Russ Intro_сайт (1137346), страница 3
Текст из файла (страница 3)
 ÷àñòíîñòè, äëÿ bε : R2d → Rd , σε : R2d → Rd ⊗ Rd , óäîâëåòâîðÿþùèõ,ïî êðàéíåé ìåðå, òåì æå óñëîâèÿì, ÷òî è b, σ , êîòîðûå, ïðè ýòîì, â íåêîòîðîìñìûñëå áëèçêè ê b, σ ïðè ìàëûõ çíà÷åíèÿõ ε > 0, îáîçíà÷èì:((ε)(ε)(ε)(ε)(ε)dXt = bε (Xt , Yt )dt + σ(Xt , Yt )dWt ,(15)(ε)(ε)dYt = Xt dt, t ∈ [0, T ],è, àíàëîãè÷íî:(RtRtε,hε,hε,hε,h, Yφ(s))dWs ,Xtε,h = x + 0 bε (Xφ(s), Yφ(s))ds + 0 σε (Xφ(s)Rtε,hε,hYt = y + 0 Xs ds.(16)äëÿ t ∈ [0, tj ), 0 < j ≤ N , ãäå φ(t) = ti ∀t ∈ [ti , ti+1 ).Ðàññìîòðèì òàêæå îñîáûé ñëó÷àé íåïðåðûâíîñòè ïî Ãåëüäåðó äëÿ íåðàâíîìåðíîé øêàëû ñèñòåìû è îäíîðîäíûõ ïî âðåìåíè êîýôôèöèåíòîâ äëÿ ε > 0:∀q ∈ (1, +∞], ∆dε,b,q := |b(., .) − bε (., .)|Lq (R2d ) .9Êðîìå òîãî, îïðåäåëèì∆dε,σ,γ := |σ(., .) − σε (., .)|d,γ ,γãäå γ ∈ (0, 1], |.|d,γ îçíà÷àåò íîðìó Ãåëüäåðà â ïðîñòðàíñòâå Cb,d(Rd ⊗ Rd )- íåïðåðûâíûõ ïî Ãåëüäåðó îãðàíè÷åííûõ ôóíêöèé ñ ðàññòîÿíèåì d, îïðåäåëåííûì ñëåäóþùèì îáðàçîì:∀(x, y), (x0 , y 0 ) ∈ (Rd )2 , d (x, y), (x0 , y 0 ) := |x − x0 | + |y 0 − y|1/3 .γÒî åñòü, äëÿ ôóíêöèè f èç Cb,d(Rd ⊗ Rd )|f |d,γ := sup |f (x, y)|+[f ]d,γ , [f ]d,γ :=x,y∈R2dsup(x,y)6=(x0 ,y 0 )∈R2d|f (x, y) − f (x0 , y 0 )|γ < +∞.d (x, y), (x0 , y 0 )Ïîëîæèì, ∀q ∈ (1, +∞],∆dε,γ,q := ∆dε,σ,γ + ∆dε,b,q ,÷òî áóäåò â äàëüíåéøåì êëþ÷åâîé âåëè÷èíîé äëÿ îöåíêè îøèáêè.Òåîðåìà (4.3.1).
Ôèêñèðóåì T > 0. Â óñëîâèÿõ AD, äëÿ q ∈ (4d, +∞], ñóùå-ñòâóþò C := C(q) ≥ 1, c ∈ (0, 1], òàêèå ÷òî, äëÿ âñåõ 0 < t ≤ T, ((x, y), (x0 , y 0 )) ∈(R2d )2 :|(p − pε )(t, (x, y), (x0 , y 0 ))| ≤ C∆dε,γ,q pc,K (t, (x, y), (x0 , y 0 )),ãäå p(t, (x, y), (., .)), pε (t, (x, y), (., .)) - ñîîòâåòñòâåííûå ïåðåõîäíûå ïëîòíîñòè â ìîìåíò âðåìåíè t óðàâíåíèé (12), (15), ñòàðòóþùèõ èç (x, y) â ìîìåíò âðåìåíè 0.Òåîðåìà (4.3.5).
Ôèêñèðóåì T > 0 è îïðåäåëèì âðåìåííóþ øêàëó Λh :={(ti )i∈[[1,N ]] }, N ∈ N∗ .  óñëîâèÿõ AD, ñóùåñòâóþò C ≥ 1, c ∈ (0, 1] òàêèå,÷òî äëÿ âñåõ 0 < tj ≤ T, ((x, y), (x0 , y 0 )) ∈ (R2d )2 :|pεh − ph |(tj , (x, y), (x0 , y 0 )) ≤ C∆dε,σ,γ pc,K (tj , (x, y), (x0 , y 0 )),ãäå pεh (t, (x, y), (., .)), ph (t, (x, y), (., .)) - ñîîòâåòñòâóþùèå ïåðåõîäíûå ïëîòíîñòè â ìîìåíò âðåìåíè t ðåøåíèé óðàâíåíèé (13), (16), ñòàðòóþùèõ èç òî÷êè (x, y) â ìîìåíò âðåìåíè 0.Óêàçàííûå âûøå äâå Òåîðåìû áóäóò ïîäðîáíî îáñóæäàòüñÿ â Ðàçäåëå 4.3.1.Àíàëèç óñòîé÷èâîñòè ïåðåõîäíûõ ïëîòíîñòåé áóäåò èñïîëüçîâàí, êàê è âðàáîòå [KM17], äëÿ òîãî, ÷òîáû îöåíèòü ñëàáóþ îøèáêó, àññîöèèðîâàííóþ ñîñõåìîé Ýéëåðà, ââåäåííîé ðàíåå â ðàáîòå [LM10] äëÿ óðàâíåíèé âèäà (12).
