$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only if $$ \liminf_{c\rightarrow \infty}\;\mathbb{E}\left[\mathcal{E}(-N)_{T_c}1_{\{T_c<\infty\}}\right]=0.$$

Do I have to use Girsanov? How can I prove this?

Any helps are appreciated.

  • $\begingroup$ For sufficiency I would try using Doob's optional sampling theorem, given you have an uniformly integrable martingale, and the fact you have a bounded stopping time. Moreover remember that the stochastic exponential is a martingale iff $\mathbb{E}[\mathcal{E}(-N)_t]=1$ for all $t$. For necessity you can try using the limiting condition and the optional sampling theorem applied to the supermartingale $\mathcal{E}(-N)_t$. $\endgroup$ – Pasriv May 20 '16 at 8:50
  • $\begingroup$ Is it right that I get from Doob's optional stopping theorem I get $\mathbf{E}[\mathcal{E}(-N)_{T_c}]=\mathbf{E}[\mathcal{E}(-N)_0]=1$. But how do I get the $\liminf$ condtition? And how should I start with the other direction? $\endgroup$ – Mathfreak May 20 '16 at 14:08

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