Assignment 11

Assigned 2024-04-03, due 2024-04-10 at noon on Gradescope

Question 1

Let $W$ be a two dimensional Brownian motion. Is $\displaystyle \int_0^t \frac{W^1_s}{\abs{W_s}^2} \, dW^1_s$ a martingale? Justify.

Question 2

Let $b = (b^1, b^2)$ be a two dimensional process, and $W$ be a two dimensional Brownian motion. \begin{equation*} Z_t = \exp\paren[\Big]{ -\sum_{i = 1}^2 \int_0^t b^i_s \, dW^i_s - \frac{1}{2} \int_0^t \abs{b_s}^2 \, ds }\,. \end{equation*} Compute $dZ$.

Question 3

Let $W$ be a two dimensional Brownian motion and $b\colon [0, \infty) \times \R^2 \to \R^2$ be a continuous function. For $i, j \in \set{1, 2}$ let $\sigma_{i,j} \colon [0, \infty) \times \R^2 \to \R$ be a continuous function. Suppose $X$ is a stochastic process that satisfies $X_0 = x \in \R^2$ and \begin{equation*} dX^i_t = b_i(t, X_t) \, dt + \sum_{j = 1}^2 \sigma_{i, j}(t, X_t) \, dW^j_t\,. \end{equation*} Let $f \colon \R^2 \to \R$ be a $C^{2}$ function. Compute $\lim_{t \to 0} \frac{1}{t}(\E f(X_t) - f(x) )$.

Question 4

(Lévy’s theorem in $d$ dimensions) Let $M$ be a $d$-dimensional process such that:

  1. $M$ is a continuous martingale, with $M_0 = 0$.
  2. The joint quadratic variation satisfies: $\displaystyle d\jqv{M^i, M^j}_t = dt$ if $i = j$, and $d\jqv{M^i, M^j}_t = 0 \, dt$ if $i \neq j$.

Show that $M$ is a $d$-dimensional Brownian motion.