Assignment 10

In light of your midterm on 2024-04-03, this homework is not due. Many of the problems cover material on the midterm and are good practice. Some of the problems will be on your regular homework (due 2024-04-10).

Question 1

Let $b$ be a continuous adapted process and define \begin{equation} X_t = \paren[\Big]{W_t + \int_0^t b_s \, ds} \exp\paren[\Big]{ - \int_0^t b_s \, dW_s - \frac{1}{2}\int_0^t b_s^2 \, ds} \,. \end{equation} Show that $X$ is a martingale. (You may assume the required finiteness condition holds.)

Question 2

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 3

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 4

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 5

Let $W$ be a two dimensional Brownian motion and define \begin{equation*} B_t = \int_0^t \frac{W^1_s}{\abs{W_s}} \, dW^1_s + \int_0^t \frac{W^2_s}{\abs{W_s}} \, dW^2_s \,. \end{equation*} Show that $B$ is a Brownian motion.

Question 6

Consider a market with two stocks $S^1$, $S^2$ and a bank with interest rate $r$. The stock prices are modelled by \begin{equation*} dS^i_t = \alpha_i S^i_t \, dt + \sum_{j = 1}^2 \sigma_{i,j} S^i_t \, dW^j_t\,, \end{equation*} where $\alpha_i, \sigma_{i,j} \in \R$ are constants and $W$ is a two dimensional Brownian motion. If a security pays $g(S^1_T, S^2_T)$ at time $T$, show how you can price this security by solving a PDE. (You should also correctly state the boundary conditions for this PDE.)

Question 7

Let $W$ be a standard one dimensional Brownian motion and define $X = 2W_2 - W_3 - W_1$, $Y = W_3 - W_1$. Show that $X, Y$ are independent, however $E_2(XY) \neq E_2 X E_2 Y$.

Question 8

(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.