Kazamaki's condition

Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. In mathematics, this is particularly important if one wishes to apply Girsanov's theorem in order to perform a change of measure. Kazamaki's condition is more general than Novikov's condition.

Statement of Kazamaki's condition

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

References

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