Диссертация (1137347), страница 4

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 4)

Due to the series representation, the authors also derived alocal limit theorem with the usual convergence rate for an associated Markov chains approximations.Later, Delarue and Menozzi have published the paper [DM10] which contained the full summaryon the existing parametrix techniques including the degenerate case. Two sided bounds for the densityof the solution of a system of n differential equations of dimension d have been provided.We also would like to emphasize the paper [Men11] due to the proof of the uniqueness of themartingale problem associated to some degenerate operator inside.Even more general framework has been also treated in the paper by Menozzi and Huang [HM16]in 2016 where they considered a stable driven degenerate stochastic differential equation, whichcoefficients satisfy a kind of weak Hörmander condition.

Under mild smoothness assumptions theyproved the uniqueness of the martingale problem for the associated generator using the parametrixmethod as a tool.2.2Other developments in ParametrixBally and Kohatsu-Higa in their paper[BKH09] introduced the parametrix method using a semigroupapproach and obtain the probabilistic representation for the density of the solution to a diffusionequation or for Levy driven SDEs. It’s worth to specifically emphasize that they have describedtwo kinds of parametrix methods: the first one - “forward” and second one - “backward” parametrix.To use the first version it’s necessary to assume that the coefficients are Cb2 .

The second versionconverges if the drift coefficient is bounded and measurable and diffusion coefficient is bounded,uniformly elliptic and Hölder continuous.Further, Kohatsu-Higa with his co-authors considered an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations withHölder continuous coefficients [AKH17].As a continuation of the topic we would like to mention a paper by N. Frikha [Fri18]. Applyingthe results obtained in [AKH17] the author studied the weak approximation error of a skew diffusionwith bounded measurable drift and Hölder diffusion coefficient by an Euler-type scheme. A boundfor the difference between densities of the skew diffusion and its Euler approximation was obtainedby using the parametrix method for the skew diffusions.Moreover we would like to emphasize other two papers as nice examples of the parametrix application.

First, in [FH15] studying the development of the Richardson-Romberg extrapolation method forMonte Carlo linear estimator to the framework of stochastic optimization by means of stochastic approximation algorithm, the authors also extended the parametrix expansion results to the derivativesof the densities.Secondly, in [FKHL16] the authors obtained properties of the law associated to the first hittingtime of a threshold by a one-dimensional uniformly elliptic diffusion process and to the associated14process stopped at the threshold. The methodology relied on the parametrix method that was appliedto the associated Markov semigroup.There is also a direction for the parametrix development mostly studied by A. Pascucchi. Althoughthere is a big variety of papers done more or less in a same flavour it is worth to mention at least thekey one.

In [CFP10] authors presented their own way to handle controls for the diffusions transitiondensities approximations deriving the technique also from the classical PDE theory.2.3The parametrix method for diffusion processesTo introduce the technique we would like to start with the non-degenerate SDEs. Namely, as inChapter 1, for a fixed given deterministic final time-horizon T > 0, we consider the following multidimensional SDE:dXt = b(t, Xt )dt + σ(t, Xt )dWt , t ∈ [0, T ],(2.2)where b : [0, T ] × Rd → Rd , σ : [0, T ] × Rd → Rd ⊗ Rd are bounded coefficients that are measurablein time and W is a Brownian motion on some filtered probability space (Ω, F, (Ft )t≥0 , P).

