Which Functions of Stopping Times are Stopping Times?

Abstract

Some functions of stopping times are necessarily stopping times, but others need not be. For example, the sum $\tau_1 + \tau_2$ of two stopping times is, while for stochastic processes in continuous time, the product $\tau_1 \cdot \tau_2$ need not be. Determined here for each positive integer $n$ are those functions $\phi$ for which $\phi(\tau)$ is a stopping time for all $n$-triples of stopping times $\tau = \tau_1, \cdots, \tau_n$.