Диссертация (1137347), страница 2

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Namely, we can only exploit the γ-Hölder continuity of the coefficients which leads to anerror controlled by the incrementshhE[|b(s, Zsh ) − b(φ(s), Zφ(s))|] + E[|a(s, Zsh ) − a(φ(s), Zφ(s))|] ≤ C(b, σ)hγ/2 .In other words, the convergence rate is closer to the one associated with the strong error in (1.3).For many applications, e.g. for neuro-sciences or diffusions in random media, it is far important tohandle rougher coefficients, for instance piecewise smooth drifts in (1.1). In that case, the previouslymentioned heat-kernel bounds do not hold.

Motivated by the investigation of the related weak error forDirac masses test functions, we have developed, with V. Konakov and S. Menozzi, a sensitivity analysisof the density of (1.1) (when suitable good Gaussian bounds exist) with respect to a perturbation ofthe coefficients. This is the first main result of the Thesis which led to the publication [KKM17] andis thoroughly developed in Chapter 3.Namely, let us introduce the SDE of the form:dXt = b(t, Xt )dt + σ(t, Xt )dWt , t ∈ [0, T ],6(1.10)where b : [0, T ] × Rd → Rd , σ : [0, T ] × Rd → Rd ⊗ Rd are bounded coefficients that are measurable intime and Hölder continuous in space (this last condition will be relaxed for the drift term b). Also,a(t, x) := σσ ∗ (t, x) is assumed to be uniformly elliptic.

In particular, those assumptions guaranteethat (1.10) admits a unique weak solution, see e.g. Bass and Perkins [BP09], [Men11], from which theuniqueness to the martingale problem for the associated generator can be derived under the currentassumptions.We now introduce, for a given parameter ε > 0, a perturbed version of (1.10) with dynamics:(ε)dXt(ε)(ε)= bε (t, Xt )dt + σε (t, Xt )dWt , t ∈ [0, T ],(1.11)where bε : [0, T ] × Rd → Rd , σε : [0, T ] × Rd → Rd ⊗ Rd satisfy at least the same assumptions as b, σand being meant to be close to b, σ when ε is small.(ε)It is known that, under the previous assumptions, densities of processes (Xt )t≥0 , (Xt )t≥0 exist and satisfy some Gaussian bounds, see e.g Aronson [Aro59] or [DM10] for extensions to somedegenerate cases.In the Chapter 3 we investigate, applying the parametrix technique, how the closeness of (bε , σε )and (b, σ) is reflected on the respective densities of the associated processes.

Our stability results willalso apply to two Markov chains with respective dynamics:√Ytk+1 = Ytk + b(tk , Ytk )h + σ(tk , Ytk ) hξk+1 , Y0 = x,√(ε)(ε)(ε)(ε)(ε)(1.12)Ytk+1 = Ytk + bε (tk , Ytk )h + σε (tk , Ytk ) hξk+1 , Y0 = x,where h > 0 is a given time step, for which we denote for all k ≥ 0, tk := kh and (ξk )k≥1 - centeredi.i.d. random variables satisfying some integrability conditions. Again, the key tool will be theparametrix representation for the densities of chains and the Gaussian local limit theorem.Let us specify the following assumptions (A) which we use in Chapter 3. The parameter ε > 0below is fixed and the constants appearing in the assumptions do not depend on ε.(A1) (Boundedness of the coefficients).

Components of the vector-valued functions b(t, x), bε (t, x)and the matrix-valued functions σ(t, x), σε (t, x) are bounded. Specifically, there exist constantsK1 , K2 > 0 s.t.sup|b(t, x)| +(t,x)∈[0,T ]×Rdsup(t,x)∈[0,T ]×Rdsup|bε (t, x)| ≤ K1 ,(t,x)∈[0,T ]×Rd|σ(t, x)| +sup|σε (t, x)| ≤ K2 .(t,x)∈[0,T ]×Rd(A2) (Uniform Ellipticity). Matrices a := σσ ∗ , aε := σε σε∗ are uniformly elliptic, i.e. there existsΛ ≥ 1, ∀(t, x, ξ) ∈ [0, T ] × (Rd )2 ,Λ−1 |ξ|2 ≤ ha(t, x)ξ, ξi ≤ Λ|ξ|2 , Λ−1 |ξ|2 ≤ haε (t, x)ξ, ξi ≤ Λ|ξ|2 .(A3) (Hölder continuity in space).

