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 $M=(M_{t})_{t\geq 0}$ be a continuous local martingale with respect to a right-continuous filtration $({\mathcal {F}}_{t})_{t\geq 0}$. If $(\exp(M_{t}/2))_{t\geq 0}$ 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.
• Protter, Philip E. (2004). Stochastic Integration and Differential Equations (2nd ed.). Berlin: Springer. ISBN 978-3-540-00313-7.{{cite book}}: CS1 maint: publisher location (link)
• Kazamaki, N. (1977). "On a problem of Girsanov". Tohoku Math. J. 29 (4): 597–600.