Novikov's condition

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Novikov's condition is sufficient for application of Girsanov's theorem to certain classes of stochastic processes.

Assume that X_t,\, 0\leq t\leq T is a real valued adapted process in the probability space \left (\Omega,\mathbb{P,}\mathbb{F}\right) and W_t,\, 0\leq t\leq T is a Brownian Motion with respect to the probability measure \mathbb{P}.

If the condition

\mathbb{E}\left[e^{\frac12\int_0^T|X_t|^2\,dt} \right]<\infty

is fulfilled then the process

\ Z_t \ = e^{ \int_0^t X_s\, dW_s -\frac{1}{2}\int_0^t X_s^2\, ds},\quad 0\leq t\leq T

is a martingale under the probability measure \mathbb{P} and the filtration \mathbb{F}.

[edit] External links

Comments on Girsanov's Theorem by H. E. Krogstad, IMF 2003[1]