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 [math]\displaystyle{ M = (M_t)_{t \ge 0} }[/math] be a continuous local martingale with respect to a right-continuous filtration [math]\displaystyle{ (\mathcal{F}_t)_{t \ge 0} }[/math]. If [math]\displaystyle{ (\exp(M_t/2))_{t \ge 0} }[/math] 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.