第8章-随机积分.pdf

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17/47kJIkJIGoBackFullScreenCloseQuitLX(t)=c0I0(t)+n1Xi=0ciI(ti,ti+1(t)(8.2.1)uZT0X(t)dB(t)=n1Xi=0ciB(ti+1)B(ti)(8.2.2)dBrown$?O5(8.2.2)?Gauss?C0?8/47kJIkJIGoBackFullScreenCloseQuitV ar(ZXdB)=En1Xi=0ci(B(ti+1)B(ti)#2=En1Xi=0n1Xj=0cicjB(ti+1)B(ti)B(tj+1)B(tj)=n1Xi=0n1Xj=0cicjE(B(ti+1)B(ti)(B(tj+1)B(tj)=n1Xi=0c2i(ti+1 ti)9/47kJIkJIGoBackFullScreenCloseQuit?4?2?X(t).?L?d?ciCi5“OiFti?.pFt=B(u),0 u t.u,dBrown$?5?Ei(B(ti+1)B(ti)|Fti=iEB(ti+1)B(ti)|Fti=0dEi(B(ti+1)B(ti)=0.10/47kJIkJIGoBackFullScreenCloseQuit 8.2.1?X(t),0 t TL=30,T?0=t0 t1 ti,i=0,1,n 1X(t)=1I0(t)+n1Xi=0iI(ti,ti+1(t)(8.2.3)dIt oRT0XdBZT0X(t)dB(t)=n1Xi=0i(B(ti+1)B(ti)(8.2.4)11/47kJIkJIGoBackFullScreenCloseQuitL?Cve5.(1)555XJX(t),Y(t)LKZT0(X(t)+Y(t)dB(t)=ZT0X(t)dB(t)+ZT0Y(t)dB(t)p,.(2)ZT0Ia,b(t)dB(t)=B(b T)B(a 0)Ia,b(t)ma,b?5.(3)555XJE2i ,(i=0,1,n 1)KEZT0X(t)dB(t)=012/47kJIkJIGoBackFullScreenCloseQuit(4)?555XJE2i ,(i=0,1,n 1)KE?ZT0X(t)dB(t)?2=ZT0EX2(t)dty5(1),(2)(3)?g1ypy5(4).|Cauchy-Schwarz?E|i(B(ti+1)B(ti)|qE(2i)EB(ti+1)B(ti)2 13/47kJIkJIGoBackFullScreenCloseQuituV arZT0XdB=En1Xi=0i(B(ti+1)B(ti)#2=En1Xi=0i(B(ti+1)B(ti)n1Xj=0j(B(tj+1)B(tj)=n1Xi=0E2i(B(ti+1)B(ti)2+2XijEij(B(ti+1)B(ti)(B(tj+1)B(tj)(8.2.5)dBrown$?O59ui?b|n1.5.7(1)kEij(B(ti+1)B(ti)(B(tj+1)B(tj)=014/47kJIkJIGoBackFullScreenCloseQuit(8.2.5)?.dBrown$?5?V arZT0XdB=n1Xi=0E2i(B(ti+1)B(ti)2=n1Xi=0E?E?2i(B(ti+1)B(ti)2|Fti?=n1Xi=0E?2iE?(B(ti+1)B(ti)2|Fti?=n1Xi=0E2i(ti+1 ti)=ZT0EX2(t)dt.15/47kJIkJIGoBackFullScreenCloseQuitk?c?O?y3?*?ALa.8.2.2?X(t),t 0LFt,t 0“6XJ t,X(t)Ft?KX(t)FtAAA?PV=nh,h30,)?ALvERT0h2(s)ds o16/47kJIkJIGoBackFullScreenCloseQuit?Ue*?V:k-h Vk.z ,h(,)Y.K3Lnn=Xjh(tj,)1tj,tj+1)(t)V?n z ZT0(h n)2dt 0ddk.n?ERT0(h n)2dt 0.17/47kJIkJIGoBackFullScreenCloseQuitg-h Vk.y3hn Vk.,n,hn(,)Y,?EZT0(h hn)2dt 0(8.2.6)”?|h(t,)|M,(t,).hn(t,)=Zt0n(s t)h(s,)dspnRKY?k?x/(1n,0),n(x)=0Rn(x)dx=1.Kz ,hn(,)Y|hn(t,)|M.dh Vwhn V,?n z ,kZT0(hn(s,)h(s,)2ds 0d2g|k.n?(8.2.6).18/47kJIkJIGoBackFullScreenCloseQuit?f V3k.?hn V?EZT0(f(t,)hn(t,)2dt 0-hn(t,)=n,e f(t,)n|n=?.19/47kJIkJIGoBackFullScreenCloseQuity3?It oRT0f(t)dB(t),f V?Xe.8.2.3?f V(0,T)Kf?It oZT0f(t,)dBt(w)=limnZT0n(t,)dBt(),(L2(P)4)(8.2.7)pn?L?S?n ,E?ZT0(f(t,)n(t,)2dt?0.