Assignment 4

Assigned 2024-02-07, due 2024-02-14 at noon on Gradescope

Question 1

  1. If $X \sim N(0, \sigma^2)$ find the characteristic function of $X^2$. (Recall if $\alpha \in \C$, with $\Re(\alpha) > 0$, then $\int_{-\infty}^\infty e^{-\alpha x^2} \, dx = \sqrt{\pi / \alpha}$.)

  2. Let $T > 0$, and $P$ be the uniform partition $P = \set{0 = t_0, t_1 = T/N, t_2 = 2T/N, \dots, t_{N} = T}$. Let $\xi_j = (W_{t_{j+1}} - W_{t_j})^2 - (t_{j+1} - t_j)$, and $S_N = \sum_0^{N-1} \xi_j$. Find $\varphi_{S_N}(\lambda) = \E e^{i \lambda S_N}$, and $\lim_{N \to \infty} \varphi_{S_N}(\lambda)$.

    [This is another way of computing the quadratic variation of Brownian motion.]

Question 2 (withdrawn)

I did this in class on 2024-02-07 (assuming $B$ is differentiable). The same proof works for this question. As a replacement, do question 2’ instead.

  1. Let $B$ be a continuous process with finite first variation. Show $\qv{B}_T = 0$.

  2. Use the previous part to show that $V_{[0, T]}(W) = \infty$.

Question 2’

Let $B, M$ be continuous adapted processes such that $B$ has finite first variation. Let $X_t = X_0 + B_t + M_t$. Show that $\qv{X}_t = \qv{M}_t$.

Question 3

  1. Find functions $f, g$ so that $\displaystyle W_t^4 = \int_0^t f(s, W_s) \, ds + \int_0^t g(s, W_s) \, dW_s\,. $

  2. Compute $\E W_t^4$ explicitly as a function of $t$.

  3. Find a function $h$ so that $\displaystyle \qv{W^4}_t = \int_0^t h(s, W_s) \, ds$.

Question 4

Compute $\displaystyle \E \brak[\Big]{ \paren[\Big]{\int_0^t e^{-2 s} \, dW_s}^4}$.

Question 5

Determine whether the following identities are true or false, and justify your answer.

  1. $\displaystyle e^{2t} \sin(2 W_t) = 2 \int_0^t e^{2s} \cos(2 W_s) \, dW_s$.

  2. $\displaystyle \abs{W_t} = \int_0^t \sign(W_s) \, dW_s$.

    [Recall $\sign(x) = 1$ if $x > 0$, $\sign(x) = -1$ if $x < 0$ and $\sign(x) = 0$ if $x = 0$.]