Also,a(t, x) := σσ ∗ (t, x) is assumed to be uniformly elliptic, precisely, there exists λ0 ≥ 1 s.t. for (t, x, ξ) ∈22[0, T ] × Rd × Rd we have λ−10 |ξ| ≤ ha(t, x)ξ, ξi ≤ λ0 |ξ| where |.| stands for the Euclidean norm.To begin with, we assume that there exists the so-called transition density of (2.2) which is afundamental solution of the equation ∂p∂t + Lt p = 0, where Lt is the generator of (2.2) (to get thisone can assume Lipschitz continuity for coefficients in time and space, i.e.). Precisely, as in Chapter1, for all φ ∈ C02 (Rd , R), z ∈ Rd :Lt φ(z) =1Tr(a(t, z)Dz2 φ(z)) + hb(t, z), ∇z φ(z)i.2(2.3)The existence of the transition density P(Xs ∈ dy|Xt = x) = p(s, t, x, y)dy has been proved in thebook [Fri64], for example.As it was mentioned in Chapter 1, we are interested in the approximation of the solution Xt forthe SDEdXt = b(t, Xt )dt + σ(t, Xt )dWt , t ∈ [0, T ],(2.4)For given (s, x) ∈ R+ × Rd , we use the standard Markov notation (Xts,x )t≥s to denote the solution of(2.4) starting from x at time s.Assume that (Xts,x )t≥s has for all t > 0 a smooth density p(s, t, x, .) (which is the case if thecoefficients are smooth see e.g.

Friedman [Fri64]). We would like to estimate this density at a givenpoint y ∈ Rd . To this end, we introduce the following Gaussian inhomogeneous process with spatialvariable frozen at y. For all (s, x) ∈ [0, T ] × Rd , t ≥ 0 we set:Z tyX̃t = x +σ(u, y)dWu ,swhich literally means we freeze the coefficient σ() at the terminal point y ∈ Rd , where the density ofX̃ty is evaluated.

However it is worth to remark that the approach used by Levi [Lev07] and [IKO62]applied the freezing procedure to the initial point (which seems to be even more natural decision).The disadvantage of such a solution is that one needs to assume additional regularity in time. Alsothe specific form without a trend coefficient is used due to the boundedness assumption on b(t, Xt ).RtIn the general case one should add s b(u, y)du to the definition of the frozen process.The density of the frozen process p̃y , which exists due to the uniform ellipticity assumptions onσ, seems to be the most natural "proxy" for the initial density p(s, t, x, y).15Assume for the beginning smoothness for coefficients in (2.4) and that (Xt )t>0 has a smoothdensity.

The density of the frozen process satisfies the Kolmogorov Backward equation:∂u p̃y (u, t, z, y) + L̃yu p̃y (u, t, z, y) = 0, s ≤ u < t, z ∈ Rd ,(2.5)p̃y (u, t, ., y) → δy (.),u↑twhere for all ϕ ∈ C02 (Rd , R), z ∈ Rd :L̃yu ϕ(z) =1Tr σσ ∗ (u, y)Dz2 ϕ(z) ,2stands for the generator of X̃ y at time u.On the other hand, since we have assumed the density of X to be smooth, it must satisfy theKolmogorov forward equation (see e.g.

Dynkin [Dyn65]). For a given starting point x ∈ Rd at times,∂u p(s, u, x, z) = L∗u p(s, u, x, z) = 0, s < u ≤ t, z ∈ Rd ,(2.6)p(s, u, x, .) → δx (.),u↓swhere L∗u stands for the formal adjoint (which is again well defined if the coefficients in (2.2) aresmooth) of the generator of (2.2) (see (2.3)).Using the Dirac convergences in (2.5) and (2.6) one can derive that:Z tZ(p − p̃y )(s, t, x, y) =du∂udzp(s, u, x, z)p̃y (u, t, z, y).RdsAfter a formal differentiating:Z t Zy(p − p̃ )(s, t, x, y) =dudz (∂u p(s, u, x, z)p̃y (u, t, z, y) + p(s, u, x, z)∂u p̃y (u, t, z, y)) .sRdEquations (2.5), (2.6) yield the formal expansion below which is initially due to McKean andSinger [MS67].Z t Z(p − p̃y )(s, t, x, y) =dudz L∗u p(s, u, x, z)p̃y (u, t, z, y) − p(s, u, x, z)L̃yu p̃y (u, t, z, y)sRdZ t Z=dudzp(s, u, x, z)(Lu − L̃yu )p̃y (u, t, z, y),(2.7)sRdWe eventually take the adjoint for the last equality.

