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
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$\begingroup$ Ah, sorry, I misread. $\endgroup$– Nate EldredgeApr 19, 2020 at 13:12
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