# Slight variant of Novikov condition

by user6247850   Last Updated August 14, 2019 15:20 PM

In a paper I found that presents an alternate proof of Novikov's condition, they use the fact that for a continuous local martingale $$M$$ if there exists $$\varepsilon > 0$$ such that $$\mathbb{E}[e^{\frac{(1+\varepsilon)}{2}\langle M,M \rangle_\infty}]<\infty$$ then $$\mathbb{E}[e^{M_\infty - \frac 12 \langle M,M\rangle_\infty}]=1$$ without a proof. This fact obviously follows immediately from Novikov's condition, but since they're using it to prove Novikov's condition there must be a way of proving it independently. Since they don't present a proof it seems like it must be a standard result, but I haven't been able to find a proof of it anywhere. Does anyone know where this result comes from or how to prove it?

The paper is "A simple proof a result of A. Novikov" by N.V. Krylov.

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