moments of absorption time of simple random walk with 1 barrier.

Question

let $S$ be a simple random walk starting at 0 with probability $p$ of going to the right, and $q$ of going to the left and with drift to the right $p>q$. Define $T=\inf \{n>0: S_n = 1\}.$ The derivation of the first two moments are often seen in the literature and can also be found here. Does $T$ have all moments? is $ \mathbb{E}\exp(cT)<\infty$ for some $t>0$ ? Any reference is appricated

Answer 1

For the simple random walk you can find an explicit distribution of $T$ in volume 1 of Feller's 'An introduction to probability theory and its applications'. The distribution of $T$ can be derived from the ballot theorem about the number of paths $\{0=s_0, s_1,..., s_n = 1 \}$ from $(0,0)$ to $(n,1)$ such that $|s_{i+1} - s_i| = 1$ and $s_i \leq 0$. By Theorem 1* (III.2, page 71 in the second edition) this number is

$$ \frac{1}{n} {{n}\choose{\frac{n+1}{2}}}. $$ Hence $\mathbb{P}\{ T = n \} = \frac{1}{n} {{n}\choose{\frac{n+1}{2}}} p^{\frac{n+1}{2}}q^{\frac{n-1}{2}}$ for odd $n$, and $\mathbb{P}\{ T = n \} = 0$ for even $n$. In particular, $T$ has an exponential moment.

Alternatively look in the index for first passage in Bernoulli trials and random walks or ruin probabilities.