Giving Drift to Brownian Motion

There are two ways to make a fair game unfair: add a deterministic drift, or change the probabilities. For Brownian motion, these are exactly the same thing. A BM with drift $m$ under $\mathbb{P}$ is the same in law as a standard BM under the measure $\mathbb{Q}$ defined by $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_t} = e^{mB_t - m^2 t/2}$ .

This exponential martingale

M_t = e^{mB_t - m^2 t/2}

is the concrete Radon-Nikodym density from the previous note specialized to Brownian motion. Its SDE is

dM_t = m M_t \, dB_t

— it is the Doléans-Dade exponential of

mB_t

, and it is a

\mathbb{P}

-martingale (with

\mathbb{E}^{\mathbb{P}}[M_t] = 1

for all

t

). Under the new measure

\mathbb{Q}

, the original BM

B_t

acquires drift

m

and becomes a BM with drift.

In finance,

m = -(\mu-r)/\sigma

is the negative of the Sharpe ratio (or market price of risk). Under

\mathbb{Q}

, every asset earns the risk-free rate

r

, not

\mu

— and the exponential martingale is exactly the density converting between the two measures.

Learning objectives. By the end of this note you should be able to:

Define the exponential martingale $M_t = e^{mB_t - m^2 t/2}$ and state its SDE.
Verify $\mathbb{E}^{\mathbb{P}}[M_t] = 1$ (so the new measure is well-defined).
Prove $B_t$ has drift $m$ under $\mathbb{Q}$ using the conditional MGF.
Apply the measure change to compute the first-passage probability for BM with drift.
Identify the market price of risk in the GBM context.

In this lesson

The Exponential Martingale
Defining the New Measure
Brownian Motion with Drift Under $\mathbb{Q}$
Finance Application: GBM Drift Removal
First-Passage Probabilities via Measure Change

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