Диссертация (1137347)
Текст из файла
National Research University Higher School of EconomicsInternational Laboratory of Stochastic Analysis and Its Applicationsas a manuscriptAnna KozhinaParametrix Method and its Applications in Probability TheoryPhD thesisfor the purpose of obtainingPhilosophy Doctor in Mathematics HSEAcademic supervisor:Valentin Konakov Doctor ofScience, professorMoscow - 2018ContentsGlossary31 Introduction42 Parametrix technique2.1 Review . . . .
. . . . . . . . . . . . . . . . . . .2.2 Other developments in Parametrix . . . . . . .2.3 The parametrix method for diffusion processes2.4 Markov Chains . . . . . . . . . . . . . . . . . .........................................................................13131415183 Stability of diffusion transition densities3.1 Stability results . . . . . . . . . .
. . . . . .3.1.1 Assumptions and Main Results. . . .3.1.2 On Some Related Applications. . . .3.1.3 Derivation of formal series expansion3.1.4 Stability of Parametrix Series. . . . .3.1.5 Stability for Markov Chains. . . . .. . . . . .. . . . . .. . . . . .densities .. . . . . .. . . . . .................................................................................................20202123252933...............................................................................................................................................................................................................................................................45454546464748494950565666737487.
.. .. .for. .. .4 Degenerate diffusions4.1 Introduction . . . . . . . . . . . . . . . . . . .4.1.1 Hypoellipticity . . . . . . . . . . . . .4.1.2 Kolmogorov’s example . . . . . . . . .4.1.3 Degeneracy and Hörmander conditions4.1.4 General models . . . . . . . . . . . . .4.1.5 Preliminary results from [KMM10] . .4.2 Parametrix in the degenerate case . . . .
. .4.2.1 Assumptions . . . . . . . . . . . . . .4.2.2 Parametrix expansion. Diffusion . . .4.3 Stability results . . . . . . . . . . . . . . . . .4.3.1 Stability for perturbed diffusions . . .4.4 Weak error . . . . . . . . . . . . . . . . . . .4.5 Global error . . . . . . . . . . . . .
. . . . . .4.5.1 Proof of Theorem 4.5.1 . . . . . . . .4.6 Appendix . . . . . . . . . . . . . . . . . . . .2..........................................................................................................GlossaryWs≥0B(a, b)C γ/2,γ ([0, T ], Rk )G(z)IdQr (z)Γ(a)pc (u, z)pc,K (t, (x, y), (x0 , y 0 ))|| · ||γBrownian motionBeta functionγ, γ/2 - Hölder continuous functions from Rk toRDensity of the standard Gaussian vector of R2dIdentity matrixR d×d1:= cr (1+|z|)r , Rd dzQr (z) = 1Gamma function|z|2cd/2:= (2πu)d/2 exp(−c 2u )ih02|y 0 −y−(x+x0 )t/2|2cd 3d/2+3−c |x −x|:= (2πt32 )d exp4ttHölder norm on the space of Hölder continuousbounded functions3Chapter 1IntroductionModelling of many natural phenomena is still a challenging task.
Data and observations which wereceive from the real word usually contain a lot of inaccuracies and noisy factors. Using deterministicmodels only often makes predictions inefficient and imprecise. Thus, researchers in many fields areforced to apply models with additional randomness inside.A possible way to model the uncertainty is to describe dynamics of the process in terms ofStochastic Differential Equations (SDEs further). We are interested in studying Brownian SDEs ofthe following formZ tZ tZt = z +b(s, Zs )ds +σ(s, Zs )dWs ,(1.1)00where (Ws≥0 ) is an Rk -valued Brownian motion on some filtered probability space (Ω, F, (Ft )t≥0 , P),Zt is Rm valued, with m ∈ N possibly different of k.
