B_s} \right).
  2.  Show that there exists a complex Brownian motion (\beta_t)_{t \ge 0} such that
    B_t=z \exp{\left( \beta_{\int_0^t \frac{ds}{\rho_s^2} }\right)},
    where \rho_t =| B_t |.
  3. Show that the process (\rho_t)_{t \ge 0} is independent from the Brownian motion (\gamma_t)_{t \ge 0}=(\mathbf{Im} ( \beta_t))_{t \ge 0}.
  4. We denote \theta_t=\mathbf{Im}\left( \int_0^t \frac{dB_s}{B_s} \right) which can be interpreted as a winding number around 0 of the complex Brownian motion paths. For r>| z|, we consider the stopping time
    T_r =\inf \{ t \ge 0, | B_t | = r \}.
  5. Compute for every r>| z|, the distribution of the random variable
    \frac{1}{\ln (r/|z|)}\theta_{T_r}.
  6. Prove Spitzer theorem: In distribution, we have the following convergence
    \frac{ 2 \theta_t}{\ln t} \to_{+\infty} C,
    where C is a Cauchy random variable with parameter 1 that is a random variable with density \frac{1}{\pi (1+ x^2)}.


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