I'm having a hard time solving this problem. Would appreciate any hint! (and special thanks to Byron Schmuland for answering my previous 2 questions. This 3rd question is different.)

Let $e_t$: $e_1$,$e_2$,$\ldots$ be i.i.d. normal mean 0 and variance 1. Let $X_t := e_1+\ldots+e_t$, for $t=1,2,\ldots$ and $X_0 := 0$. (So we have a discrete-time random walk whose steps are i.i.d. $\mathcal{N}(0,1)$)

Define passage time $T^\ast := \inf\{t>0 : X_t < -t\}$.

What is the CDF of $T^\ast$, i.e. what is $\operatorname{Prob}(T^\ast \leqslant t)$ ?

(Equivalently: let $e^\prime_t \sim \mathcal{N}(1,1)$, and $T^\ast$ above is simply $\inf\{t>0 : X^\prime_t < 0\}$.) Is there a closed form solution? (Remember we have zero correlation, just non-zero mean)

Some reference: http://dl.dropbox.com/u/4260685/orthant.pdf


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    $\begingroup$ All the questions you have asked so far seem to be from the same subject. Is this homework? $\endgroup$ – Aryabhata Sep 10 '10 at 2:08