Exercise: Drift Change for a Single Brownian Motion

Prerequisites: Girsanov's Theorem, Itô's Lemma

Problem

Let $W_t$ be a standard Brownian motion under $\mathbb{P}$ on $[0, T]$. Consider a constant drift $\theta \in \mathbb{R}$ and the Doléans-Dade exponential

1. Apply Itô's lemma to $f(W_t, t) = \exp(-\theta W_t - \tfrac{1}{2}\theta^2 t)$ and derive the SDE $dZ_t = -\theta Z_t\,dW_t.$ Confirm that $Z_t$ is a $\mathbb{P}$-martingale (no drift term).

2. Verify $\mathbb{E}^{\mathbb{P}}[Z_T] = 1$ directly. (Hint: use the MGF of $W_T \sim \mathcal{N}(0, T)$.)

3. Define $\mathbb{Q}$ by $d\mathbb{Q}/d\mathbb{P} = Z_T$, and let $\tilde W_t := W_t + \theta t$. Compute the $\mathbb{Q}$-distribution of $\tilde W_T$ by: $\mathbb{Q}(\tilde W_T \le x) = \mathbb{E}^{\mathbb{P}}[Z_T\cdot \mathbf{1}_{W_T + \theta T \le x}].$ Change variables to express this as an integral of the $\mathcal{N}(0, T)$ density against an exponential factor, and recognise the result as $\Phi(x/\sqrt T)$ — i.e. $\tilde W_T \sim \mathcal{N}(0, T)$ under $\mathbb{Q}$. Hence $\tilde W_T$ has the distribution of standard BM at time $T$.

4. Explain in two sentences what Girsanov has done: what changed (drift) and what stayed the same (volatility / quadratic variation).

Hint

For part 3, the integrand becomes $\frac{1}{\sqrt{2\pi T}}\exp(-(t + \theta T)^2/(2T)) - \ldots$ (a Gaussian density shifted in mean). Completing the square is the key algebraic step.