Solution: Drift Change for a Single Brownian Motion

Exercise: Drift Change for a Single Brownian Motion

Part 1

$f(w, t) = \exp(-\theta w - \tfrac{1}{2}\theta^2 t)$ . Partial derivatives:

$f_t = -\tfrac{1}{2}\theta^2 f$
$f_w = -\theta f$
$f_{ww} = \theta^2 f$

Itô:

dZ_t = \left(f_t + \tfrac{1}{2}f_{ww}\right)dt + f_w\,dW_t = \left(-\tfrac{1}{2}\theta^2 f + \tfrac{1}{2}\theta^2 f\right)dt + (-\theta f)\,dW_t = -\theta Z_t\,dW_t.

Drift term vanishes —

Z

is a (local) martingale; since

\theta

is constant,

|\theta| \le C

trivially, so Novikov holds, and

Z

is a true

\mathbb{P}

-martingale.

Part 2

\mathbb{E}^{\mathbb{P}}[Z_T] = e^{-\theta^2 T/2}\cdot \mathbb{E}^{\mathbb{P}}[e^{-\theta W_T}] = e^{-\theta^2 T/2}\cdot e^{\theta^2 T/2} = 1. \quad \checkmark

(Used MGF of $W_T \sim \mathcal{N}(0, T)$ at $-\theta$ : $\mathbb{E}[e^{-\theta W_T}] = e^{\theta^2 T/2}$ .)

Part 3

\mathbb{Q}(\tilde W_T \le x) = \int_{-\infty}^\infty e^{-\theta w - \theta^2 T/2}\cdot \mathbf{1}_{w + \theta T \le x}\cdot \frac{1}{\sqrt{2\pi T}}e^{-w^2/(2T)}\,dw.

Indicator becomes $\mathbf{1}_{w \le x - \theta T}$ . The exponent in the integrand:

-\theta w - \tfrac{\theta^2 T}{2} - \tfrac{w^2}{2T} = -\tfrac{1}{2T}(w^2 + 2\theta T w + \theta^2 T^2) = -\tfrac{(w + \theta T)^2}{2T}.

So the integrand is $\frac{1}{\sqrt{2\pi T}}\exp(-(w + \theta T)^2/(2T))$ — density of $\mathcal{N}(-\theta T, T)$ evaluated at $w$ . Change variable $u = w + \theta T$ :

\mathbb{Q}(\tilde W_T \le x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi T}}e^{-u^2/(2T)}\,du = \Phi(x/\sqrt T).

Hence $\tilde W_T \sim \mathcal{N}(0, T)$ under $\mathbb{Q}$ . ✓

Part 4

Girsanov has shifted the mean (drift) of the process

W_t

from

0

-\theta t

under

\mathbb{Q}

— equivalently, made

\tilde W_t := W_t + \theta t

a zero-drift BM under

\mathbb{Q}

. The variance / quadratic variation

\langle \tilde W\rangle_t = t

is unchanged — Girsanov does not change volatility.

Takeaways

Doléans-Dade exponentials are the natural Radon-Nikodym derivatives for Girsanov. The form $\exp(-\int \theta\,dW - \tfrac{1}{2}\int \theta^2\,ds)$ is forced by the requirement that $Z$ be a martingale with $\mathbb{E}[Z] = 1$ .
For constant $\theta$ , Novikov holds trivially. The full machinery of Novikov is needed only for unbounded $\theta$ (stochastic volatility, interest rates, etc.).
Drift change is a change of measure; volatility is an intrinsic property. Calibrating a volatility surface to market prices works; re-calibrating the drift is always possible (it's just a choice of measure).
The sign convention $-\theta$ vs $+\theta$ depends on which direction you're going. Ask: under the new measure $\mathbb{Q}$ , what is $W_t$ ? If $\tilde W_t = W_t + \theta t$ is the $\mathbb{Q}$ -BM, then $W_t = \tilde W_t - \theta t$ has drift $-\theta$ under $\mathbb{Q}$ .