Note carefully that the differentiation underthe integral is also here formal since we would need to justify that it can actually be performed usingsome growth properties of the density and its derivatives which we a priori do not know. Let us nowintroduce the notationZ t Zf ⊗ g(s, t, x, y) =dudzf (s, u, x, z)g(u, t, z, y)sRdfor the time-space convolution and let us define p̃(s, t, x, y) := p̃y (s, t, x, y), that is in p̃(s, t, x, y) weconsider the density of the frozen process at the final point and observe it at that specific point. Wenow introduce the parametrix kernel:H(s, t, x, y) := (Ls − L̃s )p̃(s, t, x, y) := (Ls − L̃ys )p̃y (s, t, x, y).16(2.8)With those notations equation (2.7) rewrites:(p − p̃)(s, t, x, y) = p ⊗ H(s, t, x, y).From this expression, the idea then consists in iterating this procedure for p(s, u, x, z) in (2.7) introducing the density of a process with frozen characteristics in z which is here the integration variable.This yields to iterated convolutions of the kernel and leads to the formal expansion:p(s, t, x, y) =∞Xp̃ ⊗ H (r) (s, t, x, y),(2.9)r=0where p̃ ⊗ H (0) = p̃, H (r) = H ⊗ H (r−1) , r ≥ 1.

Obtaining estimates on p from the formal expression(2.9) requires to have good controls on the right-hand side. The remarkable property of this formalexpansion is now that the right-hand-side of (2.9) only involves controls on Gaussian densities.The convergence of the series in (2.9) is in some sense standard (see e.g. [Men11] or Friedman[Fri64]). We recall for the sake of completeness the key steps.From direct computations, there exist c1 ≥ 1, c ∈ (0, 1] s.t.

for all T > 0 and all multi-indexα, |α| ≤ 8,∀0 ≤ u < t ≤ T, (z, y) ∈ (Rd )2 , |Dzα pe(u, t, z, y)| ≤wherepc (t − u, y − z) =c1pc (t − u, y − z),(t − u)|α|/2(2.10)c |y − z|2cd/2,exp−2 t−u(2π(t − u))d/2stands for the usual Gaussian density in Rd with 0 mean and covariance (t − u)c−1 Id . From (2.10),the boundedness of the drift and the Lipschitz continuity in space of the diffusion matrix we readilyget that there exists c1 ≥ 1, c ∈ (0, 1],|H(u, t, z, y)| ≤c1 (1 ∨ T 1/2 )pc (t − u, z − y).(t − u)1/2Now we present the property which usually called the smoothing property of the parametrix kernel.Let us illustrate this by deriving the time-singularity of the first order-convolution.11|ep ⊗ H(s, t, x, y)| ≤ ((1 ∨ T 1/2 )c1 )2 B( , 1)pc (t − s, y − x)(t − s) 22((1 ∨ T 1/2 )c1 )2 Γ( 12 )1=pc (t − s, y − x)(t − s) 2 ,Γ( 23 )R1where for a, b > 0, B(a, b)= 0 t−1+a (1 − t)−1+b dt stands for the β− function, and using as wellthe identity B(a, b) = Γ(a)Γ(b)Γ(a+b) for the last inequality,where Γ(a) stands for the Gamma function.Iterating the convolution operation, the exponent in time will grow with each iteration:rYr1 i+1B( ,)pc (t − s, y − x)(t − s) 222i=1r1/2((1 ∨ T )c1 )r+1 Γ( 12 )r=pc (t − s, y − x)(t − s) 2 .Γ(1 + 2r )|ep ⊗ H (r) (s, t, x, y)| ≤ ((1 ∨ T 1/2 )c1 )r+1These bound due to the asymptotic of the Gamma function readily yield the convergence of theseries as well as a Gaussian upper-bound.

Характеристики

Список файлов диссертации