FROM BOUNDARY CROSSING OF NON-RANDOM FUNCTIONS TO BOUNDARY CROSSING OF STOCHASTIC PROCESSES

Abstract

One problem of wide interest involves estimating expected crossing-times. Several tools have been developed to solve this problem beginning with the works of Wald and the theory of sequential analysis. Deriving the explicit close form solution for the expected crossing times may be difficult. In this paper, we provide a framework that can be used to estimate expected crossing times of arbitrary stochastic processes. Our key assumption is the knowledge of the average behavior of the supremum of the process. Our results include a universal sharp lower bound on the expected crossing times. Furthermore, for a wide class of time-homogeneous, Markov processes, including Bessel processes, we are able to derive an upper bound E[a(Tr)]≤2r, which implies that sup r>0|((E[a(Tr)]−r)/r)|≤1, where a(t)=E[sup tXt] with {Xt}t≥0 be a non-negative, measurable process. This inequality motivates our claim that a(t) can be viewed as a natural clock for all such processes. The cases of multidimensional processes, non-symmetric and random boundaries are handled as well. We also present applications of these bounds on renewal processes in Example 10 and other stochastic processes.