(8.2.8)53SK?LvV?5?=ve?V?.It o?2?2?Lh(s):s 0 a:V=nh:hB F?ALT 0vRT0h2(s)ds ,a.s.o.20/47kJIkJIGoBackFullScreenCloseQuit 8.2.1?fYR10f(B(t)dB(t).B(t)kY?f(B(t)30,1Y,l?R10f(B(t)dB(t)k.,?f?k(vk)k?.X(1)?f(t)=t,KduR10EB2(t)dt ,ER10B(t)dB(t)=0d58.2.1(3)(4)kE?Z10B(t)dB(t)?2=Z10EB2(t)dt=12(2)?f(t)=et2dR10eB2(t)dB(t).,3?du?t 14,Ee2B2(t)=R10Ee2B2(t)dt=R10Ef2(B(t)dt .?3.21/47kJIkJIGoBackFullScreenCloseQuit 8.2.2 J=R10tdB(t)?.R10t2dt tFt=B(s),0 s tA?,It oJ?EJ=0,?EJ2=R10t2dt=13.8.2.3?O?R10(1 t)dB(t)?.R10(1 t)2dt It o,0X Vot TRt0X(s)dB(s)?.?tRt0X(s)dB(s)Ct?LY(t),Y(t)=Zt0X(s)dB(s)(8.3.1)yIt oY(t)3Y?=3YLZ(t)?k?tkY(t)=Z(t),a.s.dly3?bY?.?!?L?5.23/47kJIkJIGoBackFullScreenCloseQuity3y?5.nnn 8.3.1?X(t)VR0EX2(s)ds KY(t)=Zt0X(s)dB(s),0 t T?Y?(=suptTEY2(t).yd8.2.3?5Y(t)=RT0X(s)dB(s),0 t T?k?9?.XJX(t)LKd(8.2)?y?E?ZtsX(u)dB(u)|Fs?=0,s 0?|f(x)|K,uRT0Ef2(B(s)ds KT.dn8.3.1,dK?y.nJ?E?.38.2!y?L?It o?C./ken.26/47kJIkJIGoBackFullScreenCloseQuitnnn 8.3.2 XJX?RT0X2(s)ds Kt,Y(t)=Rt0X(s)dB(s)?C.=Y(t),0 t TGaussL?Cov(Y(t),Y(t+u)=Zt0X2(s)ds,u 0Y(t)?.27/47kJIkJIGoBackFullScreenCloseQuity?Rt0EX2(s)ds=Rt0X2(s)ds 1231/47kJIkJIGoBackFullScreenCloseQuitd0,t?kY(tni+1)Y(tni)=(0(B(tni+1)B(tni),XJtni tni+1121(B(tni+1)B(tni),XJ12 tni12Y,Y(t)=limn1Xi=0(Y(tni+1)Y(tni)2=20limXti12(B(tni+1)B(tni)2=20B,B(12)+21B,B(12,t)=Zt0X2(s)dsp?4?n 03Ve?4.33/47kJIkJIGoBackFullScreenCloseQuituBrown$B(t)?It oY1(t)=Rt0X1(s)dB(s)Y2(t)=Rt0X2(s)dB(s)duY1(t)+Y2(t)=Rt0(X1(s)+X2(s)dB(s)Y1Y2?ggg?CCC?Y1,Y2(t)=12(Y1+Y2,Y1+Y2(t)Y1,Y1(t)Y2,Y2(t)dn8.3.3kY1,Y2(t)=Zt0X1(s)X2(s)ds.34/47kJIkJIGoBackFullScreenCloseQuit8.4It oIt o?COK?N-?XDynkinFeynman-Kac9dIt o?.Brown$30,t?gC?t=3V?elimn0n1Xi=0(B(tni+1)B(tni)2=tptni0,t?n=max0in1(tni+1 tni).35/47kJIkJIGoBackFullScreenCloseQuitl/LZt0(dB(s)2=Zt0ds=t(dB(t)2=dt/ken.nnn 8.4.1?gk.Ytni0,t?K?ni(B(tni+1),B(tni)(=B(tni),B(tni+1)m?)Ve?4limn0n1Xi=0g(ni)(B(tni+1)B(tni)2=Zt0g(B(s)ds(8.4.1)36/47kJIkJIGoBackFullScreenCloseQuityk?ni=B(tni)dg?Y5?kn1Xi=0g(B(tni)(tni+1 tni)Zt0g(B(s)ds(8.4.2)yn1Xi=0g(B(tni)(B(tni+1)B(tni)2n1Xi=0g(B(tni)(tni+1tni)0(8.4.3)3L2.PBi=B(tni+1)B(tni),ti=tni+1 tniKdBrown$?O5?37/47kJIkJIGoBackFullScreenCloseQuitEn1Xi=0g(B(tni)(Bi)2 ti)#2=EE n1Xi=0g2(B(tni)(Bi)2 ti)2|Fti!