Let $x$ be a positive number and $X_n$ be real-valued submartingale such that $X_0 = x$. I am interested in upper bounds on probability $ \psi(x) = \mathsf{P}_x\left\{\inf\limits_{n\geq 0}X_n \leq 0\right\}. $
You are welcome to discuss this question with any assumptions you like - maybe, not too strict.
One of my ideas was the following. If $Y_n = \frac{1}{X_n}$ is a supermartingale, then Doob's inequality can be used: $ \mathsf{P}_y\left\{\sup\limits_{n\geq 0}Y_n \geq N\right\}\leq \frac{y}{N}, $ but here we need to have $Y$ a non-negative process, which is not our case.
Edited: The formulation of Doob's inequality [Shiryaev: Probability, p. 492]. If $Z$ is a supermartingale then $ \mathsf P\{\sup\limits_{n\geq 0} |Z_n|\geq \delta\}\leq\frac{C}{\delta}\sup\limits_{n\geq 0} \mathsf E[|Z_n|] $ for some $C\leq 3$.
If $Z$ is a non-negative supermartingale then $C$ can be taken equal to $1$ and the expectation on the right-hand side attains its maximum at the time moment $n=0$. That leads to the inequality I've formulated in the first version of the question.
With regards to random walks: I am not quite sure in you statement since for the Lundberg inequality there exists such a bound. More precisely, if random walk is given by $ X_n = X_{n-1}+A_n $ where $A_n$ are i.i.d. such that $\mathsf E A_n >0$ and there exists $r>0$ such that $\mathsf E\mathrm e^{-rA_1} = 1$, then $\psi(x)\leq \mathrm e^{-rx}$.
I've just eventually faced such inequality from Financial Mathematics and wonder if there are known bounds for supermartingales (since $X_n$ is a supermartingale in the latter example) - they have not to be exponential of course.
P.S. I put this question also here: https://mathoverflow.net/questions/68509/bounds-for-submartingale