随机利率下有违约风险的最优投资组合

:042727104(2005)0320382206:2004205224:(101310310);(10325101);(20030246004):(1979),,..1,2(1.,200433;2.,313000):,,Cox2Ingersoll2Ross,,Hamilton2Jacobi2Bellman.:Cox2Ingersoll2Ross;;;;:O211.6:AMerton11971,Merton.KornKraft2Vasicek3HoLee4.,Lipschitz,5.VasicekHo2Lee,2,.Kraft6,Heston,,.Hou7,Vasicek.Vasicek,:..,Cox2Ingersoll2Ross8(CIR),.6,,,.1(,G,P),W()=(W1(),W2(),W3())T,W1,W2,W3.F={Ft}t[0,T]W()2,,T(0,+).t[0,T,FtG.,4:.,QP,r()Q.r()CIR:dr(t)=k1(1-r(t))dt+1r(t)dW1(t),(1)44320056()JournalofFudanUniversity(NaturalScience)Vol.44No.3Jun.2005k1,1,1,2k1121.B(t):dB(t)=r(t)B(t)dt.1T1(T1T),B(t,T1):dB(t,T1)B(t,T1)=(r(t)+1(t)11r(t))dt+1(t)1r(t)dW1(t).(2)1r().1()1(t)=-2(el1(T1-t)-1)2l1+(k1+11+l1)(el1(T1-t)-1)(3),l1=(k1+11)2+221.(,G,Q),.Q(=0)=0,t0,Q(t)0.H(t)=1{t},Ht=(H(u);ut),t[0,T],H={Ht}t[0,T]2.G=FH,Gt=FtHt,t[0,T,G2.1F2h()M(t)=H(t)-t0h(u)dut[0,T](4)G2.h()F2(hazardrate).[9],1T1(T1T),P(t,T1)=1{t}EQ(e-T1t(r(s)+h(s)(s))dsFt),(5)F2()(0,1,Q2a.s.,(lossrategivende2fault).()=h()(),()(creditspread).(t)=1,Pt[0,T1,,,,()=h().()CIR:d(t)=k2(2-(t))dt+2(t)dW2(t),(6)k2,2,2,2k2222.1T1:dP(t,T1)P(t-,T1)=(r(t)+1(t)11r(t)+2(t)22(t))dt+1(t)1r(t)dW1(t)+2(t)2(t)dW2(t)-dM(t),(7)2().M()(4),2()2(t)=-2(el2(T1-t)-1)2l2+(k2+22+l2)(el2(T1-t)-1)(8),l2=(k2+22)2+222.1,,.,(7)M().110,:dP(t,T1)P(t,T1)=(r(t)+1(t)11r(t)+2(t)22(t))dt+3833:1(t)1r(t)dW1(t)+2(t)2(t)dW2(t).S():dS(t)S(t)=(r(t)+3(t)3(t))dt+3(t)dW3(t),3(),3().2[0,T,TT1.X(),X(0)=x00.(t)=(1(t),2(t),3(t))T,t[0,T,,1(),2()3()[0,T],1-()T1v,1v=(1,1,1)T.,X():dX(t)X(t)=r(t)+(t)T((t)-r(t)1v)]dt+(t)T(t)dW(t),(9)(t)=r(t)+1(t)11r(t)r(t)+1(t)11r(t)+2(t)22(t)r(t)+3(t)3(t),(t)=1(t)1r(t)001(t)1r(t)2(t)2(t)0003(t).2t0[0,T),{(t)}t[t0,T],(i){(t)}t[t0,T];(ii)X(t0)=xt00,(9){X(t)}t[t0,T],X(t)0,Pt[t0,T;(iii)kNETt0(s)kds+.(10)(t0,xt0)A(t0,xt0).,4,.U(x)=xx0,(0,1).(P)x00,3()A(0,x0)EU(X3(T))=max()A(0,x0),EU(X(T)).(11)(P)A(0,x0),3()A(0,x0)(11),3()(P)A(0,x0).,(P)R3.Y()=(X(),r(),())T.(9),(1)(6),Y():dY(t)=(t,Y(t),(t))dt+(t,Y(t),(t))dW(t),(12)=X(r+T(-r1v))k1(1-r)k2(2-),=X(1+2)11rX222X331r00020.483()44,(P)Hamilton2Jacobi2Bellman:maxR3¬G(t,x,r,)=0t[0,T,x,r,(0,+),G(T,x,r,)=xx,r,(0,+).(13)¬:¬G(t,x,r,)=Gt+12x2((1+2)22121r+222222+2323)Gxx+1221rGrr+1222G+x(1+2)121rGxr+x2222Gx+x(r+(1+2)111r+2222+333)Gx+(k11-r)Gr+(k22-)G.HJB(13).Gxx0,,3:31=-GxxGxx111-GxrxGxx11-32,32=-GxxGxx222-GxxGxx12,33=-GxxGxx33.