#=En1Xi=0g2(B(tni)E?(Bi)2 ti)2|Fti?#=2En1Xi=0g2(B(tni)(ti)2#2nEn1Xi=0g2(B(tni)ti 0,(?n 0)38/47kJIkJIGoBackFullScreenCloseQuitd3?en1Xi=0g(B(tni)(Bi)2 ti)0?Pn1i=0g(B(tni)(B(tni+1)B(tni)2Pn1i=0g(B(tni)(tni+1tni)k?4Rt0g(B(s)ds.?ni(B(tni+1),B(tni)?n 0n1Xi=0(g(ni)g(B(tni)(Bi)2maxi(g(ni)g(B(tni)n1Xi=0(B(tni+1)B(tni)2(8.4.4)39/47kJIkJIGoBackFullScreenCloseQuitdgB?Y5k maxi(g(ni)g(B(tni)0,a.s.dBrown$?gC?Pn1i=0(B(tni+1)B(tni)2 t.u?n 0Pn1i=0(g(ni)g(B(tni)(Bi)20.dPn1i=0g(ni)(Bi)2Pn1i=0g(B(tni)(Bi)2k?V?4Rt0g(B(s)ds.40/47kJIkJIGoBackFullScreenCloseQuity3It o.nnn 8.4.2 XJf?gYK?tkf(B(t)=f(0)+Zt0f0(B(s)dB(s)+12Zt0f00(B(s)ds(8.4.5)y(8.4.5)?.?0,t?tnikf(B(t)=f(0)+n1Xi=0f(B(tni+1)f(B(tni)f(B(tni+1)f(B(tni)ATaylor?41/47kJIkJIGoBackFullScreenCloseQuitf(B(tni+1)f(B(tni)=f0(B(tni)(B(tni+1)B(tni)+12f00(ni)(B(tni+1)B(tni)2ni(B(tni+1),B(tni).uf(B(t)=f(0)+n1Xi=0f0(B(tni)(B(tni+1)B(tni)+12n1Xi=0f00(ni)(B(tni+1)B(tni)2(8.4.6)-n 0?4K(8.4.6)?1uIt oRt0f0(B(s)dB(s).|n8.4.1(8.4.6)?1?uRt0f00(B(s)ds.42/47kJIkJIGoBackFullScreenCloseQuit(8.4.5)Brown$?It oddwBrown$?LIt okk.C?YL.aLIt oL/ke.8.4.1 XJLY(t),0 t TLY(t)=Y(0)+Zt0(s)ds+Zt0(s)dB(s),0 t T(8.4.7)L(t)(t)v(1)(t)A?RT0|(t)|dt ,a.s.(2)(t)V.KY(t)It oLLL.43/47kJIkJIGoBackFullScreenCloseQuitkIt oL(8.4.7)P?/dY(t)=(t)dt+(t)dB(t),0 t T(8.4.8)(t)X,(t)*X,6uY(t)B(t)$?B(s),s t(X(t)=cos(M(t)+t)p M(t)=maxstB(s).a-?/(t)=LY(t)6ut3e(8.4.8)AU?dY(t)=(Y(t)dt+(Y(t)dB(t),0 t T(8.4.9)XJ/LIt o(8.4.5)Cd(f(B(t)=f0(B(t)dB(t)+12f00(B(t)dt(8.4.10)44/47kJIkJIGoBackFullScreenCloseQuit 8.4.1 d(eB(t).)f(x)=exIt odf0(x)=ex,f00(x)=exd(eB(t)=eB(t)dB(t)+12eB(t)dtu,X(t)=eB(t)k/dX(t)=X(t)dB(t)+12X(t)dt45/47kJIkJIGoBackFullScreenCloseQuit37KA?dS(t)dS(t)=S(t)dt+S(t)dB(t)?.XJa(t)L3t?o3?mm0,TS?ZT0a(t)dS(t)=ZT0a(t)S(t)dt+ZT0a(t)S(t)dB(t)en?uIt oL?It o46/47kJIkJIGoBackFullScreenCloseQuitnnn 8.4.3?X(t)ddX(t)=(t)dt+(t)dB(t)?It oLg(t,x)0,)R?gY.KY(t)=g(t,X(t)EIt oLdY(t)=gt(t,X(t)dt+gx(t,X(t)dX(t)+122gx2(t,X(t)(dX(t)2(8.4.11)(dX(t)2=(dX(t)(dX(t)Ue5KOdt dt=dt dB(t)=dB(t)dt=0,dB(t)dB(t)=dt47/47kJIkJIGoBackFullScreenCloseQuit=(8.4.11)U?dY(t)=?gt(t,X(t)+gx(t,X(t)(t)?dt+gx(t,X(t)(t)dB(t)+122gx2(t,X(t)2(t)dtAO/XJg(t,x)=g(x)x?(8.4.12)zdY(t)=g0(X(t)(t)+12g00(X(t)2(t)dt+g0(X(t)(t)dB(t)(8.4.12)