随机过程讲义(中科院-孙应飞)

20092010n1),,(PΣΩTt∈),(ωtX),,(PΣΩ}{Tt∈);,(tXXT=ωRT⊂TXRTtX→Ω×:),(ω),(⋅⋅XΩ×TTt∈),(⋅tXΩΩ∈ω),(ω⋅XTt∈),(ωtX)(ωtX)(tXT1},2,1,0{0L==NT2},2,1,0L±±{=T3],[baT=a0∞−b∞+{});(TttX∈SS1},{TH=Ω20092010=T2H,cos)(tttXπ),(∞+−∞∈t2/1}{}{==TPHP{)},(,)(∞+−∞∈ttX2),(,)cos()(∞+−∞∈+=ttAtXθωAω)2,0(~πθU3{});(+∞∞−ttXtAtXωcos)(=ω]1,0[~UA1)(tX23ωπωπωπωπ2/,/,4/3,4/,0=t)kt(X4nXn5[],0t)(tN}0),({≥ttNn{nS},2,L1,=nSn{}L,2,1,{=nSn}0),(≥ttN621220092010T1T2TS)(tX1)(tX2)(tXabcdabMarkovcdefgPoissionhMarkovMarkovMarkovMarkovMarkov320092010{});(TttX∈a{});(TttX∈)}({)(ˆ)(tXEtmtX==µb{});(TttX∈})]()({[)(ˆ)(22ttXEtDtXXXµσ−==c{});(TttX∈)]}()()][()({[ˆ),(ttXssXEtsCXXXµµ−−=d{});(TttX∈)}()({ˆ),(tXsXEtsRX=e)]}}()([{exp{ˆ),,,;,,,(112121nnnnXtXutXujEtttuuu++=LLLφ{}1,,,,),,,,;,,,(212121≥∈nTttttttuuunnnXLLLφ{});(TttX∈)()(),()()()}()({)]}()()][()({[ˆ),(tstsRtstXsXEttXssXEtsCXXXXXXXXµµµµµµ⋅−=⋅−=−−=22)]([),(),()()(tttRttCtDtXXXXXµσ−===711)(tX)1(),2/1(XX2)(tX))1(),2/1(XX(32009201082{});(TttX∈{});(TttY∈a{});(TttX∈{});(TttY∈)]}()()][()({[ˆ),(ttYssXEtsCYXXYµµ−−=b{});(TttX∈{});(TttY∈)}()({ˆ),(tYsXEtsRXY=)()(),(),(tstsRtsCYXXYXYµµ⋅−={});(TttX∈});({TttY∈Tts∈,0),(=tsCXY)}({)}({)()(),(tYEsXEtstsRYXXY⋅=⋅=µµ{});(TttX∈{});(TttY∈{});(TttX∈Nn∈∀)1(niTti≤≤∈∀})(,,)(,)({),,,;,,,(22112121nnnnXxtXxtXxtXPtttxxxF≤≤≤=LLL20092010{}1,,,,),,,,;,,,(212121≥∈nTttttttxxxFnnnXLLL{});(TttX∈1(),,2,1nL),,,(21njjjL),,,;,,,(),,,;,,,(21212121nnjjjjjjXnnXtttxxxFtttxxxFLLLL=2nm),,,;,,,(),,,,,,;,,,,,,(212112121mmXnmmmXtttxxxFtttttxxxFLLLLLL=+∞+∞+12{});(TttX∈Kolmogorov{}1,,,,),,,,;,,,(212121≥∈nTttttttxxxFnnnXLLL);({}TttX∈{}1,,,,),,,,2121≥;,,,(21∈nTttttttnnLL);(xxxFnXL{}TttX∈})(,,)(,)({),,,;,,,(22112121nnnnXxtXxtXxtXPtttxxxF≤≤≤=LLL{});(TttX∈{});(TttX∈{});(TttX∈{});(TttY∈Nm∈,nTttnt∈,,,21LTtttm∈′′′,,,21Lmn+20092010))(,),(),(),(,),(),((2121mntYtYtYtXtXtX′′′LL(,)(,,)({,,;,,,;,,,(111212121nnnnXYtYxtXxtXPyytttxxxF′≤≤=LLLL});(TttX∈});(TttY∈n+Nmn∈,t,21});(TttX∈});(TttY∈(),,,;,,,(,,,;,,,;,,,(2121212121YnnXnnXYyFtttxxxFyytttxxxF⋅=LLLLL});(TttX∈});(TttY∈});(TttX∈);({ttY∈});({TttX∈});(TttY∈δδ)(tf∫∫∞+∞−→∞+∞−=tdtftlim)()(0εεδδ≤=εεδεtttt,00,10,0)()(tεδδ)(tδ0ε})(,,)),,,;,121mmmnytYyttty≤′≤′′′LL{{mtTtn∈,,LTtttm∈′′′,,,21L{{),,,;,,,),,,;212121mnmntttyyttty′′′′′′LLL{{{}T{4δ1Diractdtft)()(≤ε2009201011)(0==∫∫∞+∞−εεεδtdtdt1)(=∫∞+∞−tdtδ1)(tδ0=t∞0≠t01=)(∫∞+∞−tdtδ2δδ)(tf)0()()(ftdtftI=∫δI0=t)0()()(ftdtft=∫∞+∞−δ)()()(00tftdtftt=−∫∞+∞−δ2δXL,2,1}{===ipxXPiiδX∑∞=−=1)()(iiixxpxfδδX∫∑∑∞−∞=≤−==≤=xiiixxiudxuppxXPxFi1)(}{)(δδ2009201051