Lévy's Characterisation of Brownian Motion

Girsanov's theorem — which changes the drift of a Brownian motion and underlies risk-neutral pricing — claims: if

W_t^{\mathbb{P}}

is a

\mathbb{P}

-BM and we define

W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s \, ds

, then under the new measure

\mathbb{Q}

, the process

W^{\mathbb{Q}}

is a Brownian motion. But why is

W^{\mathbb{Q}}

a BM under

\mathbb{Q}

? Checking each defining property (independence of increments, normality, continuity) directly would be laborious. The elegant answer comes from Lévy's characterisation:

A continuous local martingale $M$ starting at 0 with $[M, M]_t = t$ is a standard Brownian motion.

This is a theorem, not a definition. It says: among all processes, the condition " $M$ is a continuous local martingale with $[M,M]_t = t$ " is equivalent to " $M$ is SBM." The proof uses only characteristic functions (the Fourier side) and Itô's formula (the algebra) — no direct verification of independence or normality required.

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

State Lévy's characterisation theorem precisely (including in $d$ dimensions).
Prove the one-dimensional case via the exponential martingale argument.
Apply Lévy's theorem to verify that a Girsanov-transformed process is a BM.
Explain why continuity is essential in the theorem (the compensated Poisson process provides a counter-example for discontinuous processes).
Interpret Lévy's theorem as a uniqueness result: BM is the unique continuous local martingale with quadratic variation $t$ .

In this lesson

The Theorem
Proof (One-Dimensional Case)
Interpretation and Applications

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