Диссертация (1137347), страница 3

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

Текст из файла (страница 3)

Specifically, there exists a constant K s.t.sup|b(x, y)| +(x,y)∈R2d|σ(x, y)| ≤ K.sup(x,y)∈R2d(AD2) (Uniform Ellipticity).The matrix a := σσ ∗ is uniformly elliptic, i.e. there exists Λ ≥ 1, such that ∀(x, y, ξ) ∈ (Rd )3 ,Λ−1 |ξ|2 ≤ ha(x, y)ξ, ξi ≤ Λ|ξ|2 .(AD3) (Hölder continuity in space).For some γ ∈ (0, 1] , κ, we haveγγ/3≤ κ |x − x0 | + |y − y 0 |.|b(x, y) − b(x0 , y 0 )| + |σ(x, y) − σ(x0 , y 0 )|We say that assumption (AD) holds when conditions (AD1)-(AD3) are in force.Under the above mentioned assumptions, we now introduce perturbed versions of (1.13) and(1.14). Namely, for bε : R2d → Rd , σε : R2d → Rd ⊗ Rd satisfying at least the same assumptions asb, σ and being meant to be close to b, σ for small values of ε > 0 one denote:((ε)(ε)(ε)(ε)(ε)dXt = bε (Xt , Yt )dt + σ(Xt , Yt )dWt ,(1.16)(ε)(ε)dYt = Xt dt, t ∈ [0, T ],and similarly:(Xtε,h = x +Ytε,h = y +Rtbε (X ε,h , Y ε,h )dsR 0t ε,h φ(s) φ(s)Xs ds.0+Rt0ε,hε,hσε (Xφ(s), Yφ(s))dWs ,(1.17)for t ∈ [0, tj ), 0 < j ≤ N , where φ(t) = ti ∀t ∈ [ti , ti+1 ).Considering a specific kind of Hölder continuity associated with the intrinsic scales of the systemand the time-homogeneous case we set for ε > 0:∀q ∈ (1, +∞], ∆dε,b,q := |b(., .) − bε (., .)|Lq (R2d ) .We also define∆dε,σ,γ := |σ(., .) − σε (., .)|d,γ ,10γwhere for γ ∈ (0, 1], |.|d,γ stands for the Hölder norm in space on Cb,d(Rd ⊗ Rd ), which denotes thespace of Hölder continuous bounded functions with respect to the distance d defined as follows:∀(x, y), (x0 , y 0 ) ∈ (Rd )2 , d (x, y), (x0 , y 0 ) := |x − x0 | + |y 0 − y|1/3 .γNamely, a measurable function f is in Cb,d(Rd ⊗ Rd ) if|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 )We eventually set ∀q ∈ (1, +∞],∆dε,γ,q := ∆dε,σ,γ + ∆dε,b,q ,which will be the key quantity governing the error in our results.Theorem 4.3.1.

Fix T > 0. Under AD, for q ∈ (4d, +∞], there exist C := C(q) ≥ 1, c ∈ (0, 1] s.t.for all 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 )),where p(t, (x, y), (., .)), pε (t, (x, y), (., .)) respectively stand for the transition densities at time t ofequations (1.13), (1.16) starting from (x, y) at time 0.Theorem 4.3.5.

Fix T > 0 and let us define a time-grid Λh := {(ti )i∈[[1,N ]] }, N ∈ N∗ . Under AD,there exist C ≥ 1, c ∈ (0, 1] s.t. for all 0 < tj ≤ T, ((x, y), (x0 , y 0 )) ∈ (R2d )2 :|pεh − ph |(tj , (x, y), (x0 , y 0 )) ≤ C∆dε,γ,q pc,K (tj , (x, y), (x0 , y 0 )),where pεh (t, (x, y), (., .)), ph (t, (x, y), (., .)) respectively stand for the transition densities at time t ofequations (1.14), (1.17) starting from (x, y) at time 0.These two theorems will be restated and discussed in Section 4.3.1.The sensitivity analysis will then be applied, in the flavour of [KM17] to investigate the weak errorassociated to a specific Euler scheme which had already been considered in [LM10] for equationsof type (1.13). However, to perform the analysis we need to change assumptions (AD) slightly.Precisely, we have to assume more about Hölder properties of coefficients than in (AD).Instead of (AD3), we assume for some γ ∈ (0, 1] , 0 < κ < ∞ it holds:γγ/2|b(x, y) − b(x0 , y 0 )| + |σ(x, y) − σ(x0 , y 0 )| ≤ κ |x − x0 | + |y − y 0 |.and denote that as (ÂD3).

