Say $M_t$ is a discrete time supermartingale on $\mathbb{R}$, with $M_0 > 0$ and $\mathbb{E}[M_t | \mathscr{F}_{t-1}] \leq M_{t-1} - \alpha$ for some $\alpha > 0$. Let $\tau := \min(t : M_t <0)$. We want to know $\mathbb{E}[\tau]$.
I know how to handle this when $M_t$ is something like a Brownian motion with drift, but I wonder if we can say anything in the discrete time case?
Thank you --
You'd have to make the question more specific for it to have an answer.
If all you know is that $\mathbb E[M_t | \mathscr F_{t-1}] \leq M_{t-1} - \alpha$, then this is too permissive to be able to say anything concrete. (Consider two deterministic walks that each start at $10$; one takes a step downward at each time increment, and the other takes a half-step downward at each time increment. Both satisfy the supermartingale condition with $\alpha = 0.1$, for instance.)
The inequality is the issue here more than the $\alpha$. The reason that you're able to do anything concrete with Brownian motion with drift isn't that it's a sub/super-martingale, it's that you can add something convenient to it to get it to become a martingale. That's the level of specificity you'd want to have here. If your condition was instead that $\mathbb E[M_t | \mathscr F_{t-1}] \fbox = M_{t-1} - \alpha$ (for all $t$), then you'd have a situation where $M_t + t \alpha$ was a genuine martingale, and you could use martingale tactics like the Optional Stopping Theorem to proceed. Without that level of specificity, the best you'll be able to hope for is bounds on things like $\mathbb E[\tau]$.
Edited to add: Even in the case where you can specify the expected drift, you might have to settle for bounds. You can get much cleaner results in cases when you know $M_{\tau} = -1$ almost surely (for instance) rather than just knowing that $M_{\tau} < 0$.