Novikov's condition

In probability theory, Novikov's condition is the sufficient condition for a which takes the form of the Radon-Nikodym derivative in Girsanov's theorem to be a martingale. If satisfied together with other conditions, Girsanov's theorem may be applied to a Brownian motion stochastic process to change from the original measure to the new measure defined by the Radon-Nikodym derivative.

This condition was suggested and proved by Alexander Novikov. There are other results which may be used to show that the Radon-Nikodym derivative is a martingale, such as the more general criterion Kazamaki's condition, however Novikov's condition is the most well-known result.

Assume that ${\displaystyle (X_{t})_{0\leq t\leq T}}$ is a real valued adapted process on the probability space ${\displaystyle \left(\Omega ,({\mathcal {F}}_{t}),\mathbb {P} \right)}$ and ${\displaystyle (W_{t})_{0\leq t\leq T}}$ is an adapted Brownian motion:

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