第7章-Brown运动.pdf

1/59kJIkJIGoBackFullScreenCloseQuitAAALLL?III 0,X(t)l?N(0,2t).KX(t),t 0Brown$WienerLLL.PB(t),t 0 W(t),t 0.XJ=1IOBrown$XJ 6=1KX(t)/,t 0IOBrown$.?”5IOBrown$?/.du3ABy/e?5Brown$?dy3NL?.7/59kJIkJIGoBackFullScreenCloseQuit5557.1.1 Brown$ke5?LB(t),t 0.(1)(?O)B(t)B(s)N(0,t s)=B(t)B(s)l0,?t s?.?s=0B(t)B(0)N(0,t).(2)(O)B(t)B(s)uL?L?G?B(u),0 u s.(3)(?Y5)B(t),t 0 t?Y.55557.1.1vkbB(0)=0dux?Brown$k?rN:PBx(t).?7.1.1?u0?Brown$B0(t).Bx(t)x=B0(t)(7.1.2)8/59kJIkJIGoBackFullScreenCloseQuit(7.1.2)Ue?7.1.2 Brown$?mmmggg555.d5Bx(t)x+B0(t)?Iu0?Brown$?XBrown$u0?Brown$.7.1.2?X(t),t 0LXJ?kmC?=PX(t1)x1,X(t2)x2,X(tn)xn|X(0)=0=PX(t1)x1+x,X(t2)x2+x,X(tn)xn+x|X(0)=xKdLmmmggg?.9/59kJIkJIGoBackFullScreenCloseQuituBrwon$?VO?f.7.1.1?B(t),t 0IOBrown$OPB(2)0 PB(t)0,t=0,1,2.)duB(2)N(0,2)PB(2)0=12.B(0)=0PB(t)0,t=0,1,2=PB(t)0,t=1,2=PB(1)0,B(2)0.,B(1)B(2)?d57.1.1(2)(3)B(2)B(1)B(1)p?IO?Cu|)B(2)=B(1)+(B(2)B(1)k10/59kJIkJIGoBackFullScreenCloseQuitPB(1)0,B(2)0=PB(1)0,B(1)+(B(2)B(1)0=PB(1)0,B(2)B(1)B(1)=Z0PB(2)B(1)x(x)dx=Z0(x)d(x)pOLIO?.dCO?Z0(x)(x)dx=Z0(x)d(x)=Z112ydy=3811/59kJIkJIGoBackFullScreenCloseQuitXJLlxmB(0)=xKB(t)N(x,t)uPxB(t)(a,b)=Zba12te(yx)22tdy.pVPx?eIxLLux.?pt(x,y)=12te(yx)22t(7.1.3)Brown$?=V.|O59=VO?Brown$?kPxB(t1)x1,B(tn)xn=Zx1pt1(x,y1)dy1Zx2pt2t1(y1,y2)dy2Zxnptntn1(yn1,yn)dyn(7.1.4)12/59kJIkJIGoBackFullScreenCloseQuit?Brown$?5k?gC?.7.1.3 Brown$?gC?B,B(t)?tnini=0H?0,t?n=max0in1(tni+1tni)0 Ve?4B,B(t)=B,B(0,t)=limn0n1Xi=0|B(tni+1)B(tni)|2(7.1.5)13/59kJIkJIGoBackFullScreenCloseQuiteBrown$?5.l0?TBrown$?g*?Brown$3m0,T?yyy.Brown$?A?k?B(t),0 t Tke5(1)t?Y(2)3?m(mo?)N?;(3)3?:?(4)3?m(mo?)C?;(5)?t30,t?gC?ut.5(1)(3)Jn)(4)l(5)?(3SK).14/59kJIkJIGoBackFullScreenCloseQuitnnn 7.1.1 B,B(t)=t.y?m0,t?tnini=0?Pnn.PSn=n1Xi=0B(tni+1)B(tni)2KESn=n1Xi=0EB(tni+1)B(tni)2=n1Xi=1(tni+1 tni)=t2dIO?o?V arSn=V arn1Xi=0(B(tni+1)B(tni)2=n1Xi=0V ar(B(tni+1)B(tni)2=n1Xi=03(tni+1 tni)2 3max(tni+1 tni)t=3tn15/59kJIkJIGoBackFullScreenCloseQuitXn=1V arSn dNn?EXn=1(Sn t)2#dPn=1(Sn t)2,a.s.ukSn t 0,a.s.?B,B(t)=t.16/59kJIkJIGoBackFullScreenCloseQuit7.2GaussLLLGaussLLLkk?L.?!?8?yBrown$A?GaussL.k|K1.3.2Nyen.nnn 7.2.1?X N(1,21),Y N(2,22)pK(X,X+Y)N(,).=(1,1+2)0?