The coefficients b, σ are Rm and Rm ⊗ Rk valuedrespectively and such that a unique weak solution to (1.1) exists.Equation (1.1) appears in many applied fields varied from physics to finance. Let us mentionHamiltonian mechanics [Tal02], financial mathematics [JYC10] and biologic “simple epidemic model”([Bai17]; [BY89]).Except from some very specific cases, the SDE (1.1) cannot be solved explicitly and it thereforeseems natural to investigate some related approximation procedures. The Euler - Maruyama method(usually simply called the Euler method), introduced in the current SDE framework in [Mar55], isstill one of the simplest effective computational methods.
Let us fix a finite time horizon T > 0. Fora given integer N , representing the number of time steps to be considered along the time interval[0, T ], introducing the time step h = T /N and for all t ∈ [0, T ]:Z tZ thhZth = z +b(φ(s), Zφ(s))ds +σ(φ(s), Zφ(s))dWs ,(1.2)00Studying the accuracy of the approximation of the scheme proposed in (1.2) for the initial SDE(1.1) two main types of errors are usually considered. The first one to be investigated (see e.g.[Mar55], Gikhman and Skorokhod [GS67], [GS82]) was the so-called strong error. Namely, for allp ∈ [1, +∞), with the usual Markovian notations for the processes Zsh , Zs , started from z at themoment 0 it holds:ES (T, z, h, p) := Ez [ sup |Zsh,0,z − Zs0,z |p ]1/p.(1.3)s∈[0,T ]When the coefficients in (1.1) are Lipschitz continuous in space and at least 1/2-Hölder continuousin time, it is easily seen from usual stochastic analysis techniques, namely Itô’ s formula Burkholder4Davis-Gundy inequalities and the Gronwall, Lemma that:∃Cp (T, b, σ), ES (T, z, h, p) ≤ Cp (T, b, σ)h1/2 .On the other hand, in many applications, such as the pricing and hedging of financial derivatives,only so called weak error, introduced in (1.1) and (1.2), is of interest.
For a suitable test function f(we remain here a bit vague about the function space to which f belongs to), one introduces:EW (T, z, h, f ) := Ez [f (ZTh,0,z )] − Ez [f (ZT0,z )].(1.4)There are two sets of assumptions which guarantee that the convergence rate for EW (T, z, h, f ) isactually of order h. Namely, if(i) b, σ, f are smooth and without any specific non-degeneracy assumptionsor(ii) b, σ enjoy some structure property (i.e.
the generator associated with (1.1) is elliptic or hypoelliptic) and some smoothness, and for f that enjoys suitable growth conditions (and that can even be aDirac mass)then|EW (T, z, h, f )| = |Ez [f (ZTh )] − Ez [f (ZT )]| ≤ C(T, f, σ, b)h.(1.5)Ez [f (ZTh )]and theIn the both cases the main tool for the analysis is the correspondence betweensolution of a second order parabolic PDE. This correspondence is provided by the Feynman-Kac representation formula.
Precisely, under the above assumptions we have that, with the usual Markoviannotations, v(t, z) := E[f (ZTt,z )] solves((∂t + Lt )v(t, z) = 0, (t, z) ∈ [0, T [×Rm ,(1.6)v(T, z) = f (z), z ∈ Rm ,where1 Lt v(t, z) = hb(t, z), ∇z v(t, z)i + Tr a(t, z)Dz2 v(t, z) , a(t, z) := σσ ∗ (t, z),2is the generator associated with (1.1). Assuming some smoothness on v, one can writeEW (T, z, h, f )=E[f (ZTh,0,z )] − E[f (ZT0,z )] =N−1XE[v(ti+1 , Zth,0,z) − v(ti , Zth,0,z)]i+1i(1.7)i=0=N−1XhZEti+1tii=0n∂s v(s, Zsh,0,z ) + ∇z v(s, Zsh,0,z )b(ti , Zth,0,z)i−1 h Z ti+1 no i NXoi1h,0,z2h,0,z+Tr(Dz v(s, Zs )a(ti , Zti )) ds =E∂s v + Ls v (Zsh,0,z )ds2tii=0h Z ti+1 n+ E∇z v(s, Zsh,0,z ) · (b(ti , Zth,0,z) − b(s, Zsh,0,z ))iti+=o i1Tr(Dz2 v(s, Zsh,0,z )(a(ti , Zth,0,z) − a(s, Zsh,0,z ))) dsi2N−1 h Z ti+1 nXE∇z vε (s, Zsh,0,z ) · (b(ti , Zth,0,z) − b(s, Zsh,0,z ))ii=0+tio i1h,0,zTr(Dz2 v(s, Zsh,0,z )(a(ti , Zth,0,z)−a(s,Z)))ds ,si25(1.8)exploiting the PDE satisfied by v for the last equality and Itô formula for the third equality.
