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I'm looking for an example of two stopping times $\sigma\leq\tau$ and a martingale $M$ that is bounded in $L^{1}$ but not uniformly integrablem for which the equality $$E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$$ a.s. fails.

My idea was to use the martingale $$e^{aW_{t}-a^2t/2}$$ with $a$ not $0$ and $W$ a Brownian Motion, as this is a martingale but not uniformly integrable.

Could anyone give me an example of stopping times for which the equality fails and why?

I've been quite stuck on this, so help is much appreciated.

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If $M_t=\exp\left(aW_{t}-\frac12a^2t\right)$ with $a\gt0$, $\sigma=0$ and $\tau=\inf\{t\geqslant1\mid W_t=0\}$.

Then $\tau$ is almost surely finite because the random set $\{t\geqslant0\mid W_t=0\}$ is almost surely unbounded, $M_\sigma=1$ and $M_\tau\leqslant\exp\left(-\frac12a^2\right)$ almost surely hence $E[M_\tau\mid\mathcal F_\sigma]\lt M_\sigma$ almost surely.

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