For some γ ∈ (0, 1] , κ < ∞, we have for all t ∈ [0, T ],|σ(t, x) − σ(t, y)| + |σε (t, x) − σε (t, y)|γ≤ κ |x − y| .Observe that the last condition also readily gives, thanks to the boundedness of σ, σε , that a, aε arealso uniformly γ-Hölder continuous.For a given ε > 0, we say that assumption (A) holds when conditions (A1)-(A3) are in force.Let us now introduce, under (A), quantities that will bound the difference of the densities in our7main results below.

Set for ε > 0:∆ε,b,∞ :={|b(t, x) − bε (t, x)|},sup(t,x)∈[0,T ]×Rd∀q ∈ (1, +∞), ∆ε,b,q := sup kb(t, .) − bε (t, .)kLq .t∈[0,T ]Since σ, σε are both γ-Hölder continuous, see (A3), we also define∆ε,σ,γ := sup |σ(u, .) − σε (u, .)|γ ,u∈[0,T ]where for γ ∈ (0, 1], || · ||γ stands for the usual Hölder norm in space on Cbγ (Rd , Rd ⊗ Rd ) (space ofHölder continuous bounded functions, see e.g. Krylov [Kry96]) i.e.

:|f |γ := sup |f (x)| + [f ]γ , [f ]γ :=supx6=y,(x,y)∈(Rd )2x∈Rd|f (x) − f (y)|.|x − y|γWe eventually set for q ∈ (1, +∞],∆ε,γ,q := ∆ε,σ,γ + ∆ε,b,q .Theorem 3.1.1. Fix ε > 0 and a final deterministic time horizon T > 0. Under assumptions (A),specified before, for q > d, there exist C := C(q) ≥ 1, c := c(q) ∈ (0, 1] s.t. for all 0 ≤ s < t ≤T, (x, y) ∈ (Rd )2 :pc (t − s, y − x)−1 |(p − pε )(s, t, x, y)| ≤ C∆ε,γ,q ,where p(s, t, x, .), pε (s, t, x, .) respectively stand for the transition densities at time t of equations(1.10), (1.11) starting from x at time s.

Also, we denote for a given c > 0 and for all (u, z) ∈ R+ ×Rd ,|z|2cd/2pc (u, z):= (2πu)d/2 exp(−c 2u ). If q = ∞, the constants C, c do not depend on q.This and the next theorem will be restated and discussed in Section 3.1.1.Before stating our results for Markov Chains we introduce two kinds of innovations in (1.12).Namely:(IG) The i.i.d. random variables (ξk )k≥1 are Gaussian, with law N (0,Id ). In that case the dynamicsin (1.12) correspond to the Euler discretization of equations (1.10) and (1.11).(IP) For a given integer M > 2d + 5 + γ, the innovations (ξk )k≥1 are centered and have C 5 density fξwhich has, together with its derivatives up to order 5, at most polynomial decay of order M .

Namely,for all z ∈ Rd and multi-index ν, |ν| ≤ 5:|Dν fξ (z)| ≤ CQM (z),R1dzQr (z) = 1.where we denote for all r > d, z ∈ Rd , Qr (z) := cr (1+|z|)r,RdTheorem 3.1.2. Fix ε > 0 and a final deterministic time horizon T > 0. For h = T /N, N ∈ N∗ ,we set for i ∈ N, ti := ih. Under (A), assuming that either(IG) or (IP) holds, and for q > d thereexist C := C(q) ≥ 1, c := c(q) ∈ (0, 1] s.t. for all 0 ≤ ti < tj ≤ T, (x, y) ∈ (Rd )2 :χc (tj − ti , y − x)−1 |(ph − phε )(ti , tj , x, y)| ≤ C∆ε,γ,q ,where ph (ti , tj , x, .), phε (ti , tj , x, .) respectively stand for the transition densities at time tj of the MarkovChains Y and Y (ε) in (1.12) starting from x at time ti .

