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Stochastic Calculus, Filtering, and Stochastic Control

StochasticCalculus, Filtering, andStochast icControlLecture Notes(Thisversion:May29,2007)RamonvanHan delSpring2007 PrefaceTheselecturenoteswerewrittenforth ecourseACM217:AdvancedTopicsin Stochas-ticAnalysisat Caltech;thisyear(2007), anintroductorycourseonthesubject,andasth ereareonlysomany weeksina term, suf cientdodevelopa largeclassofinterestingmodels,andto developsomestochasticcontroland ,andmany otherfascinating(anduseful)topics,arelef tfora , thestochasticcontrolportionofthesenotesc oncentratesonveri- cationtheorems, hope,however, thattheinterestedreaderwillbeencouragedt oprobea littledeeperandultimatelytomove have noillusionsaboutthestateofthesenotes they werewrittenratherquickly,sometimesatther ateofa chaptera have nodoubtthatmany errorsremaininthetext;at theveryleastmany oftheproofsareextremelycompact,andshould bemadea littleclearerasis be ttingofa pedagogical(?) I have anotheropportunitytoteachsucha course,I willgoover thenotesagainindetailandattemptthenecess arymodi ,however, youhave any commentsat allaboutthesenotes questions,suggestions,omis-sions,general comments,andparticularlymistakes Iwouldlove assumethatthereaderhashada basiccourseinprobabil-itytheoryat thelevel of,say, GrimmettandStirzaker [GS01] orhigher(ACM116/216shouldbesuf cient).

May 29, 2007 · tic Analysis at Caltech; this year (2007), the topic of this course was stochastic calcu- ... 7.2 Nonlinear ltering for stochastic differential equations . . . . . . . . 177 ... This is not at all obvious (we have only shown convergence in. Introduction 3 distribution for x ed time t), nor is the resolution of this problem entirely straightfor-

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