Stopping a Sum During a Success Run

Abstract

Let $\{Z_i\}$ be i.i.d., let $\{\varepsilon_i\}$ be i.i.d. Bernoulli, independent of $\{Z_i\}$, let $T_0 = z$ and $T_n = \varepsilon_n(T_{n-1} + Z_n)$ for $n \geqq 1$. Under a moment condition, optimal stopping rules are found for stopping $T_n - nc$ where $c > 0$ (the cost model), and for stopping $\beta^nT_n$ where $0 < \beta < 1$ (the discount model). Special cases are treated in detail. The cost model generalizes results of N. Starr, and the discount model generalizes results of Dubins and Teicher.