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Our stability results will also apply to two Markov chains withrespective dynamics:√Ytk+1 = Ytk + b(tk , Ytk )h + σ(tk , Ytk ) hξk+1 , Y0 = x,√(ε)(ε)(ε)(ε)(ε)(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 - centered i.i.d.
random variables satisfying some integrability conditions.Again, the key tool will be the parametrix representation for the densities of chainsand the Gaussian local limit theorem.Let us specify the following assumptions (A) which we use in Chapter 3. Theparameter ε > 0 below is fixed and the constants appearing in the assumptions donot 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 constants K1 , K2 > 0 s.t.sup|b(t, x)| +sup(t,x)∈[0,T ]×Rdsup|bε (t, x)| ≤ K1 ,(t,x)∈[0,T ]×Rd(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 exists Λ ≥ 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 allt ∈ [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 ofσ, σε , that a, aε are also 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 bound5the difference of the densities in our main 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 of Hölder continuous bounded functions, see e.g.
Krylov [Kry96]) i.e. :|f |γ := sup |f (x)| + [f ]γ , [f ]γ :=x∈Rdsupx6=y,(x,y)∈(Rd )2|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. Underassumptions (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 timet of equations (10), (11) starting from x at time s. Also, we denote for a given|z|2cd/2c > 0 and for all (u, z) ∈ R+ × Rd , pc (u, z):= (2πu)d/2 exp(−c 2u ).
If q = ∞, theconstants 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 (12). Namely:(IG) The i.i.d. random variables (ξk )k≥1 are Gaussian, with law N (0,Id ). In thatcase the dynamics in (12) correspond to the Euler discretization of equations (10)and (11).(IP) For a given integer M > 2d + 5 + γ, the innovations (ξk )k≥1 are centered andhave C 5 density fξ which has, together with its derivatives up to order 5, at mostpolynomial decay of order M .
Namely, for all z ∈ Rd and multi-index ν, |ν| ≤ 5:|Dν fξ (z)| ≤ CQM (z),1where we denote for all r > d, z ∈ Rd , Qr (z) := cr (1+|z|)r,6RRddzQr (z) = 1.Theorem (3.1.2). Fix ε > 0 and a final deterministic time horizon T > 0. Forh = T /N, N ∈ N∗ , we set for i ∈ N, ti := ih. Under (A), assuming thateither(IG) or (IP) holds, and for q > d there exist 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 attime tj of the Markov Chains Y and Y (ε) in (12) starting from x at time ti . Also:- 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 mentionedabove to study the weak error of the Euler scheme approximations in the paper[KM17]. To investigate the weak error for rough drifts, the idea in [KM17] is tomollify the drifts. The difference between the density of the initial diffusion andthe one with mollified coefficients is precisely controlled by the previous result.The same occurs for the Euler scheme case. It therefore remains to control thedifference between the densities of the mollified diffusion and the scheme which canbe addressed from previous results of [KM02] provided, that high order derivatives(which explode with the mollifying parameter) are sharply controlled.Motivated by the extension of the previous study, we continue with the weakerror controls for the case of rough coefficients to Kolmogorov’s degenerate SDEsin Chapter 4.
Namely, we specify the model in (1) writing Zt = (Xt , Yt ) with:(dXt = b(Xt , Yt )dt + σ(Xt , Yt )dWt ,(13)dYt = Xt dt, t ∈ [0, T ],where b : R2d → Rd , σ : R2d → Rd ⊗ Rd are bounded coefficients that are Höldercontinuous in space (this condition will be relaxed for the drift term b) and W isa Brownian motion on some filtered probability space (Ω, F, (Ft )t≥0 , P). In (13),T > 0 is a fixed deterministic final time. Also, a(x, y) := σσ ∗ (x, y) is assumed tobe uniformly elliptic.We point out that those assumptions (specified below) are actually sufficient toguarantee weak uniqueness for the solution of equation (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 the7coefficients are constants.
The parametrix approach in that framework has thenbeen applied by various authors, Weber [Web51], Sonin [Son67] and the more recent[KMM10] under various kinds of assumptions. Adapting the techniques introducedin the last quoted work, which deals with Lipschitz coefficients, it is now possibleto consider the Hölder setting for the degenerate Kolmogorov diffusions of type(13). The sensitivity analysis naturally extends to this framework. These aspectsare detailed in Chapter 4 (see as well the published article [Koz16]).Precisely, let us introduce the Euler scheme for the SDE (13) first.
For a fixedN and T > 0 we define a time grid {0, t1 , . . . , tN } with a given step h := T /N , i.e.ti = ih, for i = 0, . . . , N and the scheme(RtRthhhhXth = x + 0 b(Xφ(s), Yφ(s))ds + 0 σ(Xφ(s), Yφ(s))dWs ,R(14)thhYt = y + 0 Xs ds.where φ(t) = ti ∀t ∈ [ti , ti+1 ). Observe that the above scheme is in fact welldefined even though the non degenerate component of the scheme itself appearsin the integral. On every time-step the increments of (Xth , Yth )t∈[ti ,ti+1 ] , i ≥ 0 areactually Gaussian.
They indeed correspond to a suitable rescaling of the Brownianincrement and its integral on the considered time step, see also Remark 4.2.3.Let us also denote for a given c > 0 and for all (x, y), (x0 , y 0 ) ∈ R2d theKolmogorov-type density 0|x − x|2|y 0 − y − (x + x0 )t/2|2cd 3d/20 0exp −c+3. (15)pc,K (t, (x, y), (x , y )) :=(2πt2 )d4tt3The subscript K in the notation pc,K (t, (x, y), (x0 , y 0 )) stands for Kolmogorovlike equations.We would like to emphasize that in Chapter 4 we are considering time-homogeneouscoefficients b, σ under the following assumptions:(AD1) (Boundedness of the coefficients).The components of the vector-valued function b(x, y) and the matrix-valuedfunction σ(x, y) are bounded measurable.
Specifically, there exists a constant Ks.t.sup(x,y)∈R2d|b(x, y)| +sup|σ(x, y)| ≤ K.(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).8For 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 versionsof (13) and (14).
Namely, for bε : R2d → Rd , σε : R2d → Rd ⊗ Rd satisfying atleast the same assumptions as b, σ and being meant to be close to b, σ for smallvalues of ε > 0 one denote:((ε)(ε)(ε)(ε)(ε)dXt = bε (Xt , Yt )dt + σ(Xt , Yt )dWt ,(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 ,(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 intrinsicscales of the system and 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 ,γwhere for γ ∈ (0, 1], |.|d,γ stands for the Hölder norm in space on Cb,d(Rd ⊗ Rd ),which denotes the space of Hölder continuous bounded functions with respect tothe 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 :=sup(x,y)6=(x0 ,y 0 )∈R2dx,y∈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.9Theorem (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 of equations (13), (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.
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