Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $X_0 =0$. The stochastic exponential of $X$ is $\mathcal{E}(X) := \exp(X-\frac{1}{2}\langle X \rangle)$. There are several conditions to guarantee that $\mathcal{E}(X)$ is a true martingale.

  1. Novikov proved $\mathbb{E}[\exp(\frac{1}{2}\langle X \rangle_t)]<\infty, t \geq 0$ is sufficient.
  2. Kazamaki and Sekiguchi proved, $\sup_\tau \mathbb{E}[|X_\infty-X\tau||\mathcal{F}_\tau]<\infty$, where the $\sup$ operates on all stopping times $\tau \geq 0$, is also sufficient. It's also called the BMO condition.

I want to know the comparability between them. I find that 2 does not imply 1, from the Example 4 in the paper, on the transformation of some classes of martingales by a change of law, N. Kazamaki and T.Sekiguchi, 1979.

Now, does 1 imply 2? Any proofs or counter-examples? Many thanks in advance!

  • $\begingroup$ maybe Burkholder-Davis-Gundy and then Jensen's . $\endgroup$ – OOESCoupling Apr 11 '18 at 18:12

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