On Wald’s Equation and First Exit Times for Randomly Stopped Processes with Independent Increments

Abstract

The general theme of this paper is the study of the properties of randomly stopped processes with independent increments with CADLAG (right continuous with left limits) paths with values in general Banach spaces. First we present a general version of (de la Peña and Eisenbaum (1997))(1) which extends the Burkholder-Gundy inequality for randomly stopped Brownian motion to general Banach spaces and processes with CADLAG paths and independent increments. We continue by providing a proof (from first principles) of the upper bound of (1) in the case the process has continuous paths. We apply our results to extend Wald’s equation to this type of process and to obtain information on first exit times of the processes involved. An example involving the joint behavior of several stocks in a market underlines the importance of the fact that the results hold in general Banach spaces. This fact is also used to obtain bounds for the L P-norms of randomly stopped Bessel and related processes. The issue of constants in exponential bounds is also addressed.KeywordsBrownian MotionMoment Generate FunctionExit TimeStandard Brownian MotionContinuous PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.