=21212121+22!17/59kJIkJIGoBackFullScreenCloseQuitnnn 7.2.1 Brown$m(t)=0?(s,t)=min(t,s)?GaussL.yduBrown$?0,?(s,t)=CovB(t),B(s)=EB(t)B(s)et sKEB(t)B(s)=s.2dn98B?B(t)?k?.18/59kJIkJIGoBackFullScreenCloseQuit 7.2.1?B(t)Brown$B(1)+B(2)+B(3)+B(4)?.)X=(B(1),B(2),B(3),B(4)0dn7.2.1Xl?k1 1 1 11 2 2 21 2 3 31 2 3 4-A=(1,1,1,1)KAX=X1+X2+X3+X4=B(1)+B(2)+B(3)+B(4)k?AA0=30?.uB(1)+B(2)+B(3)+B(4)l0,?30?.19/59kJIkJIGoBackFullScreenCloseQuit 7.2.2 B(14)+B(12)+B(34)+B(1)?.)Y=(B(14),B(12),B(34),B(1)0.Y?Xk?/.?14.d AY=B(14)+B(12)+B(34)+B(1)l0,?304?.20/59kJIkJIGoBackFullScreenCloseQuit 7.2.3 VPR10B(t)dt 23.)kI?Brown$kYz5,RiemannR10B(t)dt 3.IR10B(t)dt?.dRiemann?l%CPB(ti)ti?4?.pti0,1?:ti=ti+1 ti.X?ti=in?n=4%C=?C./aq/yk%C?d?4?.uR10B(t)dt?.e5OR10B(t)dt?.21/59kJIkJIGoBackFullScreenCloseQuitV ar?Z10B(t)dt?=Cov?Z10B(t)dt,Z10B(s)ds?=E?Z10B(t)dtZ10B(s)ds?=Z10Z10EB(t)B(s)dtds=Z10Z10CovB(t)B(s)dtds=Z10Z10min(t,s)dtds=13?,R10B(t)dt N(0,13).uVP?Z10B(t)dt 23?=P?3Z10B(t)dt 2?=1(2)p(x)IO?.22/59kJIkJIGoBackFullScreenCloseQuit7.3Brown$?555?!?Brown$X?A?kY?.LX(t),t 0?XJt,E|X(t)|0,kEX(t+s)|Ft=X(t),a.s.(7.3.1)pFt=X(u),0 u tdX(u),0 u t)?“?(7.3.1)A?7,?3?k?yka.s.23/59kJIkJIGoBackFullScreenCloseQuitnnn 7.31?B(t)Brown$K(1)B(t)?(2)B(t)2 t?(3)?u,expuB(t)u22t?.ykdB(t+s)B(t)Ft?5?g(x)kEg(B(t+s)B(t)|Ft=Eg(B(t+s)B(t)(7.3.2)dBrown$?,B(t)N(0,t),B(t),EB(t)=0,2d5?EB(t+s)|Ft=EB(t)+(B(t+s)B(t)|Ft=EB(t)|Ft+EB(t+s)B(t)|Ft=B(t)+EB(t+s)B(t)=B(t)24/59kJIkJIGoBackFullScreenCloseQuitl?(1)?y.duEB2(t)=t B2(t).u?B2(t+s)=B(t)+B(t+s)B(t)2=B2(t)+2B(t)B(t+s)B(t)+B(t+s)B(t)2EB2(t+s)|Ft=B2(t)+2EB(t)(B(t+s)B(t)|Ft+E(B(t+s)B(t)2|Ft=B2(t)+s(7.3.3)p|?B(t+s)B(t)Ft?5k0g(x)=x2A(7.3.2).3(7.3.3)?(t+s)K(2)?y.25/59kJIkJIGoBackFullScreenCloseQuitB(t)N(0,t)?1EeuB(t)=etu2/2 0,kPX(t+s)y|Ft=PX(t+s)y|X(t),a.s.(7.4.1)KX(t)MarkovL,pFt=X(u),0 u t.5(7.4.1)Markov5.29/59kJIkJIGoBackFullScreenCloseQuitnnn 7.4.1 Brown$B(t)kMarkov5.y1N?B(t+s)3Fte?3B(t)e?.EeuB(t+s)|Ft=euB(t)Eheu(B(t+s)B(t)|Fti=euB(t)Eheu(B(t+s)B(t)i(eu(B(t+s)B(t)uFt)=euB(t)eu2s2(B(t+s)B(t)N(0,s)=euB(t)Eheu(B(t+s)B(t)|B(t)i=Eheu(B(t+s)|B(t)iB(t)kMarkov5.30/59kJIkJIGoBackFullScreenCloseQuitYG?MarkovLX(t)?=V3sL?uG?x?eL3t?F(y,t,x,s)=PX(t)y|X(s)=x3Brown$?e?F(y,t,x,s)=Zy1p2(t s)e(ux)22(ts)duBrown$?