Exercise: Apply Itô — $d(e^{W_t})$ and $\mathbb{E}[e^{W_t}]$

Prerequisites: Itô's Lemma, Brownian Motion, Expectation and Variance

Problem

Let $W_t$ be a standard Brownian motion and set $Y_t = e^{W_t}$.

1. Apply Itô's lemma to $V(W) = e^W$ and write down the SDE satisfied by $Y_t$. Identify the drift and the diffusion separately, and read off the Itô correction.

2. Take expectations of the SDE in part 1. Since the $dW_t$ term is a mean-zero Itô integral, you obtain an ODE for $m(t) := \mathbb{E}[Y_t]$. Solve it with initial condition $m(0) = 1$.

3. Verify your answer by computing $\mathbb{E}[e^{W_t}]$ directly from the fact that $W_t \sim \mathcal{N}(0, t)$ (use the moment-generating function of a normal).

4. Is $Y_t = e^{W_t}$ a martingale? If not, write down the simplest rescaling $\tilde Y_t = e^{\alpha W_t + \beta t}$ that is a martingale, and explain which role Itô's lemma played in the correction.

Hint

For part 1: $V'(W) = e^W$ and $V''(W) = e^W$ — both equal to $V(W)$ itself, which makes the algebra very clean.