應(yīng)用隨機(jī)過(guò)程 期末復(fù)習(xí)資料

第一章隨機(jī)過(guò)程的基本概念一、隨機(jī)過(guò)程的定義例nn列X，X，X12nnn例nnXnn例p以tt例t示tttPt(,,)tTX):Et例例例為}例)E3TT取R,t+當(dāng)T取Zt+1為4二、有限維分布與Kolmogorov定理P{X(t)x}F(t,x)隨機(jī)過(guò)程的二維分布：F(x,x)P{X(t)x,X(t)x},t,tTt,t1121122122nFx,xx){t)x,t)xt)xt,ttTt,t,t12n1122nn12n12n(x,x,x),t,t,tT,nt{Ft,tt12n12n12n(,,)jjj12n(x,x,x)F(x,x,x)Ft,t,tt,t,tjjj12njjjn12n12n12F(x,x,)F(x,x)t,t,ttt,t1m1m1mm1n1m定理：設(shè)分布函數(shù)族{F(,,xx1),,,tt,Tnxtt,tt2n12n12n性和相容性，則必存在一個(gè)隨機(jī)過(guò)程t，使(x,x,x),t,t,tT,nt{Ft,tt12n12n12nt稱(t)E[X(tXtT，[()]2EXtt(,)()(()(tXtt，t,tTttEXt121X12X212(t,t)差函數(shù)；稱Var[X(tR(s,t)E[X(s)X(ts,tT為過(guò)程的方差函數(shù)；稱X()Xt()tVatb中X和VX00(t)(t,t)。和X12三、隨機(jī)過(guò)程的基本類型,,,,tttTtt,Xt()(),tXt12n12n21X(t)X(t)tnn1)dt,tt112212Poisson過(guò)程2.1Poisson過(guò)程{NttNt)0到tANt)0st{Ntt(s)Nt)Nt)N(s)且(s,t]AN0（N(0)tt的tnt0切s(()())PNtsNsne,nt!因?yàn)镹ts)N(s)s的分布只依賴于與區(qū)間起點(diǎn)s無(wú)關(guān)，tnP(Nt)n)et,n!mt)t)t是例510，n05{(2)N{(2),PNPNNeee!n0t)例Nt]過(guò)程就是{Ntt4(412)n!{NN(0)}e(412)nE[NN(0)]ne!n0定義2.1.2：計(jì)數(shù)過(guò)程N(yùn)t),t0Poisson過(guò)程，如果滿足：(i)Nt)(ii)過(guò)程有平穩(wěn)獨(dú)立增量(iii)存在當(dāng)h時(shí)，PN(h)1ho(h);(iv)h時(shí)，PN(h)2o(h).12{Ntt例A的pM(t)表示到時(shí)刻t{M(ttp的例k與設(shè)0。XnTnTn0nX和TnnXnTn和的n,則計(jì)數(shù)過(guò)程{NttXX12的例9例3sst{|()PTsNt1tn2Nt)nn,TT,Tn12!(,,fttt)0ttt，12nt12nnt]例的t(t)[()]EtTTiiii12.3Poisson過(guò)程的推廣{N(t),t(t)0(t0)過(guò)(i)Nt)(ii){Ntt(iii)PN(th)Nt)1t)ho(h(iv)PN(th)Nt)2o(h{N(t),t(t)0(t0)N(0)（2）{N(t),t過(guò)s0N(ts)N(t)t，mtsmt)()udutstmt)(sds的t02.3.1：設(shè){N(t),t(t)0(t0)t0*()Nt(Nmt()),{*()}則Nt1的1例55均20{X(t),ttN(t){(),Ntt{,2,}YiXt)Yiii1{N(t),t例過(guò)程{N(t),tY[0,]t時(shí)i{X(t),t間保險(xiǎn)公司需要賠付的總金額就是一個(gè)復(fù)合過(guò)程，其中N(t)Xt)Y。ii1,,例SS的12(n2,)SYSnnn(n2,)Y[0,]Stnn例的(t)Xt見(jiàn){X(t),tN(t){(),tXt)Yt0，Ntii1(t)X[][([]，XtYEY2E[Xt)]tEY]211i例例2.3.6：設(shè)顧客以每分鐘60Nt{(),的t{N(t),t2.3.3：設(shè){N(t),t[]2EE[N(t)]tE[]；[()]VarNt[][]tEtVar例,P()p,121P()1pq，0p1tn21第三章Markov鏈3.1基本概念{X,n2}n2n0,ij,i,i,i，01n1{PX|jX,,,}i=PXXi{|jX}有iXiXi表n100n1n1nn1nn2Eni，稱{.{PX|}{X,n2}jXi為鏈n1nnp，它代表處于狀態(tài)ij,jp=PX{|}injXin1n,p（ijEppppppppp001020011121021222（,p（ijE）p0,，ijE；piE。pijE例SSS23pppppp111213212223001例0~nn率p1000000q0p00000000q0p0000001例100100000q0p00000000q0p0000001例0，2,000000pqp000000q0000q0p0000000qp2．n{PX|jX},;ijEmn為1pn()iijmnm()npPn()(n)為nij0ij1ijpp(0)ij()和pnijijn,mi,jE有m()pmn()ppn()ijikkjkEP()PP(nPPP(n2)Pnn例0~nn1率ppqn224例pqr1是2nXn{X,n0,}1n例1121310330例AAAB(0.30)C(0.20)D(0.20)B(0.02)B(0.60)B(0.10)B(0.20)C(0.02)C(0.06)C(0.70)C(0.10)DD(0.04)D(0.00)D(0.50)A3.2狀態(tài)的分類及性質(zhì)00n(,)ijijEijpn()ijiijij與。