A Unified Approach for Drawdown (Drawup) of一种统一的抽取（提取）方法 时间齐次马尔可夫过程

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1、AUnifiedApproachforDrawdown(Drawup)ofTime-HomogeneousMarkovProcessesDavidLandriault∗BinLi†HongzhongZhang‡February28,2017AbstractDrawdown(resp.drawup)ofastochasticprocess,alsoreferredasthereflectedprocessatitssupremum(resp.infimum),haswideapplicationsinmanyareasincludingfinanci

2、alriskmanagement,actuarialmathematicsandstatistics.Inthispaper,forgeneraltime-homogeneousMarkovprocesses,westudythejointlawofthefirstpassagetimeofthedrawdown(resp.drawup)process,itsovershoot,andthemaximumoftheunderlyingprocessatthisfirstpassagetime.Byusingshort-timepathwisea

3、nalysis,undersomemildregularityconditions,thejointlawofthethreedrawdownquantitiesisshowntobetheuniquesolutiontoanintegralequationwhichisexpressedintermsoffundamentaltwo-sidedexitquantitiesoftheunderlyingprocess.ExplicitformsforthisjointlawarefoundwhentheMarkovprocesshasonl

4、yone-sidedjumpsorisaL´evyprocess(possiblywithtwo-sidedjumps).Theproposedmethodologyprovidesaunifiedapproachtostudyvariousdrawdownquantitiesforthegeneralclassoftime-homogeneousMarkovprocesses.Keywords:Drawdown;Integralequation;Reflectedprocess;Time-homogeneousMarkovprocessMSC

5、(2000):Primary60G07;Secondary60G401IntroductionWeconsideratime-homogeneous,real-valued,non-explosive,c`adl`agMarkovprocessX=(Xt)t≥0withstatespaceR1definedonafilteredprobabilityspace(Ω,F,F=(Ft)t≥0,P)withacompleteandright-continuousfiltration.Throughout,wesilentlyassumethatXsat

6、isfiesthestrongMarkovarXiv:1702.07786v1[q-fin.MF]24Feb2017property(seeSectionIII.8,9ofRogersandWilliams[33]),andexcludeMarkovprocesseswithmonotonepaths.ThefirstpassagetimeofXabove(below)alevelx∈RisdenotedbyT+(−)=inf{t≥0:X>(<)x},xtwiththecommonconventionthatinf∅=∞.∗Department

7、ofStatisticsandActuarialScience,UniversityofWaterloo,Waterloo,ON,N2L3G1,Canada(dlandria@uwaterloo.ca)†CorrespondingAuthor:DepartmentofStatisticsandActuarialScience,UniversityofWaterloo,Waterloo,ON,N2L3G1,Canada(bin.li@uwaterloo.ca)‡DepartmentofIEOR,ColumbiaUniversity,NewYo

8、rk,NY,10027,USA(hz2244@columbia.edu)1Thestatespacecansometimesberelaxedtoanopenintervalof