Thus, we say that assumption (ÂD) holds when conditions (AD1),(AD2),(ÂD3) are in force.Theorem 4.4.1. Fix T > 0. Under assumptions (ÂD) for any test function f ∈ C β,β/2 (R2d )(β−Hölder in the first variable and β/2−Hölder in the second variable functions) for β ∈ (0, 1], thereexists C > 0, such that:|E(x,y) [f (XTh , YTh )] − E(x,y) [f (XT , YT )]| ≤ Chγ/2 (1 + |x|γ/2 ).where γ ∈ (0, 1] stands for the Hölder index of γ, γ/2 (γ for the variable x, γ/2 for y) Höldercontinuous time-homogeneous functions b, σ.11The theorem will be restated in Section 4.4.We also would like to present our control for the direct difference of transition densities p(t, (x, y), (x0 , y 0 ))and ph (t, (x, y), (x0 , y 0 )). The result below is in clear contrast with the one of Theorem 4.4.1 for theweak error, i.e. when additionally one considers an integration of a Hölder function w.r.t.

the final(or forward variable). We finally can reach a global error of order hβ , β < γ − 1/2 which is close tothe expected one in hγ/2 when γ goes to 1.Theorem 4.5.1. Fix a final time horizon T > 0 and a time step h = T /N, N ∈ N∗ for the Eulerscheme. Under assumptions (ÂD), for γ ∈ (1/2, 1] and β ∈ (0, γ − 12 ), for all t in the time gridΛh := {(ti )i∈[[1,N ]] } and (x, y), (x0 , y 0 ) ∈ R2d there exist C := (T, b, a, β), c > 0 such that :|p(t, (x, y), (x0 , y 0 )) − ph (t, (x, y), (x0 , y 0 )|≤ Ch (1 + (|x| ∧ |x0 |))1+γ )βsuppc,K (s, (x, y), (x0 , y 0 )),(1.18)s∈[t−h,t]where pc,K (s, (x, y), (x0 , y 0 )) stands for the Kolmogorov-type Gaussian density (1.15) at time s.The theorem will be discussed in Section 4.5.12Chapter 2Parametrix technique2.1ReviewThe parametrix method is a classical method in order to construct fundamental solutions for parabolictype partial differential equations using an expansion argument.

This method allows for coefficientsto be less regular than in the Malliavin Calculus approach. However, the methodology is restrictedto caseswith underlying process being Markov.The parametix approach has been established in the beginning of the XX century as a perturbationtechnique for partial differential equations theory by Levi [Lev07]. The original method has been usedfor approximations of the elliptic linear differential equation solution. In a nutshell the idea consistedof the appropriate separation of the “main” part and controlling the “remainder” of a particularrepresentation of the solution of the pde in terms of series of functions. A common choice for theprincipal part consists in considering the solution of the underlying equation with constant coefficients.The technique has been further developed by Hadamard [Had23]. The important modificationsof the parametrix method, introduced by Il’in et.

all in 1962 [IKO62], Friedman in 1964 [Fri64] andMcKean and I. Singer in 1967 [MS67], provided the way to use it for SDEs theory. The main point isthat the transition density of the SDE can be found through the fundamental solution of the Cauchyproblem for the corresponding generator. Parametrix in a nutshell allows to get a representation ofthe SDE transition density as a sum, where each term contains the transition density of the moresimple Gaussian process.As far as we know one of the first investigation of the parametrix method for Markov chainswas presented in the paper by Konakov and Mammen in 2000 [KM00] where the authors studiedtriangular arrays of Markov chains Xn (k), that converge weakly to a diffusion process (for n → ∞).However local limit theorems for homogeneous Markov chains with continuous state space have beenalready given in Konakov and Molchanov [KM85] but the last article was not really well-known beingpublished in Russian.

In the article [KM00] Konakov and Mammen applied the parametrix methodfor parabolic PDEs and a modification of the method – for discrete time Markov chain. As the resultthey achieved the convergence rate of order O(n−1/2 ) for transition densities of triangular array ofMarkov Chains to the transition density of the limiting diffusion.After that, in 2001, the same authors considered the situation of triangular arrays of Markov random walks that can be approximated by an accompanying sequence of diffusion processes. The mainresult consisted in proving that normalized transition probabilities differ from transition densities inthe diffusion model by rate O(n−1/2 ).

In particular, local limit theorems for the case of the Markovrandom walks has been stated and proved.In 2002 Konakov and Mammen studied the approximation of the density of the diffusion by13the density of the Euler discretization with discretisation step n1 in [KM02]. Namely, the authorsconsidered the Euler scheme with equidistant partitions, where for 0 ≤ k ≤ n − 1, Yn k+1=nYn nk + n1 m Yn nk + √1n εn k+1withappropriateconditionson"innovations"ε.Usingthennparametrix approach they obtained an asymptotic expansion in powers of n1 .In 2010, Konakov, Menozzi and Molchanov presented the paper [KMM10], where parametrixmethod has been adapted for a larger class of processes - namely for degenerate diffusions with rank2:(RtRtXt = x + 0 b(Xs , Ys )ds + 0 σ(Xs , Ys )dWs ,Rt(2.1)Yt = y + 0 Xs ds.The authors derived the density representation in terms of parametrix series and provided the explicitGaussian upper and partial lower bounds.

Характеристики

Список файлов диссертации