Лекционный курс (1134109), страница 6
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Åñëè (ξn , Fn )n≥0 - ìàðòèíãàë, òî ξn = 4Xn = Xn −Xn−1 , 4X0 = 0. Íàäî ïðîâåðèòü, ÷òî E(Xn |Fm ) = Xm ⇐⇒ E(4Xn |Fn−1 ) =0.)Îïð.: ïðîöåññ {An , n ≥ 0} -íàçûâàåòñÿ ïðåäñêàçóåìûì, åñëè An ∈ Fn−1 /B(R).(Ïèøóò (An , Fn−1 ).)Ò Å Î Ð Å Ì À (Äóá). Ïóñòü (Xn , Fn ) - íåêîòîðûé ñëó÷. ïðîöåññ,E|Xn | < ∞. Òîãäà Xn = Mn + An , ãäå Mn - ìàðòèíãàë, An - ïðåäñêàçóåìûéïðîöåññ è A0 ≡ 0, F−1 = {∅, Ω}. Òàêîå ðàçëîæåíèå åäèíñòâåííî.Óïðàæíåíèå. P (εk = 1) = P (εk = −1) = 1/2 , Sn = ε1 + ...
+ εn , S0 = 0.nPÄîêàçàòü, ÷òî |Sn | =sgn(Sk−1 )4Sk + Ln (0), ãäå Ln (0) - ÷èñëî íóëåé,k=1{k = {1, ..., n}, Sk−1 = 0}. Ýòî äèñêðåòíûé âàðèàíò ôîðìóëû Òàíàêà.Ëåêöèÿ 8Äîêàçàòåëüñòâî:(òåîðåìû Äóáà)◦ Ïóñòü (*) ðàçëîæåíèå ñïðàâåäëèâî. Òîãäà 4Xn = 4Mn +4An .
E(4Xn |Fn−1 ) =E(4Mn |Fn−1 ) + E(4An |Fn−1 ) = 4An , ò.ê.29âî-ïåðâûõ E(4Mn |Fn−1 ) = 0, ò.ê. ýòî ìàðòèíãàë-ðàçíîñòü.âî-âòîðûõ A - ïðåäñêàçóåìàÿ, à çíà÷èò Fn−1 -èçìåðèìà.PÈòàê, 4An = E(4Xn |Fn−1 ), ò.ê. A0 = 0, òî An =E(4Xn |Fn−1 ). Åäèíñòâåííîñòüäîêàçàíà.Îáðàòíî. Ïîëîæèì A0 ≡ 0nPAn =E(4Xk |Fk−1 ) - ïðåäñêàç. Ïóñòü Mn = Xn − An , òîãäà 4Mn =k=14Xn − 4An = 4Xn − E(4Xn |Fn−1 )E(4Mn |Fn−1 ) = 0, •Çàìå÷àíèå. Èç äîêàçàòåëüñòâà âèäíî, ÷òîX -ñóáìàðòèíãàë [E(Xn+1 |Fn ) ≥ Xn ]mA - íå óáûâàåò [4An ≥ 0]Ïðèìåð. Ïóñòü ε1 , ε2 , . .
. - í.î.ð.ñ.â. εk = ±1 ñ âåð.12S0 = 0, Sn = ε1 + . . . + εn . Ïðîöåññ Xn = |Sn | ñ h(x) = |x| âûïóêëà âíèç. Àïðîöåññ Xn - ñóáìàðòèíãàë.4An = E(4Xn |Fn−1 ) = E(Xn |Fn−1 ) − Xn−1Xn = |Sn |E(|Sn | | Fn−1 ) = E(|Sn−1 + εn | | Fn−1 )Çàïèøåì áåç ìîäóëÿ |Sn−1 +εn | = (Sn−1 +εn )I{Sn−1 > 0}+|εn |I{Sn−1 = 0}−− (Sn−1 + εn )I{Sn−1 < 0}.|εn | = 1E(|Sn | | Fn−1 ) = E(Sn−1 I{Sn−1 > 0} | Fn−1 ) ++ E(εn I{Sn−1 > 0} | Fn−1 ) + E(I{Sn−1 = 0} | Fn−1 −− E(Sn−1 I{Sn−1 < 0} | Fn−1 ) − E(εn−1 I{Sn−1 < 0} | Fn )Â ýòîì âûðàæåíèè E(εn I{Sn−1 > 0} | Fn−1 ) = 0 è E(εn−1 I{Sn−1 < 0} |Fn ) = 0.
