process Z which solves the stochastic differential equation The measure Q constructed above is not equivalent to P on as this would only be the case if the Radon–Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not. On the other hand as long as Novikov's condition is satisfied the measures are equivalent on . Additionally, then combining this above observation in this case, we see that the process for is a Q Brownian motion. This was Igor Girsanov's original formulation of the above theorem. Application to finance This theorem can be used to show in the Black–Scholes model the unique risk-neutral measure, i.e. the measure in which the fair value of a derivative is the discounted expected value, Q, is specified by Application to Langevin equations Another application of this theorem, also given in the original paper of Igor Girsanov, is for stochastic differential equations. Specifically, let us consider the equation where denotes a Brownian motion. Here and are fixed deterministic functions. We assume that this equation has a unique strong solution on . In this case Girsanov's theorem may be used to compute functionals of directly in terms a related functional for Brownian motion. More specifically, we have for any bounded functional on continuous functions that This follows by applying Girsanov's theorem, and the above observation, to the martingale process In particular, we note that with the notation above, the process is a Q Brownian motion. Rewriting this in differential form as we see that the law of under Q solves the equation defining , as is a Q Brownian motion. In particular, we see that the right-hand side may be written as , where Q is the measure taken with respect to the process Y, so the result now is just the statement of Girsanov's theorem. A more general form of this application is that if both admit unique strong solutions on , then for any bounded functional on , we have that See also References External links Notes on Stochastic Calculus which contain a simple outline proof of Girsanov's theorem. Stochastic processes Mathematical theorems Mathematical finance