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Suppose I have a simple continuous time random walk starting at $0$ at time $0$ with Poisson transition rate 1 and probability $p$ the jump is $+1$ and probability $1-p$ thejump is $-1$. Suppose $p < 1/2$ so the drift is negative. I am interested in $q_t$, the probability that the random walk will be nonnegative at some time after t. Is there a nice closed form expression ?

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  • $\begingroup$ The question is not the probability being nonnegative AT a particular time t but AT or AFTER a particular time t. $\endgroup$
    – David T.
    Apr 19, 2020 at 5:07
  • $\begingroup$ Ah, sorry, I misread. $\endgroup$ Apr 19, 2020 at 13:12

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