A Note on the Variance of a Stopping Time

Abstract

Let $\{S_n = \sum^n_1 X_i\}_{n\geq 0}$ be a random walk with positive drift $\mu = EX_1 > 0$ and finite variance $\sigma^2 = \operatorname{Var} X_1$. Let $\tau(b) = \inf\{n \geq 1: S_n > b\}, R_b = S_{\tau(b)} - b, M = \min_{n\geq 0} S_n, \tau^+ = \tau(0)$ and $H = S_\tau +$. Lai and Siegmund show that $\operatorname{Var} \tau(b) = b\sigma^2/\mu^3 + K/\mu^2 + o(1)$ as $b \rightarrow \infty$, but give an unpleasant expression for the constant $K$. Using the identity $\int Eh(R_{-y}) dP(M \leq y) = E^+_\tau h(H)/E\tau^+$, the expression for $K$ can be simplified to a form that depends only on moments of ladder variables.