Novikov's condition

In probability theory, Novikov's condition is the sufficient condition for a stochastic process 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 ( X t ) 0 ≤ t ≤ T is a real valued adapted process on the probability space ( Ω , ( F t ) , P ) and ( W t ) 0 ≤ t ≤ T is an adapted Brownian motion:

If the condition

is fulfilled then the process

is a martingale under the probability measure P and the filtration F . Here E denotes the Doléans-Dade exponential.