Ò.ê.1.E(ξη|A) = ηE(ξ|A), åñëè η - èçìåðèìà îòíîñèòåëüíî A è E|ξη| < ∞ E|ξ| <∞.2. Åñëè ξ è A íåçàâèñèìû, òî E(ξ|A) = Eξ .Ïî ýòèì ñâîéñòâàì, E(εn I{Sn−1 < 0} | Fn−1 ) == I{Sn−1 > 0} E(εn |Fn−1 )kÒàêèì îáðàçîì,Eεn = 0E(|Sn | | Fn−1 ) = Sn−1 I{Sn−1 > 0} + I{Sn = 0} − Sn−1 I{Sn−1 < 0}|Sn | = (Sn−1 + εn )I{Sn−1 > 0} + I{Sn−1 = 0} − (Sn−1 + εn )I{Sn−1 < 0}4An = |Sn | − E(|Sn | | Fn−1 ) = εn I{Sn−1 > 0} − εn I{Sn−1 < 0} == (4Sn )sgn(Sn−1 )nPAn =4Sk sgn(Sk−1 ) (A0 = |Sn | = 0)k=130Mn = Ln (0) =nPI{Sk−1 = 0} =k=1=êîëè÷åñòâî{k ∈ {0; . .
. ; k − 1}, Sk = 0}Äîêàçàíà ôîðìóëà (äèñêðåòíûé âàðèàíò ôóðìóëû Òàíàêà )nP|Sn | =4Sk sgn(Sk−1 ) + Ln (0)k=1qn→∞2E|Sn | = ELn (0) ELn (0) ∼πnÏî ÖÏÒSn D√→nZ ∼ N (0, 1)DÂñïîìíèì ËÅÌÌÓ Yn → Y.h -íåïðåðûâíîå îòîáðàæåíèå. ÒîãäàDh(Yn ) → h(Y ). Âîçüìåì â êà÷åñòâå h(x) = |x| ⇒|Sn | D√ → |Z|nDÀ ÷òî çíà÷èò →:Ef (Yn ) → Ef (Y ) ∀f íåïðåðûâíîé è îãðàíè÷åííîé.DÅñëè ξn → ξ è {ξn } - ðàâíîìåðíî èíòåãðèðóåìû, òîEξn → Eξ , ÷òî æå çíà÷èò ðàâíîìåðíàÿ èíòåãðèðóåìîñòü:lim sup E(|ξn |I{|ξn | > c}) = 0. Ðàññìîòðèì åùå äîñòàòî÷íîå óñëîâèå ðàâíîìåðíîék→∞èíòåãðèðóåìîñòè (ð.è.):sup E|ξn |1+δ < ∞ äëÿ íåêîòðîãî δ > 0, ò.ê.nE(|ξn |I{|ξn | > c}) ≤µE|ξn |1+δCδ- ïî íåðàâåíñòâó ×åáûøåâà.¶2ESn2DSnDε1 + . . . + Dεn1 + ...
+ 1====1nnnnq³´|2√nÑëåäîâàòåëüíî, E |S→E|Z|=πnE|Sn |√n=Ë Å Ì Ì À. Ïóñòü Y = {Yt , t ≥ 0}- ïðîöåññ ñ íåçàâèñèìûìè ïðèðàùåíèÿìè,ò.÷. EeαYt < ∞ è Eeα(Yt −Ys ) < ∞ äëÿ íåêîòîðîé α ∈ R è âñåõ s, t ∈ [0; +∞)eαYtÎïðåäåëèì Zt = EeαYt , t ≥ 0.Ïðîöåññ Z = {Zt , t ≥ 0} ÿâëÿåòñÿ ìàðòèíãàë-ðàçíîñòüþ òîãäà è òîëüêîòîãäà, êîãäൠαYt ¶eEeαYtE=αYe sEeαYsÄîêàçàòåëüñòâî.◦ E(Zt |FS ) = Zs s ≤ t¡¡¢¢E eαYs eα(Yt −Ys ) |FsE eαYt |Fs==E(Zt |Fs ) =EeαYtEeαYtµ¶eαYseαYt ?=E= ZsEeαYteαYs31À òàê êàê Yt − Ys è F íåçàâèñèìû, òî E³eαYteαYs´=EeαYtEeαYs•Çäåñü îïÿòü äàåòñÿ îïðåäåëåíèå ìîäåëè Êðàìåðà-Ëóíäáåðòà.Xt (ω)XYt = y0 + ct −ηj (ω)j=1t ≥ 0.{ξj } è {ηj } - íåçàâèñèìû.
