Beyond Wald's Equation and the Optional Sampling Theorem

Abstract

This paper establishes a conservation identity for mean-zero martingales stopped by extended-valued stopping times. For any mean-zero martingale $\{M_n\}$ and any extended-valued stopping time $T$ satisfying $E|M_T|I(T<\infty)<\infty$, the quantity $L\equiv E[M_T I(T<\infty)]$ exists and equals $\lim_n E[-M_n I(T>n)]$, a limit which always exists. The optional sampling theorem for stopping times and uniformly integrable martingales -- and Wald's equation for mean-zero random variables, as its i.i.d.\ specialization -- is recovered with a little extra effort, in which case the limit also vanishes. The identity itself remains in force whether or not $L=0$, and whether or not $P(T<\infty)=1$. Two corollaries and an application derived from this identity provide information on the rate of decay of the tail probability of the stopping time. Moreover, a necessary and sufficient condition is presented to characterize when $E|M_T|I(T<\infty)$ is finite. The characterization applies more generally whenever $|M_n|$ is a sequence of random variables, each having finite expectation. A third theorem provides sufficient conditions ensuring that certain exceedance-level, potentially extended-valued, stopping times are finite with probability one. It further implies that $\limsup M_n=\infty$ almost surely. We demonstrate these results through examples and explore their implications for different families of martingales. Our findings extend classical results in martingale theory and provide new insights into the behavior of stopped martingales, especially when the expected value of the stopped martingale on the set where the extended-valued stopping time $T$ is finite differs from the expected value of the martingale at time 1.