jii；jjiiijjkik，例SSS123{:nnpn()()ddii的ii1d1diiiidp(nd)ii0idpd()ii,jdi)d(j)。i,j3.2.4：對(duì)于任何狀態(tài)i(n)njfiijff(0)ijijij(){Xj,Xj,kn1|X},n1nnk0令1jf1jff()fnn1：fijii(n)iin1iii，則稱iii1ii1。，fii例E11200021002P13003110022ip()inn01。p()n1fn0,j1n3.2.1：對(duì)任意狀態(tài)i及p(n)f()p(nl)。ll1j1。fi且iji例E}111p,p,p,iE22200i,i1i03.3極限定理與平穩(wěn)分布1pp例：設(shè)Pq1qlimP(n)n1plim例=P(2n)003n1plim=P(2n)002ndd,0idP()當(dāng)，iniiilim0Pn()iin0ii設(shè)P(n)iinlim0j有iEPn()ijnd,,,j有ijiEP()nj0j為非常返狀態(tài)或零常返狀態(tài)j為正常返狀態(tài)1n對(duì)i,jE,有l(wèi)imP()kdnijnk1j而若例設(shè)E2100100000000001212P1212000121200i。ip,jEpppj,ijjiE1i有l(wèi)imP,jE(n)ijjn1jjlim0,()jEP()njijn例0.50.500P0.50.500.50.5例6例k1nPn1limP2nn例13.4Markov鏈的應(yīng)用：考慮一個(gè)能產(chǎn)生同類后代的個(gè)體組成的群體，每一個(gè)體生命結(jié)束時(shí)以概率p(j2)jjXX01Xn鏈nXnn1XXZ0nnii1iZniEXini0{}群體消亡P0011設(shè)p則003.5連續(xù)時(shí)間Markov鏈鏈過(guò)程{XttEs,t0i,jE及有t)j|),)u}t)j|){Xt),t(s,t){X(ts)j|X(s)}p(s,t)p(s,t)Pp(s,t)(s,t)pt)與sp,ij(t)p(t)P{Xt),t0iiE)。以iiiiiip1ipijijE1{Nt),t，取值為}例的第2章可知，它在任意一個(gè)狀態(tài)i1到i例的例的例的sX記t{Xt),tt)的1nsnnsnsnnp(t)ij(t)0pijpt)1ijE()pts()()ptps—kE：1pt)limqtt0p(t)(2)limqtt0qqiiijjiqqiiijjii,jE,t0且qqji()()()p'tqptqptijikiiijkipt)qpt)qpt)ki例例過(guò)程{Xt),t例121010000002102120141414040P0001000001012100200pn()1ijfffp()nn1ij1pn1(n)jjn1Xnn{X,n0}Pn{X,n0}n1N0和0101第四章更新過(guò)程4.1更新過(guò)程的定義及若干分布,XX12{Nt),tXn[]nEX()xdFx，Xi=Tnnn0i1(t)sup{n:Tt}N0nT1T1T2tnnntnt例j4.2更新方程(t)Mt)mt)若M。(t)由M=()Ft=(())()ft。nFt知nnn1n1n1(t)F(t)f是nn(t)mt)M和tM(t)F(t)M(ts)dF(s)0tm(t)f(t)m(ts)f(s)ds0ft)Ft)。tK(t)H(t)K(ts)dF(s)0t),Ft)Ft)Ht),Ft)0。HtHt)tK(t)H(t)H(ts)dM(s)0(t)Ft)M是[]EX例i[ET][][][()EXXXEXENtN(t)112N(t)114.3更新定理Mt)11[]0(),。t記EXnt0d{}1PXXn0d[]記EXna0若FatMtaMt()()。。d若Fdn在}P[]記()()()EXhtt滿足：(1)ht非負(fù)不增；(2)htdt＜。n0Ht)H(t)h(t)H(tx)dF(x)t01()0hx若F()Ht0tdh(c)若F0cd()Hcn1n0例1X（iiY（ii例的1r.u(n)例N(n)MM(n)limnn例1YY…12t()YNt1[Ntt1Y]t1例以()rtTttN(t)1t()()SttT為tStN(t)Lundberg—Cramer4.4破產(chǎn)論tN(t)，t0Ut)uXkk1XkkNt)tXkE[X][1F(x)]dx；F(x){X}，x110{Nt),t0}({,的XkkTt:Ut)u)T|U(0)})c>0e[1F(x)]dx(r)E[e]edF(x)1rrxrxrxx00c()1rrx31(0);1()eLundbergu0,uLundberg—Cramer近似：存在正常數(shù)C，使得(u).()uulim1~Ce,即CeRuu習(xí)題Nt)<nT>tnNt)nTtnNt)>nT<tn{,1,2,(,)Xii}n的t)NNt)1[],uEXtNt)tu1limtnt1323{{2}，PXPXii{N(1)k}{Nk}{Nk}第五章Brown運(yùn)動(dòng)5.1基本概念與性質(zhì){Xt),t(0)0X（2）{Xt),tXt))tNt2則稱{Xt),t為{Bt),tWt),t或：如果Xt),0tBt),t0st,Bt)B(s)~Nts)0st,Bt)B(s)Buus(Bt),t0是tB(0)0的Bt。()xxB0t)Bt)xx()2(2)0例Bt),tpB和pBtt[B,Bt)：,且ntnii0maxtt0nnni1i0in1