(Ìîäåëü ñòðàõîâàíèÿ)η1 Iη2 I¾-η3 Itξ1 (ω) ξ2 (ω)ξ3 (ω)âûïëàòûct - âçíîñûÌîìåíò ðàçîðåíèÿ τ = inf{t : Yt < 0}. Âîïðîñ òàêîé: îöåíèòü P (τ < ∞) ≤?Äîêàæåì, ÷òî Y = {Yt , t ≥ 0} - ïðîöåññ ñ íåçàâèñèìûìè ïðèðàùåíèÿìè:Yt1 Yt2 − Yt1 . . . Ytm − Ytm−1kkξ1ξ2mQξ1 , . . . , ξm - íåçàâèñèìû ⇔ Eeiν1 ξ1 +...+iνm ξm =eiνk ξkk=1∀νk ∈ R k = 1, . . .
, mEeiξk = Eeiνk (Ytk −Ytk−1 ) =[áåç îãðàíè÷åíèÿ îáùíîñòè Yt ==∞ P∞Pr=o j=oNtPj=0ηj ]Eeiνk (Ntk −Ntk−1 ) I{Ntk−1 = j}I{Ytk − Ytk−1 = r} =Pj+r=∞ P∞PiνkEeηlI{Ntk−1 = j}I{Ytk − Ytk−1 = r} =l=j+1r=o j=o[åñëè ηj è ξj íåçàâèñèìû]Pj+r==∞ P∞Pr=o j=o∞PiνkEel=j+1ηl(λtk−1 )j −λtk−1 λr (tk −tk−1 )r −λ(tk −tk−1 )eej!r!rk−1 ))(Eeνk η1 )r (λ(tk −te−λ(tk −tk−1 ) =k!r=0iν η1= e−λ(tk −tk−1 )+λ(tk −tk−1 )Ee k = eλ(tk −tk−1 )(EeÍî ëó÷øå ïðîâåðèòü âûêëàäêè çäåñü -32iνk η1−1).=Óïðàæíåíèå. Âû÷èñëèòüµEZt =eαYtEeαYteαYteαYs¶= Eeα(Yt −Ys ), a = const, t ≥ 0Ë Å Ì Ì À. {Xt , t ≥ 0} - ìàðò., èìåþùèé ï.í. íåïðåðûâíûå ñïðàâàòðàåêòîðèè. Òîãäà äëÿ ëþáûõ îãðàíè÷åííûõ îïöèîíàëüíûõ ìîìåíòîâ (îòíîñèòåëüíî(Ft )t≥0 ) τ è σ :EXτ = EXσëåììà áóäåò åùå ðàç ñôîðìóëèðîâàííà è äîêàçàííà íà ñëåäóþùåé ëåêöèè.Óïðàæíåíèå.
X - èìååò ï.í. íåïðåðûâíûå ñïðàâà òðàåêòîðèè, G - îòêðûòîå.Òîãäà τG = inf{t ≥ 0 : Xt ≤ G} - îïöèîíàëüíûé ìîìåíò îòíîñèòåëüíî(FtX )t≥0 .Ñëåäñòâèå. τ = inf{t ≥ 0, Yt < 0} = inf{t ≥ 0, Yt ∈ (−∞; 0)} - îïöèîíàëüíûéìîìåíò. (Ñì. "Ïîñëåäîâàòåëüíûé ñòàòèñòè÷åñêèé àíàëèç"Øèðÿåâ)Ëåêöèÿ 9Ïðåäïîëîæåíèå. (*) ψ(v) = Eevη1 < ∞ ∀v ∈ R(ηi ≥ 0). Óñëîâèå ñïðàâåäëèâî,åñëè |ηi | < constÏðîïóùåíî íåìíîãî. Íóæíî äîïå÷àòàòü.Ñëåäîâàòåëüíî, evYt = etg(v)−vy0Ee−vYs = esg(v)−vy0µ −vYt ¶eEe−vYt(t−s)g(v)E=e=−vYseEe−vYs−vYte−vYt −tg(v)+vy0Èòàê, Zt = Ee−vYT = eÅñëè Zt ìàòðèíãàë, òî constZt - òîæå ìàðò. Òàêèì îáðàçîì, Xt = e−vYt −tg(v) , t ≥0 - ìàðòèíãàë.τ = inf{t > 0, Yt < 0} = inf{t > 0, Yt ∈ (−∞; 0)}.