=VvF(y,t,x,s)=F(y,ts,x,0).PB(t)y|B(s)=x=PB(t s)y|B(0)=x(7.4.2)?s=0F(y,t,x,0)kpt(x,y)=12te(yx)22t31/59kJIkJIGoBackFullScreenCloseQuit(7.4.2)?5Brown$?555=m?Cz.ud(7.1.4)Brown$?kk?.e?Brown$?rMarkov5duB(t)?.7.4.2 XJKCT?QT:0,tkT t Ft=B(u),0 u tKTuB(t),t 0?.rMarkov5SMarkov5?mtT5“O.ey?uBrown$?rMarkov5n.32/59kJIkJIGoBackFullScreenCloseQuitnnn 7.4.2?TuBrown$B(t)?kFT=A F,A T t Ft,t 0KPB(T+t)y|FT=PB(T+t)y|B(T)a.s.=Brown$B(t)krMarkov5.ddnw,XJB(t)=B(T+t)B(T)(7.4.3)KB(t)u0?Brown$uFT.33/59kJIkJIGoBackFullScreenCloseQuit7.5Brown$?CCC999?uuuTxPBrown$gx?=Tx=inft 0,B(t)=x?x 0OPTx t,PB(t)x.d?VPB(t)x=PB(t)x|Tx tPTx t+PB(t)x|Tx tPTx t(7.5.1)eTx tKB(t)30,t?,xd5?PB(t)x|Tx t=1234/59kJIkJIGoBackFullScreenCloseQuit2dY5B(t)Uxux(7.5.1)12.dPTx t=2PB(t)x=22tZxeu2/2tdu=22Zx/tey2/2dy(7.5.2)ddPTx =limtPTx t=22Z0ey2/2dy=135/59kJIkJIGoBackFullScreenCloseQuitddPTx 00,XJ u 0(7.5.3)36/59kJIkJIGoBackFullScreenCloseQuit|(7.5.2)?ETx=Z0PTx tdt=Z0 1 22Zx/tey2/2dy!dt=22Z0Zx/t0ey2/2dydt=22Z0ey2/2dyZx2/y20dt=2x22Z01y20ey2/2dy2x2e1/22Z101y2dy=37/59kJIkJIGoBackFullScreenCloseQuitdTx,A?7,k?k?.*/wBrown$V1 x?m?.5PTx =1Brown$?5.duu:a?Brown$a+B(t)?pB(t)u0?Brown$PaTx =P0Txa =1=Brown$l?:ux?V1.?x 00,u 0(7.5.4)38/59kJIkJIGoBackFullScreenCloseQuit,k?CBrown$30,t?M(t)=max0stB(s)?de?x 0kPM(t)x=PTx t=2PB(t)x=22Zx/tey2/2dyJ?Brown$30,t?m(t)=min0stB(s)?.XJ?B()=0KBrown$?:.ken.39/59kJIkJIGoBackFullScreenCloseQuitnnn 7.5.1?Bx(t)ux?Brown$KBx(t)3(0,t)?k:?V|x|2Zt0u32ex22udu.yXJx 0?yaq?Brown$?=?y.41/59kJIkJIGoBackFullScreenCloseQuit|d(Jyen.nnn 7.5.2 By(t)3m(a,b)?k:?V2arccosrabyPh(x)=PBy3(a,b)?k:|B(a)=x.dMarkov5PBy3(a,b)?k:|B(a)=xPBx3(0,b a)?k:.dV42/59kJIkJIGoBackFullScreenCloseQuitPBy3(a,b)?k:=ZPBy3(a,b)?k:|B(a)=xP(dx)=Zh(x)P(dx)=r2aZ0h(x)ex22adx=r2aZ0 x2ex22adxZba0u32ex22udu=1aZba0u32Z0 xex2(12u+12a)dxdu=1aZba0u32aua+udu=2aZba0u12a+udu=2arctanb aa=2arccosrab43/59kJIkJIGoBackFullScreenCloseQuitu?Brown$?u.nnn 7.5.3?By(t),t 0Brown$KPBy(t)3(a,b)vk:=2arcsinrab.44/59kJIkJIGoBackFullScreenCloseQuite0?Brown$3tc?:93t?1:?.-t=sups t,B(s)=0=tc?:t=infs t,B(s)=0=t?1:5?t?t(?y).d?ukPt x=PB 3(x,t)vk:=2arcsinrxtPt y=PB 3(t,y)vk:=2arcsinstyPt x,t y=PB 3(x,y)vk:=2arcsinrxy45/59kJIkJIGoBackFullScreenCloseQuit7.6Brown$?AAACCCzzz7.6.1BrownxxxdBrown$,a3n7K?L)BrownxL.7.6.1?B(t),t 0Brown$.