For afunctionf in Cb2+β (Rk , R), ∀β ∈ (0, 1] the spatial derivatives of v up to order two are globally boundedon [0, T ]. Through Taylor like expansions, whenever (i) or (ii) holds, one can control (1.8), derivingthat each contribution in (1.8) has the order h2 . This leads to the error of order h achieved aftersumming from 0 to N − 1.In case (i), which was considered among the others in the seminal paper by Talay and Tubaro[TT90], the smoothness of v is simply derived via stochastic flow techniques.
In case (ii) let usmention that in the hypoelliptic setting (see Section 4.1.1 for additional details on hypoellipticity),Bally and Talay [BT96a], [BT96b] established (1.5) for bounded Borel functions f and Dirac masses.It respectively bases on the controls of Kusuoka and Stroock [KS84], [KS85] for the derivatives of thedensity of the diffusion process. We carefully mention that, for this method, which anyhow allows toconsider a broad class of potential degeneracies, to apply, the coefficients are assumed to be smooth.The estimates on the tangent processes and Malliavin matrices in the works by Kusuoka and Stroockindeed require such a smoothness.
In the uniformly elliptic case yet another approach has beendeveloped by Konakov and Mammen [KM00], [KM02] which is based on parametrix expansions.Parametrix expansions, which roughly consists in approximating the density of a process withvariable coefficients by the density of the corresponding dynamics with constant coefficients, havebeen a successful tool in many fields.
In particular, when a good proxy is available (which is, forinstance, the case the coefficients b ad σ in (1.1) are non-degenerate and bounded), parametrix allowedto derive the controls required for the analysis of the weak error under rather mild assumptions. Wecan mention the work of Il’in et al. [IKO62], who derived Gaussian heat kernel for the density of (1.1)for bounded Hölder coefficients when σσ ∗ is non-degenerate. Similar bounds have been successfullyexploited by Konakov and Menozzi [KM17] to derive, in the non-degenerate Hölder continuous setting,that for b, σ ∈ C γ/2,γ ([0, T ], Rk ), γ ∈ (0, 1] and f ∈ C β (Rk , R), β ∈ (0, 1]:|EW (T, z, h, f )| = |Ez [f (Zsh )] − Ez [f (Zs )]| ≤ C(T, f, σ, b)hγ/2 ,(1.9)improving the previous result by Mikulevičius and Platen [MP91] who also obtained the bound(1.9) for a function f ∈ C 2+γ (Rk , R). This additional smoothness was due to the fact that theybased their analysis on the associated Schauder estimates (which could already be found in [IKO62]).Going directly to the heat-kernel allows to notably alleviate the smoothness assumptions on the finalcondition, which might be useful for applications.Intuitively, the above convergence rate can be explained by the fact that, in the low regularitysetting, the terms of order greater than one in the telescopic sum (1.7) cannot be expanded muchfurther.
Характеристики
Тип файла PDF
PDF-формат наиболее широко используется для просмотра любого типа файлов на любом устройстве. В него можно сохранить документ, таблицы, презентацию, текст, чертежи, вычисления, графики и всё остальное, что можно показать на экране любого устройства. Именно его лучше всего использовать для печати.
Например, если Вам нужно распечатать чертёж из автокада, Вы сохраните чертёж на флешку, но будет ли автокад в пункте печати? А если будет, то нужная версия с нужными библиотеками? Именно для этого и нужен формат PDF - в нём точно будет показано верно вне зависимости от того, в какой программе создали PDF-файл и есть ли нужная программа для его просмотра.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
