# Ruin-time in Gambler's Ruins

Could someone give me some tips?

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

Define first-hitting time $T_0 := \inf\{t>0 : X_t <0\}$.

What is the CDF of $T_0$, i.e. what is $\mathbb{P}(T_0 \leq t)$?

And second (probably more difficult) question: what is $\mathbb{E}( X_t \mid T_0 > t )$?

Thank you.

You are looking at a Brownian motion, evaluated at integer times $t$. Then defining $$T_0=\inf(t > 0, t\in {\mathbb Z}: B_t < 0)$$ and integer $t > 0$ we have $$P(T_0 > t)=P(B_1 > 0,B_2 > 0,\dots, B_t > 0)={2t\choose t}/4^t.$$
The key insight is that you should not to try to solve the probability exactly, but rather set up a recursion that the probabilities solve. Setting $p(t)=P(T_0>t)$ for integer $t\geq 0$, you can show that for any $t$ we have $\sum_{j=0}^t p(t-j)p(j)=1$.
The probabilistic justification for the recursion formula is that every path of a diffuse, symmetric, discrete-time random walk can be broken into two positive paths at the time $j$ where it achieves its minimum value.