Change of measure

If a measure $\nu$ has density $w=d\nu /d\mu$ with respect to $\mu$, thenThis identity is the basis of likelihood ratios and importance sampling.

For a Radon-Nikodym derivative $L_T\ge 0$ with $\mathbb{E}{L}_T=1$, its conditional-expectation process is a nonnegative martingale. If $L_T>0$ almost surely, it defines an equivalent probability measure. A stochastic exponential satisfying the Novikov condition is a common positive density process.