Îäíàêî, äëÿ ïðîâåäåíèÿ àíàëèçà íàì íåîáõîäèìî íåìíîãî èçìåíèòü ïðåäïîëîæåíèÿ î ìîäåëè (AD). Òî÷íåå, íàì íåîáõîäèìî èçìåíèòü óñëîâèÿ îòíîñèòåëüíîíåïðåðûâíîñòè ïî Ãåëüäåðó êîýôôèöèåíòîâ.10Âìåñòî óñëîâèÿ (AD3), ìû ïðåäïîëîæèì, ÷òî äëÿ íåêîòîðûõ γ ∈ (0, 1] ,κ < ∞,γγ/2|b(x, y) − b(x0 , y 0 )| + |σ(x, y) − σ(x0 , y 0 )| ≤ κ |x − x0 | + |y − y 0 |. ). Áóäåì ãîâîðèòü, ÷òî âûïîëíåíû óñëîè îáîçíà÷èì äàííîå óñëîâèå çà (AD3âèÿ (AD), åñëè âûïîëíåíû (AD1),(AD2), (AD3).Òåîðåìà (4.4.1).
Çàôèêñèðóåì T > 0.  óñëîâèÿõ (AD)äëÿ ëþáîé òåñòîâîéôóíêöèè f ∈ C β,β/2 (R2d ) (β−íåïðåðûâíîé ïî Ãåëüäåðó ïî ïåðâîé ïåðåìåííîéè β/2-íåïðåðûâíîé ïî Ãåëüäåðó ïî âòîðîé ïåðåìåííîé) äëÿ β ∈ (0, 1], ñóùåñòâóåò C > 0, òàêàÿ ÷òî:|E(x,y) [f (XTh , YTh )] − E(x,y) [f (XT , YT )]| ≤ Chγ/2 (1 + |x|γ/2 ).ãäå γ ∈ (0, 1] îáîçíà÷àåò èíäåêñ Ãåëüäåðà äëÿ γ, γ/2 íåïðåðûâíûõ ïî Ãåëüäåðó,îäíîðîäíûõ ïî âðåìåíè ôóíêöèé b, σ .Òåîðåìà áóäåò ïîäðîáíî ðàññìîòðåíà â Ðàçäåëå 4.4. Ìû òàê æå õîòåëè áûïðåäñòàâèòü îöåíêè, ïîëó÷åííûå äëÿ ðàçíîñòè ïåðåõîäíûõ ïëîòíîñòåép(t, (x, y), (x0 , y 0 )) è ph (t, (x, y), (x0 , y 0 )).
 êîíå÷íîì èòîãå, ìû ìîæåì ïîëó÷èòü îöåíêó ãëîáàëüíîé îøèáêè ïîðÿäêà hβ , β < γ − 1/2, ÷òî â íåêîòîðîìñìûñëå áëèçêî ê îæèäàåìîìó ïîðÿäêó hγ/2 , åñëè γ ñòðåìèòñÿ ê 1.Âîçìîæíî, æåëàþùèå óñèëèòü äàííûé ðåçóëüòàò ìîãóò âîñïîëüçîâàòüñÿèíûìè ïîäõîäàìè ê îöåíêå îøèáêè - èçìåíèòü àïïðîêñèìàöèîííóþ ñõåìó èëèñòàíäàðòíóþ äåêîìïîçèöèþ îøèáêè, èñïîëüçîâàííóþ â ðàáîòàõ [KM10], [KM17],[Fri18]).Òåîðåìà (4.5.1). Çàôèêñèðóåì êîíå÷íûé âðåìåííîé ãîðèçîíò T > 0 è øàã ïî , äëÿâðåìåíè h = T /N, N ∈ N∗ äëÿ ñõåìû Ýéëåðà.  ïðåäïîëîæåíèÿõ (AD)γ ∈ (1/2, 1] è β ∈ (0, γ − 21 ), äëÿ âñåõ t íà âðåìåííîé ðåøåòêå Λh := {(ti )i∈[[1,N ]] }è (x, y), (x0 , y 0 ) ∈ R2d , ñóùåñòâóþò C := (T, b, a, β), c > 0, òàêèå ÷òî:|p(t, (x, y), (x0 , y 0 )) − ph (t, (x, y), (x0 , y 0 )|≤ Chβ (1 + (|x| ∧ |x0 |))1+γ )suppc,K (s, (x, y), (x0 , y 0 )),(17)s∈[t−h,t]ãäå pc,K (s, (x, y), (x0 , y 0 )) åñòü ãàóññîâñêàÿ ïëîòíîñòü ïî òèïó Êîëìîãîðîâà,ââåäåííàÿ â (14), â ìîìåíò âðåìåíè s.Äàííàÿ Òåîðåìà îáñóæäàåòñÿ ïîäðîáíî â Ðàçäåëå 4.5.Ñïèñîê ëèòåðàòóðû[Aro59]D.
