Girsanov theorem

In probability theory, the Girsanov theorem (named after Igor Vladimirovich Girsanov) describes how the dynamics of change when the original measure is changed to an equivalent probability measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying.

Results of this type were first proved by Cameron–Martin in the 1940s and by Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).

Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.

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 ${\displaystyle \{W_{t}\}}$ be a Wiener process on the Wiener probability space ${\displaystyle \{\Omega ,{\mathcal {F}},P\}}$. Let ${\displaystyle X_{t}}$ be a measurable process adapted to the natural filtration of the Wiener process ${\displaystyle \{{\mathcal {F}}_{t}^{W}\}}$ with ${\displaystyle X_{0}=0}$.

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