-B(t)=B(t)tB(1),0 t 1KLB(t),0 t 1Brownx(Brown Bridge).46/59kJIkJIGoBackFullScreenCloseQuitBrown$GaussLBrownxGaussLnd?(.0 s t 1kEB(t)=0EB(s)B(t)=E(B(s)sB(1)(B(t)tB(1)=EB(s)B(t)tB(s)B(1)sB(t)B(1)+tsB2(1)=s ts ts+ts=s(1 t)d?dB(0)=B(1)=0=dL?:?x?Brownx?d5.47/59kJIkJIGoBackFullScreenCloseQuit7.6.2kkk?Brown$?TxBrown$B(t)gx?,x 0.-Z(t)=(B(t),et 0CZ(t)?klY.l?PZ(t)=x=PTx t=22tZxey22tdy48/59kJIkJIGoBackFullScreenCloseQuiteY?.y xPZ(t)y=PB(t)y,max0stB(s)x=PB(t)y PB(t)y,max0stB(s)x(7.6.1)yO(7.6.1)?dV?PB(t)y,max0stB(s)x=PB(t)y|max0stB(s)xPmax0stB(s)x(7.6.2)dumax0stB(s)x?duTx t9Brown$?5B(t)3Tx(x=PB(t)2xy|max0stB(s)x(7.6.3)l(7.6.2)9(7.6.3)?PB(t)y,max0stB(s)x=PBt)2x y,max0stB(s)x=PB(t)2x y(y x)d(7.6.1)kPZ(t)y=PB(t)y PB(t)2x y=PB(t)y PB(t)y 2x=12tZyy2xeu22tdu.50/59kJIkJIGoBackFullScreenCloseQuit7.6.3333?:?Brown$dY(t)=|B(t)|,t 0?LY(t),t 03?:?Brown$.?VPY(t)y=PB(t)y PB(t)y=2PB(t)y 1=22tZyeu22tdu 1y 051/59kJIkJIGoBackFullScreenCloseQuit7.6.4AAABrown$dX(t)=eB(t),t 0?LX(t),t 0ABrown$.duBrown$?1EesB(t)=ets2/2ABrown$?OEX(t)=EeB(t)=et/2V arX(t)=EX2(t)(EX(t)2=Ee2B(t)et=e2t et37K|b?dUABrown$Cz3e?fXdb.52/59kJIkJIGoBackFullScreenCloseQuit 7.6.1(?ddd)?,udu=Z0PyeB(T)K udu=Z0PB(T)logK+uydu=12TZ0Zlog(K+u)/yex2/2Tdxdu53/59kJIkJIGoBackFullScreenCloseQuit7.6.5kkk?Brown$B(t)IOBrown$X(t)=B(t)+tk?Brown$X.Nwk?Brown$m?L.edLOk,?kA,B 0,B x.P(A)=1,P(B)=0)?c1=e2Be2B e2A,c2=1e2B e2A56/59kJIkJIGoBackFullScreenCloseQuitl?p(x)=e2B e2xe2B e2Adlx=0uL3?Bck?A?Vp(0)PL3eBck,A=e2B 1e2B e2A(7.6.5)e 03-B KkPL,A=e2A(7.6.6)ddLK?2?C.e3(7.6.5)-0KkPBrown$3eBc,A=BA+B.57/59kJIkJIGoBackFullScreenCloseQuit 7.6.2?b?,0)?Brown$.3o?3dx?.3de?p(x)(x A)p(x)L?x?V.d(7.6.6)kp(x)=e2x,x 0 x?(x A)e2x?x=A+12.58/59kJIkJIGoBackFullScreenCloseQuit7.7?VVVggg555 7.7.1 B1(t),B2(t),Bd(t)p?IOBrown$KB(t)=(B1(t),B2(t),Bd(t)dBrown$XBrown$dBrown$kXe=V1(2t)d/2exp12tdXi=1(yi xi)2,x,y RdaquBrown$udBrown$kXen59/59kJIkJIGoBackFullScreenCloseQuitnnn 7.7.1?B(t)=(B1(t),B2(t),Bd(t)dBrown$Kk1.B(t),t 0?2.|B(t)|2,t 0?|x|=(x21+x2d)1/2,x Rd3.?y Rdexp(hu,B(t)i12t|u|2),t 0?dBrown$?.C hx,yi=Pdi=1xiyi,x,y Rd.