Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}$. We can interpret $\tau_{s,a,b}$ as the first time after time $s$ that the process hits $a$ or $b$.

Suppose that for all $s,a,b$ we have:

$$\mathbb{E}[X_{\tau_{s,a,b}}|\mathcal{F}_s]\leq X_s$$

Then is $X$ necessarily a local supermartingale?

At first I thought that perhaps $X$ was necessarily a supermartingale, but there are local supermartingales with this property. For example,

$$ X_t = \begin{cases} W_{\min(\frac{t}{1-t},T)} &\text{for } 0 \le t < 1\\ 1 &\text{for } 1 \le t < \infty, \end{cases}$$

where $(W_t)_{t\geq 0}$ is a Weiner process and $T=\inf\{t\geq 0:W_t=1\}$.

A discrete time analogue of this statement can be proved by induction. I'm unsure as to whether that result is useful here.

This would be an interesting result, since it would facilitate showing that certain processes are supermartingales.

Thank you.


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