(14)3HJB(13),G(t,x,r,):G(t,x,r,)=ef(t)+g(t)r+h(t)x,f(T)=g(T)=h(T)=0.,min1(1+11)2+221,1(1+22)2(15),i=iki,i=1,2.G(t,x,r,)=ef(t)+g(t)r+h(t)xt[0,T,x,r,(0,+)(16)HJB(13)C1,22.f(t)=231-(T-t)+k11Ttg(s)ds+k22Tth(s)ds,(17)g(t)=2+211-(ev1(T-t)-1)2v1+v1+k1-111-(ev1(T-t)-1),(18)h(t)=221-(ev2(T-t)-1)2v2+v2+k2-221-(ev2(T-t)-1),(19)v1=k1-1-112-211-2+211-,v2=k2-1-222-2222(1-)2.G(t,x,r,)3,31(t)=11-11(t)1+g(t)1(t)-32(t),5833:32(t)=11-22(t)2+h(t)2(t),33(t)=11-3(t)3(t).(20)(9)Lipschitz,CIR,2(20).,6,.3(15),t0[0,T,[t0,T]{p}pN,{G(p,X(p),r(p),(p))}pN,{(t)}t[t0,T].(t0,xt0)A(t0,xt0).3()[t0,T],,(10).,xt003,(9)X3(t)=xt0exptt0r(s)+3(s)T((s)-r(s)1v)-123(s)T(s)2ds+3(s)T(s)dW(s),X3(t)0,Pt[t0,T].(20)3.1(15),(16)G,{p}pN,t0pT,q1suppNEG(p,X3(p),r(p),(p))q.(21)q1,(20)ItÉ,G(t,X3(t),r(t),(t))q=d1(t)expq1-g(t)+11r(t)+q1-h(t)+22(t)+q1-1+212+1k11tt0r(s)ds+q1-222+2k22tt0(s)dsL1(t),d1(t)t,[t0,T],L1(t)=exp-12q22tt033(s)23(s)2ds+qtt033(s)3(s)dW3(s).(15),G(t,X3(t),r(t),(t))qd1(t)expqk121r(t)+qk222(t)+q1-1+212+1k11tt0r(s)ds+q1-222+2k22tt0(s)dsL1(t)=d2(t)expq1-1+212+1k11-(2-q)k21221(3)tt0r(s)ds+q1-222+2k22-(2-q)k22222(33)tt0(s)dsL2(t),d2(t)t,[t0,T],L2(t)=exp-q2k21221tt0r(s)ds-q2k22222tt0(s)ds-12q22tt033(s)23(s)2ds+qk11tt0r(s)dW1(s)+qk22tt0(s)dW2(s)+qtt033(s)3(s)dW3(s).683()44(15),q1(3)(33),G(t,X3(t),r(t),(t))qd2(t)L2(t).L2(t),[t0,T]p,EG(p,X3(p),r(p),(p))qE(d2(p)L2(p))supt[t0,T]d2(t)E(L2(p))supt[t0,T]d2(t).,(20)3A(t0,xt0).(16)G(P)A(t0,xt0),(20)3()(P)A(t0,xt0).2(15),(9),GC1,2(L)C(L)HJB(13),L=[0,T](0,)3.(t0,yt0)L(i)G(t0,yt0)Et0,yt0(X(T)),P()A(t0,xt0);(ii)G(t0,yt0)=Et0,yt0(X3(T)).(15),3,23.2,..,.3(15),(P)A(t0,xt0)(16)(20).:1MertonRC.OptimumconsumptionandportfoliorulesinacontinuoustimesmodelJ.JournalofEconomicsTheory,1971,3:3732413.2KornR,KraftH.AstochasticcontrolapproachtoportfolioproblemswithstochasticinterestratesJ.SIAMJournalofControlandOptimization,2001,40:125021269.3VasicekO.AnequilibriumcharacterisationofthetermstructureJ.JournalofFinancialEconomics,1977,5:1772188.4HoTSY,LeeSB.TermstructuremovementsandpricinginterestcontingentclaimsJ.JournalofFinance,1986,41:101121029.5FlemingWH,SonerHM.ControlledmarkovprocessandviscositysolutionsM.NewYork:Springer,1993.6KraftH.GeneralapproachtoportfolioproblemswithstochasticvolatilityandaverificationresultforhestonsmodelEB/OL.:AdynamicassetallocationperspectiveEB/OL.:3852407.9DuffieD,SingletonK.ModelingtermstructuresofdefaultriskybondsJ.ReviewofFinancialStudies,1999,12:1772