Change of Measure for Jump Processes

Girsanov's theorem changes the drift of a Brownian motion by shifting the measure. The analogous operation for a compound Poisson process (CPP) changes the Lévy measure $\mu$ to a new measure $\nu$: under the new measure $\mathbb{Q}$, the CPP still has independent stationary increments, but with jump-size distribution $\nu^\#$ instead of $\mu^\#$.

The key contrast with Brownian motion: any pair $(\mu, \nu)$ with $\mu \sim \nu$ (mutually absolutely continuous jump measures) can be realised by a measure change. This means there are infinitely many equivalent martingale measures for a jump model — the market is incomplete. The Esscher transform is a canonical parametric family of such measure changes, and the Merton (1976) model uses a specific choice to pin down a particular risk-neutral measure.

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