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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

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    @Didier Piau: Here the misprint was found - http://math.stackexchange.com/questions/47103/doobs-inequality-in-probability. In fact it doesn't influence the first inequality I've written - so I've just corrected the arguments for it. In fact the inequality for non-negative supermartingales can be also derived with an optimal stopping arguments (which I eventually did once independently of the book I quoted - that's why I was surprised about your statement).2011-06-23

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Ideas coming from Skorokhod embedding theorem may help. Let $U_n=\mathrm{e}^{-rX_n}$ for a given nonnegative $r$ and assume that $(U_n)$ is a supermartingale, that is, that $E(\mathrm{e}^{-rX_{n+1}}|\mathcal{F}_n^X)\le\mathrm{e}^{-rX_n}$ (Dubins and Savage call this condition the existence of an exponential house).

Then $U_0=\mathrm{e}^{-rx}$ and the stopping time theorem applied to the first time when $U_n\ge1$ yields $\psi(x)=P(\sup_nU_n\ge1)\le\mathrm{e}^{-rx}$ for every nonnegative $x$.

This extends the result you mentioned about sums of i.i.d. increments.