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1 Stopping Times - Columbia University

Copyrightc 2009 by Karl Sigman1 Stopping Stopping Times : DefinitionGiven a stochastic processX={Xn:n 0}, arandom time is a discrete random variableon the same probability space asX, taking values in the time set IN ={0,1,2,..}.X denotesthe state at the random time ; if =n, thenX =Xn. If we were to observe the valuesX0,X1,.., sequentially in time and then stop doing so right after some timen, basing ourdecision to stop on (at most) only what we have seen thus far, then we have the essence ofastopping time. The basic feature is that we do not know the future hence can t base ourdecision to stop now on knowing the future. To make this precise, let thetotal informationknown up to timen, for any givenn 0, be defined as all the information (events) containedin{X0.}

1. (First passage/hitting times/Gambler’s ruin problem:) Suppose that X has a discrete state space and let ibe a xed state. Let ˝= minfn 0 : X n= ig: This is called the rst passage time of the process into state i. Also called the hitting time of the process to state i. More generally we can let Abe a collection of states such

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