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It's well known that a local martingale of is a uniformly martingale if and only if it is of class D. I want to show the following: Let $L$ be a continuous local martingale, null at zero such that $\langle L\rangle $ is uniformly bounded. Then I want to prove that $\{\mathcal{E}(L)_\tau;\tau \mbox{ stopping time}\}$ is bounded in $L^p$ for every $p\in (1,\infty)$ and from there I should deduce that $\mathcal{E}(L)$ is a uniformly integrable martingale. With $\mathcal{E}(L)$ I denote the stochastic exponential.

I think the proof is straight forward. I want to use the boundedness of $\langle L\rangle $ to show the boundedness in $L^p$ of the family $\{\mathcal{E}(L)_\tau;\tau \mbox{ stopping time}\}$, i.e. I would like to have an inequality like $E[(\mathcal{E}(L)_\tau)^p] \le E[(\exp (\langle L \rangle ))^p]\le c$. From boundedness in $L^p$, with $p\in (1,\infty)$ we clearly deduce uniformly integrability of $\{\mathcal{E}(L)_\tau;\tau \mbox{ stopping time}\}$. Hence the result at the beginning solves the problem. However I'm stuck at the desired inequality. This was my thought how to solve this! Maybe this is not the right approach. However, any help would be appreciated. Thanks in advance!

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