Also:8- If (IG) holds:χc (tj − ti , y − x) := pc (tj − ti , y − x),with pc as in Theorem 3.1.1.- If (IP) holds:cdQM −(d+5+γ)χc (tj − ti , y − x) :=(tj − ti )d/2|y − x|(tj − ti )1/2 /c.Again, if q = +∞ the constants C, c do not depend on q.Continuing the research, V. Konakov and S. Menozzi applied results mentioned above to study theweak error of the Euler scheme approximations in the paper [KM17]. To investigate the weak errorfor rough drifts, the idea in [KM17] is to mollify the drifts. The difference between the density of theinitial diffusion and the one with mollified coefficients is precisely controlled by the previous result.The same occurs for the Euler scheme case. It therefore remains to control the difference betweenthe densities of the mollified diffusion and the scheme which can be addressed from previous resultsof [KM02] provided, that high order derivatives (which explode with the mollifying parameter) aresharply controlled.Motivated by the extension of the previous study, we continue with the weak error controls forthe case of rough coefficients to Kolmogorov’s degenerate SDEs in Chapter 4.

Namely, we specify themodel in (1.1) writing Zt = (Xt , Yt ) with:(dXt = b(Xt , Yt )dt + σ(Xt , Yt )dWt ,(1.13)dYt = Xt dt, t ∈ [0, T ],where b : R2d → Rd , σ : R2d → Rd ⊗ Rd are bounded coefficients that are Hölder continuous inspace (this condition will be relaxed for the drift term b) and W is a Brownian motion on somefiltered probability space (Ω, F, (Ft )t≥0 , P). In (1.13), T > 0 is a fixed deterministic final time.

Also,a(x, y) := σσ ∗ (x, y) is assumed to be uniformly elliptic.We point out that those assumptions (specified below) are actually sufficient to guarantee weakuniqueness for the solution of equation (1.13), see Remark 4.2.1.Such equations were first introduced in the seminal paper [Kol34] by Kolmogorov. In that work,he found the explicit expression of the density when the coefficients are constants. The parametrixapproach in that framework has then been applied by various authors, Weber [Web51], Sonin [Son67]and the more recent [KMM10] under various kinds of assumptions.

Adapting the techniques introduced in the last quoted work, which deals with Lipschitz coefficients, it is now possible to considerthe Hölder setting for the degenerate Kolmogorov diffusions of type (1.13). The sensitivity analysis naturally extends to this framework. These aspects are detailed in Chapter 4 (see as well thepublished article [Koz16]).Precisely, let us introduce the Euler scheme for the SDE (1.13) first. For a fixed N and T > 0 wedefine a time grid {0, t1 , . . . , tN } with a given step h := T /N , i.e.

ti = ih, for i = 0, . . . , N and thescheme(RtRthhhhXth = x + 0 b(Xφ(s), Yφ(s))ds + 0 σ(Xφ(s), Yφ(s))dWs ,Rt h(1.14)hYt = y + 0 Xs ds.where φ(t) = ti ∀t ∈ [ti , ti+1 ). Observe that the above scheme is in fact well defined even thoughthe non degenerate component of the scheme itself appears in the integral. On every time-step theincrements of (Xth , Yth )t∈[ti ,ti+1 ] , i ≥ 0 are actually Gaussian. They indeed correspond to a suitable9rescaling of the Brownian increment and its integral on the considered time step, see also Remark4.2.3.Let us also denote for a given c > 0 and for all (x, y), (x0 , y 0 ) ∈ R2d the Kolmogorov-type densitypc,K (t, (x, y), (x0 , y 0 )) := 0|x − x|2|y 0 − y − (x + x0 )t/2|2cd 3d/2exp−c.+3(2πt2 )d4tt3(1.15)The subscript K in the notation pc,K (t, (x, y), (x0 , y 0 )) stands for Kolmogorov-like equations.We would like to emphasize that in Chapter 4 we are considering time-homogeneous coefficientsb, σ under the following assumptions:(AD1) (Boundedness of the coefficients).The components of the vector-valued function b(x, y) and the matrix-valued function σ(x, y) arebounded measurable.

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