Randomized Dimension Reduction for Monte Carlo Simulations蒙特卡洛模拟的随机化降维

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1、RandomizedDimensionReductionforMonteCarloSimulationsNabilKahal´e∗December14,2017AbstractWepresentanewunbiasedalgorithmthatestimatestheexpectedvalueoff(U)viaMonteCarlosimulation,whereUisavectorofdindependentrandomvariables,andfisafunctionofdvariables.Weassumethatfdoesnotdependequallyonallitsarg

2、uments.Undercertainconditionsweprovethat,forthesamecomputationalcost,thevarianceofourestimatorislowerthanthevarianceofthestandardMonteCarloestimatorbyafactoroforderd.Ourmethodcanbeusedtoobtainalow-varianceunbiasedestimatorfortheexpectationofafunctionofthestateofaMarkovchainatagiventime-step.We

3、studyapplicationstovolatilityforecastingandtime-varyingqueues.NumericalexperimentsshowthatouralgorithmdramaticallyimprovesuponthestandardMonteCarlomethodforlargevaluesofd,andishighlyresilienttodiscontinuities.Keywords:dimensionreduction;variancereduction;effectivedimension;Markovchains;MonteCar

4、lomethods1IntroductionMarkovchainsariseinavarietyoffieldssuchasfinance,queuingtheory,andsocialnetworks.Whilemuchresearchhasbeendevotedtothestudyofsteady-statesofMarkovchains,sev-eralpracticalapplicationsrelyonthetransientbehaviorofMarkovchains.Forexample,thevolatilityofanindexcanbemodelledasaMar

5、kovchainusingtheGARCHmodel(Hull2014,Ch.23).Financialinstitutionsconductingstresstestsmayneedtoestimatetheprobabilitythatthevolatilityexceedsagivenlevelinafewyearsfromnow.Also,duetothenatureofhumanactivity,queuingsystemsinareassuchashealth-care,manufacturing,telecommunica-tionandtransportationn

6、etworks,haveoftentime-varyingfeaturesanddonothaveasteady-state.Forinstance,empiricaldatashowsignificantdailyvariationintrafficinwide-areanet-works(Paxson1994,Thompson,MillerandWilder1997)andvehicularflowonroads(Nagel,WagnerandWoesler2003).Estimatingtheexpecteddelayofpacketsinawide-areanetworkatarX

7、iv:1708.07466v2[stat.CO]13Dec2017aspecifictimeoftheday(12pm,say)couldbeusedtodimensionsuchnetworks.Similarly,esti-matingthevelocityofcarsinaregionat6pmcouldbeusedtodesigntransportationnetworks.Inthesamevein,considertheproblemofestimating