Novikov's condition
Novikov's condition is a simple test that ensures a certain exponential expression is a martingale, which you need for Girsanov's theorem to work with Brownian motion. Suppose X_t is a real, adapted process on the time interval [0, T], and W_t is a Brownian motion on the same probability space. If Novikov's condition holds, namely
E[ exp( (1/2) ∫_0^T X_s^2 ds ) ] < ∞,
then the Doléans-Dade exponential
E_t = exp( ∫_0^t X_s dW_s − (1/2) ∫_0^t X_s^2 ds )
is a martingale. This allows you to define a new probability measure P* by dP*/dP|_{F_t} = E_t. Under P*, the process W_t* = W_t − ∫_0^t X_s ds is a Brownian motion, and expectations under P* can be computed using Girsanov's theorem.
Novikov's condition was proposed by Alexander Novikov. While there are other criteria to guarantee the Radon–Nikodym derivative is a martingale (such as Kazamaki's condition), Novikov's condition is the best-known and most commonly used.
This page was last edited on 2 February 2026, at 08:02 (CET).