Kazamaki's condition

In mathematics Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's theorem is to be applied to perform a change of measure. Kazamaki's condition is more general than Novikov's condition.

Statement of Kazamaki's condition

Let ${\displaystyle M=(M_{t})_{t\geq 0}}$ be a continuous local martingale with respect to a right-continuous filtration ${\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}}$. If ${\displaystyle (\exp(M_{t}/2))_{t\geq 0}}$ is a uniformly integrable submartingale, then the Doléans-Dade exponential Ɛ(M) of M is a uniformly integrable martingale.

References

• Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian motion. New York: Springer-Verlag. ISBN 3-540-64325-7.