 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.m. musiela, m. rutkowski: martingale methods in financial modelling. 2nd ed. new york : springer-verlag, 2004. print. 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. ==history== results of this type were first proved by cameronâmartin 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). ==significance== 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. ==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 \{w_t\} be a wiener process on the wiener probability space \{\omega,\mathcal{f},p\}. let x_t be a measurable process adapted to the natural filtration of the wiener process \{\mathcal{f}^w_t\}. given an adapted process x_t with x_0 = 0 define :z_t=\mathcal{e} (x)_t,\, where \mathcal{e}(x) is the stochastic exponential (or dolÃ©ans exponential) of x with respect to w, i.e. :\mathcal{e}(x)_t=\exp \left ( x_t - \frac{1}{2} [x]_t \right ), where [x]_t is a quadratic variation for x_t. thus z_t is a strictly positive local martingale, and a probability measure q can be defined on \{\omega,\mathcal{f}\} such that we have radonânikodym derivative :\frac{d q}{d p} |_{\mathcal{f}_t} = z_t = \mathcal{e} (x )_t then for each t the measure q restricted to the unaugmented sigma fields \mathcal
Probability_theorems Stochastic_processes