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Suppose that $\{M_n\}$ is a submartingale that is bounded and we have that $\nu, \tau$ are bounded stopping times and that $\nu \leq \tau$. I would like to show that $\mathbb{E}(M_\nu) \leq \mathbb{E}(M_\tau)$. I believe that as long as I can create a relation between $M_\tau$ and $M_\nu$ in terms of a new martingale, and show that its positive, I am done. Does anyone have any hints how I can approach this?

BCLC
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user136503
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    May you use the Optional Sampling theorem for martingales? If yes, then use Doob's decomposition

    $$M_n=M_0+V_n+A_n,$$

    where $V_n$ is a UI martingale and $A_n$ is an increasing predictable process.

     –  Mar 05 '16 at 01:26
  • could you elaborate how you might do this? thanks – user136503 Mar 05 '16 at 06:14
  • @d.k.o. Can we use $n \to \infty$ in Problem 2 here? – BCLC Mar 05 '16 at 15:21

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