Girsanov theorem

In probability theory, the Girsanov theorem (named after Igor Vladimirovich Girsanov) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.:607 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 share 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 instrument.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).If X is a continuous process and W is Brownian motion under measure P thenIn finance, Girsanov theorem is used each time one needs to derive an asset's or rate's dynamics under a new probability measure. The most well known case is moving from historic measure P to risk neutral measure Q which is done—in Black–Scholes model—via Radon–Nikodym derivative: