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 .
Read more about this topic: Girsanov Theorem
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