G Aronson. The fundamental solution of a linear parabolic equationcontaining a small parameter. Illinois Journal of Mathematics, 3:580619, 1959.11[Bai17]N.T.G. Bailey. The mathematical theory of infectious diseases and itsapplications. 2nd ed. Hafner Press, New York, 2017.[BP09]R.F. Bass and E.A. Perkins. A new technique for proving uniquenessfor martingale problems.
From Probability to Geometry (I): Volume inHonor of the 60th Birthday of Jean-Michel Bismut, pages 4753, 2009.[BT96a]V. Bally and D. Talay. The law of the Euler scheme for stochasticdierential equations: I. Convergence rate of the distribution function.Probability Theory and Related Fields, 104-1:4360, 1996.[BT96b]V. Bally and D. Talay. The law of the Euler scheme for stochasticdierential equations, II.
Convergence rate of the density. Monte CarloMethods and Applications, 2:93128, 1996.[BY89]N. Becker and P. Yip. Analysis of variations in an infection rate.Australian Journal of Statistics, 31(1), 1989.[DM10]F. Delarue and S. Menozzi. Density estimates for a random noisepropagating through a chain of dierential equations. Journal ofFunctional Analysis, 2596:15771630, 2010.[Fri18]N. Frikha.
On the weak approximation of a skew diusion by an Eulertype scheme. Bernoulli, 24(3):16531691, 2018.[GS67]I. Gihman and A. Skorohod. Stochastic Dierential Equations. Naukovadumka, Kiev., 1967.[GS82]I. Gihman and A. Skorohod. Stochastic Dierential Equations andApplications. Naukova dumka, Kiev., 1982.[IKO62]A. M. Il'in, A. S. Kalashnikov, and O. A.
Oleinik. Second-order linearequations of parabolic type. Uspehi Mat. Nauk, 173(105):3146, 1962.[JYC10]M. Jeanblanc, M. Yor, and M. Chesney. Mathematical Methods forFinancial Markets. Springer Finance, London, 2010.[KKM17] V. Konakov, A. Kozhina, and S Menozzi.perturbed Diusions and Markov Chains.Statistics, 21:88112, 2017.Stability of densities forESAIM: Probability and[KM00]V. Konakov and E.
Mammen. Local limit theorems for transitiondensities of Markov chains converging to diusions. Probability Theoryand Related Fields, 117:551587, 2000.[KM02]V. Konakov and E. Mammen. Edgeworth type expansions for Eulerschemes for stochastic dierential equations. Monte Carlo Methods andApplications, 83:271285, 2002.12[KM10]V. Konakov and S.
Menozzi. Weak error for stable driven stochasticdierential equations: Expansion of the densities. Journal of TheoreticalProbability, 24-2:554578, 2010.[KM17]V. Konakov and S Menozzi. Weak Error for the Euler SchemeApproximation of Diusions with non-smooth coecients. ElectronicJournal of Probability, 22:147, 2017.[KMM10] V. Konakov, S.
Menozzi, and S. Molchanov. Explicit parametrix andlocal limit theorems for some degenerate diusion processes. Annales del'Institut Henri Poincare, Serie B, 464:908923, 2010.[Kol34]A. N. Kolmogorov. Zufallige Bewegungen (zur Theorie der BrownschenBewegung). Annals of Mathematics, 2-35:116117, 1934.[Koz16]A. Kozhina. Stability of densities for perturbed degenerate diusions.Teoriya Veroyatnostei i ee Primeneniya, 3:570579, 2016.[Kry96]N. V. Krylov. Lectures on elliptic and parabolic equations in Holderspaces. Graduate Studies in Mathematics 12. AMS, 1996.[KS84]S.
Kusuoka and D. Stroock. Applications of the Malliavin calculus.I. Stochastic analysis (Katata/Kyoto, 1982), North-Holland Math.Library, 32:271306, 1984.[KS85]S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II.Journal of the Faculty of Science, the University of Tokyo, 32:176, 1985.[LM10]V. Lemaire and S. Menozzi. On some non asymptotic bounds for theEuler scheme. Electronic Journal of Probability, 15:16451681, 2010.[Mar55]G. Maruyama. Continuous markov processes and stochastic equations.Rendiconti del Circolo Matematico di Palermo, 4:48, 1955.[Men11]S. Menozzi. Parametrix techniques and martingale problems forsome degenerate Kolmogorov equations. Electronic Communications inProbability, 17:234250, 2011.[MP91]R. Mikulevicius and E.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