Çàìåòè, ÷òî Yt - ïðîöåññ,èìåþùèé íåïðåðûâíûå ñïðàâà òðàåêòîðèè. Ïîýòîìó {τ < t} ∈ Ft - ò.å.îïöèîíàëüíûé ìîìåíò.Ë Å Ì Ì À. {Xt , t ≥ 0} - ìàðò., èìåþùèé ï.í. íåïðåðûâíûå ñïðàâàòðàåêòîðèè. Òîãäà äëÿ ëþáûõ îãðàíè÷åííûõ îïöèîíàëüíûõ ìîìåíòîâ (îòíîñèòåëüíî(Ft )t≥0 ) τ è σ :EXτ = EXσåñëè τ ≤ c è σ ≤ c (îãðàíè÷åííûå ìîìåíòû).33Äîêàçàòåëüñòâî.◦ 0 ≤ τ ∧ t ≤ t. Ñëåäîâàòåëüíî, EX0 = EXτ ∧tXt = e−vYt −tg(v) t ≥ 0Yt -íåïðåð. ñïðàâà., à tg(v) - íåïðåð.EX0 = e−vy0EXτ ∧t = Ee−vYτ ∧t −(τ ∧t)g(v) ≥≥ Ee−vYτ ∧t −(τ ∧t)g(v) I{τ ≤ t} == Ee−vYτ −τ g(v) I{τ ≤ t} ≥[−vYτ ≥ 0, ò.ê. v > 0, Yτ < 0, ò.ê.
τ - ìîìåíò âûõîäà ê îòðèö. çíà÷åíèÿì.]≥ Ee−τ g(v) I{τ ≤ t} ≥ inf e−τ s EI{τ ≤}.0≤s≤t[EI{τ ≤ t} = P ({τ ≤ t}) ≤ e−vy0 sup esg(v) ,0≤s≤te−vy0 = EX0 ]Âûáåðåì v = v0 òàê, ÷òîáû g(v0 ) = 0. Âñïîìíèì óñòðîéñòâî ôóíêöèèg(v) = λ(ψ(v) − 1) − vc×òî òàêîå ψ(v) = Eevη1 , ψ(0) = 1, ψ 0 (v) = Eη1 evη1 , ψ 0 (0) = Eη1 = a > 0 ïðåäïîëîæèì, ÷òî g 0 (v) = λψ 0 (v) − c, g 0 (0) = λa − c < 0Âîçíèêàåò c > λaEη1 - ìàò. îæèäàíèå âûïëàò. λ - ïëîòíîñòü ïóàñ.
ïîòîêà. ψ 00 (v) = E(η12 evη1 ) >0∃!v0 íà (0; +∞).Èòàê, ñóùåñòâóåò åäèíñòâåííàÿ òî÷êà, ò.÷. g(v0 ) = 0. Òîãäà P {τ ≤ t} ≤e−v0 y0 èòîãåP {τ ≤ ∞} ≤ e−v0 y0-ýòî íàçûâàåòñÿ îñíîâíîé òåîðåìà ñòðàõîâîé ìàòåìàòèêè. Îñòàëàñü ËÅÌÌÀ:Ò Å Î Ð Å Ì À (Äóá) Ïóñòü (Xn )n≥0 - ìàðòèíãàë. Ïóñòü τ è σ ìàðêîâñêèå ìîìåíòû, òàêèå ÷òî σ ≥ τ ≥ const. ÒîãäàE(Xτ |Fσ ) = XσÏóñòü {Xt , t ≥ 0} - ìàðòèíãàë, èìåþùèé íåïðåðûâíûå ñïðàâà òðàåêòîðèè.Ïóñòü τ è σ - îãðàíè÷åííûå îïöèîíàëüíûå ìîìåíòû. ÒîãäàEXσ = EXτÂâåäåì τ (n) = 2−n [2n τ + 1]σ(n) = 2−n [2n σ + 1]Òîãäà τ è σ - ìàðêîâñêèå ìîìåíòû îòíîñèòåëüíî ôèëüòðàöèè Fk2−n , k =0, 1, . .