(),YX0),(jiyx},{≥===jijipyYxXP∑===jjyYyYXEIYXEj}{)(ˆ}{)(ω}{YXEXY1)()(ωjyYI==∉=∈==})(:{,0})(:{,1)()(jjyYyYyYIjωωωωωωω2}{YXEY)(}{}{kjyYXEyYXEkj≠=≠=}{}}{}{{jjyYPyYXEYXEP====}}{}{:{jkjyYXEyYXEkD====∑∈====jDkkjyYPyYXEYXEP}{}}{}{{3}{YXEY}{}{}{}}{{XEyYPyYXEYXEEjjj====∑9),(YX}{YXE}}{{},{YXEEXE20092010XY123jp•12/274/271/277/2725/277/273/2715/2731/272/272/275/27•ip8/2713/276/2712),(yxfY)(yfY}{YXEa}{YXEYyY=}{yYXE=bD}{}}{{DYXEDYYXEE∈=∈}{YXEXY1}{YXEY}{)()(}}{{XEdyyfyYXEYXEEY∫+∞∞−===10(),,,,(~),222121σσρµµNYX)(}{1122µσσρµ−+==xxXYE)(}{1122µσσρµ−+=XXYEYxX=[]−−−−−−==−=2111222222/122)()1(21exp)1(21)(),()(µσρσµρσρσπxyxfyxfxyfXxXY20092010)]1(),([~)(22211122ρσµσρσµ−−+−=xNxyfxXY)()(}{11122µσρσµ−+===−+∞∞−=∫xdyxyyfxXYExXY)(}{11122µσρσµ−+=−XXYE3a}}{{}{YXEEXE=b∑∑===niiiniiiYXEYXE11}{ααa.s.)1(nii≤≤αc})({)(})()({YXgEYhYYhXgE=a.s.d}})({)({)}()({YXgEYhEYhXgE=eYX,}{}{XEYXE=),(~),y(xfYX}})({)({)()(})({)()()(),()(),()()()}()({YXgEYhEdyyfyhyYXgEdyyfyhdxyfyxfxgdxdyyxfyhxgYhXgEYYY=====∫∫∫∫∫∞+∞−∞+∞−∞+∞−∞+∞−∞+∞−1∫∞+∞−==dyyfxhyYXgEyhXgEY)()(})({)}()({∫+∞∞−==dyyfyYAPAPY)(}{}{∫+∞∞−=≤=≤dyyfyYxXPxXPY)(}{}{6a20092010)cos()(θω+=tAtXAωθ(),ππ−≤≤−=,0,21)(πππθxxft)(tXt)cos()(θω+=tAtXθ})cos({})({)()(ytAPytXPyFtX≤+=≤=θωAy−0)()(=yFtXAy+≥1)()(=yFtXAyA+≤−−+=+−++−=+=≤−−≤−==≤+=≤=∫∫−−−AyttAyAytdxdxtAyAytPytAPyyXPyFtAyAyttXarccos1arccosarccos2121}){arccos}arccos({})cos({})({)(arccosarccos)(πωπωππωπππθωωθπθωπωωπUAyA+≤−)(tX22/)()(1)()(yAyFyftXtX−==π)(tX+≤≤−−=,0,1)(22)(AyAyAyftXπ20092010b),(,)cos()(∞+−∞∈+Ω=ttAtXθAΩθ),(350250~UΩ)2,0(~πθU∉∈=),0(,0),0(,2)(0020AaAaAaafA)(tX)cos()(θω+=tatY,aω)2,0(~πθUa)(tY+≤≤−−=,0,1)(22)(ayayayftYπ)(tX)(tY),()(ω=Ω==aAtXtY∫∫Ω≤=≤),(},)({})({ωadFAxtXPxtXPΩ,AωωωωωdadfafdadafadFA)()(),(),(Ω==)(tX≤−=⋅⋅−==−=∫∫∫∫0022020350250202222)(,0,2100121)()(1)(0AxAxxAAdAaxadadadfafxaxfAxAtXπωπωωπΩ20092010cx0=txL,3,2,1=txppq−=1nknkiiξiξ11,}1pP=+={iξqpPi=−=−=1}1{ξnt=nXnX001==∑=XXniinξiξiξkξki≠nt=:nnnnnn−−−−−−−),2(),4(,,4,2,Lnmm1+=iξnm−mn−1−=iξknmmnmXniin=−=−×−++×==∑=2)1()()1(1ξ2knm+=22222)!2()!2(!}{knknknknknnmnmmnnqpknknnqpCqpCkXP−+−++−−+====mnk()nknk)(nk20092010dT00T)(tX}2,1,1{−−+,2+4/1}2)({}1)({}1)({}2)({=−==−==+==+=tXPtXPtXPtXPu(),00T21,tt)(tX())(1tX(,)2tXXk(t)t21,ttCt21,t21,ttcCtt21,t21,t1021Ttt−1}{=CP0}{=cCP2021Ttt≤−t21,tθ1t0=tu(),00Tθ(),101tTt−t21t20092010012020202111}{1}{}{}{0201TttdTTtPTtPTtPCPTtTtc−−=−=−−=−=+=∫−−θθθθt21t0210211}{}{201TttdTtPCPtTtc−−===∫−θθ021021}{1}{TttCPTttCPc−=−−=}{),(}{),(),(212121212121ccCXXCXXXXCPCxxfCPCxxfxxfctttttt+=−×−=∑∑−−=−−=2