文档介绍：UTLogo100Stochastic ProgrammingSchool of Management, UCASRenewal ProcessKai YaoSchool of Management, UCASKai ******@ ProgrammingSchool of Management, UCASStochastic ProcessDe?nitionLet T be a totally ordered set (. time) and let(?,A,Pr)be a probabilityspace. A stochastic process is a function Xt(ω)from T×(?,A,Pr)to theset of real numbers such that{Xt∈B}={ω∈?|Xt(ω)∈B}is an event for any Borel set B at each time stochastic process is a spectrum of random variables indexed by stochastic process is a family of “functions of timet”.Kai ******@ ProgrammingSchool of Management, UCASIndependent Increment ProcessDe?nitionA stochastic process Xtis said to have independent increments ifXt0,Xt1?Xt0,Xt2?Xt1,· · ·,Xtk?Xtk?1are independent random variables where t0is the initial time and t1,t2,· · ·,tkare any times with t0<t1<· · ·< ******@ ProgrammingSchool of Management, UCASStationary Increment ProcessDe?nitionA stochastic process Xtis said to havestationary incrementsif, for anygiven t>0, the incrementsXs+t?Xsare identically distributed randomvariables for all s> ******@ ProgrammingSchool of Management, UCASRenewal ProcessDe?nitionLetξ1, ξ2,· · ·be a sequence of positive iid random variables. De?ne S0=0and Sn=ξ1+ξ2+· · ·+ξnfor n≥1. Then the stochastic processNt=maxn≥0nn???Sn≤tois called a renewal ******@ ProgrammingSchool of Management, UCASRenewal TheoremTheoremLet Ntbe a renewal process with random interarrival timesξ1, ξ2,· · ·If thoseinterarrival times have mon probability distributionΦ, thenlimt→+∞Ntt=1E[ξ1],→+∞E[Nt]t=1E[ξ1].Kai ******@ ProgrammingSchool of Management, UCASSimulation of Renewal ProcessHowto obtain the probability distribution of a renewal process at some giventimet?Step 1: Generating a sequence of random numbers according to theprobability di