Girsanov transformations describe how Brownian motion and, more generally, local martingales behave under changes of the underlying probability measure. Let us start with a much simpler identity applying to normal random variables. Suppose that *X* and are jointly normal random variables defined on a probability space . Then is a positive random variable with expectation 1, and a new measure can be defined by for all sets . Writing for expectation under the new measure, then for all bounded random variables *Z*. The expectation of a bounded measurable function of *Y* under the new measure is

(1) |

where is the covariance. This is a vector whose i’th component is the covariance . So, *Y* has the same distribution under as has under . That is, when changing to the new measure, *Y* remains jointly normal with the same covariance matrix, but its mean increases by . Equation (1) follows from a straightforward calculation of the characteristic function of *Y* with respect to both and .

Now consider a standard Brownian motion *B* and fix a time and a constant . Then, for all times , the covariance of and is . Applying (1) to the measure shows that

where is a standard Brownian motion under . Under the new measure, *B* has gained a constant drift of over the interval . Such transformations are widely applied in finance. For example, in the Black-Scholes model of option pricing it is common to work under a *risk-neutral* measure, which transforms the drift of a financial asset to be the risk-free rate of return. Girsanov transformations extend this idea to much more general changes of measure, and to arbitrary local martingales. However, as shown below, the strongest results are obtained for Brownian motion which, under a change of measure, just gains a stochastic drift term. (more…)