.{τ (n) ≤ k2−n } = {τ < k2−n } inFk2−nτ (n) ↓ τ σ(n) ↓ σ ñèëó íåïðåðûâíîñòè ñïðàâà òðàåêòîðèé {Xt , t ≥ 0} èìååì Xτ (n) → Xτ è34Xσ(n) → Xσ . Çàìåòèì, ÷òî σ(n) < τ (n). Ïî ò. Äóáà E(Xτ (n) |Fσ(N ) ) = Xσ(n) ñèëó ñêàçàííîãî {Xσ(n) } - ÿâë. ðàâí. èíòåãð. Ïîýòîìó Xσ(n) áóäåò èçìåðèìîîòíîñèòåëüíî Fσ(m) ïðè n ≥ m. Òàêèì îáðàçîì, Xσ - áóäåò èçì.
îòíîñèòåëüíîTG = F σ(n) . Åñëè Yn ∈ A|B(R) è Yn → Y ï.í. è Y ∈ A|B(R)nE(Xτ |G) = Xσ⇒ EXτ = EXσXσ ÿâëÿåòñÿ G -èçìåðèìîé âåëè÷èíîé. Ñëåäîâàòåëüíî òðåáóåòñÿ ïðîâåðèòü:EXτ IA = EXσ IA∀A ∈ G- ýòî âûòåêàåò èç òîãî, ÷òî Xτ (n) → Xτ ï.í. è â L1Xσ(n) → Xσ ï.í. è â L1E(Xτ (n) |Fσ(n) ) = Xσ(n) •ÌÀÐÊÎÂÑÊÈÅ ÏÐÎÖÅÑÑÛ.(Àíäðåé Àíäðååâè÷ Ìàðêîâ)Ïóñòü X + {Xt , t ∈ T }, T ⊂ RXt : Ω rightarrowSt , F≥t = σ{Xs , s ≥ t, s ∈ T }Ïóñòü èìååòñÿ ôèëüòðàöèÿ F = (Ft )t∈T , ò.÷. X = {Xt , t ∈ T } ñîãëàñîâàíàñ F.Îïð.: X - ìàðêîâñêèé ïðîöåññ, åñëè ∀c ∈ F≥t P (C|Ft ) = P (C|Xt ).Óïðàæíåíèå. Åñëè St , t ∈ T - áîðåëåâñêîå ïðîñòðàíñòâî, òîE(h(X)|XS1 .
. . XSm , Xt ) = E(h(X)|Xt )(∗∗)∀s1 < . . . < sm < t < u, ãäå s,t,u ∈ T è h - ïðîèçâîëüíàÿ îãð. è èçì. Îáû÷íîT = R+ èëè T = Z+Îïð.: Ìàðêîâñêèé ïðöåññ íàçûâàåòñÿ öåïüþ Ìàðêîâà, åñëè âñå St = S èδ í.á.÷.ñ. (íå áîëåå, ÷åì ñ÷åòíî), ò.÷. S èëè {0, 1, . . . , r}, èëè Z+ .Äëÿ öåïåé Ìàðêîâà îïðåäåëåíèå (**) ⇔ P (Xn = j|XS1 = i1 , .
. . , XSm =im , Xt = i) == P (Xn = j|Xt = i), åñëè P (XS1 = i1 , . . . , XSm = im , Xt = i) 6= 0Ò Å Î Ð Å Ì À. Ïóñòü {Xt , t ≥ 0} - ïðîöåññ ñ íåçàâèñèìûìèïðèðàùåíèÿìè (ñî çíà÷åíèÿìè â Rk ). Òîãäà X - ìàðêîâñêèé ïðîöåññ (îòíîñèòåëüíîåñòåòñòâåííîé ôèëüòðàöèè).Ñëåäñòâèå. Âèííåðîâñêèé ïðîöåññ ÿâëÿåòñÿ ìàðêîâñêèì. Ïóàññîíîâñêèé ïðîöåññ- ìàðêîâñêèé.35Äîêàçàòåëüñòâî.(òåîðåìà)◦ Ðàññìîòðèì s1 < . . . < sm < t < u, σ{XS1 , . . . , XSm , Xt } = σ{XS1 , XS2 −XS1 , .
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