stochastic-calculus-solutions

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1、1StochasticCalculusforFinanceSolutionstoExercisesChapter1Exercise1.1:Showthatforeachn,therandomvariablesK(1),...,K(n)areindependent.Solution:SinceK(r)havediscretedistributions,theindependenceofK(1),...,K(n)meansthatforeachsequenceV1,...,Vn,Vi∈{U,D}wehaveP(K(1)

2、=V1,K(2)=V2,...,K(n)=Vn)=P(K(1)=V1)·...·P(K(n)=Vn).FixasequenceV1,...,Vn.Startbysplittingtheinterval[0,1]intotwointervalsI,Ioflength1,I={ω:K(1)=U},I={ω:K(1)=D}.UD2UDRepeatthesplittingforeachintervalateachstage.AtstagetwowehaveIU=IUU∪IUD,ID=IDU∪IDDandthevariabl

3、eK(2)isconstantoneachIαβ.Forexample,IUD={ω:K(1)=U,K(2)=D}.Usingthisnotationwehave{K(1)=V1}=IV1,{K(1)=V1,K(2)=V2}=IV1,V2,...{K(1)=V1,...,K(n)=Vn}=IV1,...,Vn.TheLebesguemeasureofIis1,sothatV1,...,Vn2n1P(K(1)=V1,...,K(n)=Vn)=2n.FromthedefinitionofK(r)followsdirect

4、lyP(K(1)=V1)·...P(K(n)=V)=1.n2nExercise1.2:RedesigntherandomvariablesK(n)sothatP(K(n)=U)=p∈(0,1),arbitrarySolution:Giventheprobabilityspace(Ω,F,P)=([0,1],B([0,1]),m),wheremdenotestheLebesguemeasure,wewilldefineasequenceofran-domvariablesK(n),n=1,2,....onΩ.First

5、split[0,1]intotwosubintervals:[0,1]=IU∪ID,whereIU,IDaredisjointintervalswithlengths

6、IU

7、=p,

8、ID

9、=q,p+q=1,withIUtotheleftonID..Nowset2SolutionstoExercises()Uifω∈IUK(1,ω)=.Difω∈IDClearlyP(K(1)=U)=p,P(K(1)=D)=q.Repeattheproceduresep-aratelyonIUandID,splittingeachin

10、totwosubintervalsintheproportionptoq.ThenI=I∪I,I=I∪I,

11、I

12、=p2,

13、I

14、=pq,UUUUDDDUDDUUUD

15、I

16、=qp,

17、I

18、=q2.RepeatingthisrecursiveconstructionntimesweDUDDobtainintervalsoftheformIα1,...,αr,withαieitherUorD,andwithlengthplqr−l,wherel=#{α:α=U}.iiAgainset()Uifω∈Iα1,...,αr−1,U

19、K(r,ω)=.Difω∈Iα1,...,αr−1,DIfthevalueUappearsltimesinasequenceα1,...,αr−1,then

20、Iα1,...,αr−1,U

21、=pplqr−1−l.Therearer−1differentsequencesα,...,αhavingUexactlyl1r−1atlplaces.ThenforAr={K(r)=U}wefindXr−1!r−1lr−1−lP(Ar)=P(K(r)=U)=ppqll=0r−1=p(p+q)=pAsaconsequenceisP

22、(K(r)=D)=q.TheproofthatthevariablesK(1),...,K(n)areindependentfollowsasinEx.1.1.Exercise1.3:FindthefiltrationinΩ=[0,1]generatedbytheprocessX(n,ω)=2ω1[0,1−1](ω).nSolution:SinceX(1)(ω