Girsanov Theorem - Statement of Theorem

Statement of Theorem

We state the theorem first for the special case when the underlying stochastic process is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black-Scholes model and in many other models (e.g. all continuous models).

Let be a Wiener process on the Wiener probability space . Let be a measurable process adapted to the natural filtration of the Wiener process .

Given an adapted process with define

where is the stochastic exponential (or Doléans exponential) of X with respect to W, i.e.

Thus is a strictly positive local martingale, and a probability measure Q can be defined on such that we have Radon–Nikodym derivative

Then for each t the measure Q restricted to the unaugmented sigma fields is equivalent to P restricted to

Furthermore if Y is a local martingale under P then the process

is a Q local martingale on the filtered probability space .