Диссертация (1137347), страница 15

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Namely,pc,K (t − u, (w, z), (x0 , y 0 )). (4.33)|(H − Hε )(t − u, (w, z), (x0 , y 0 ))| ≤ 1 ∨ T (1−γ)/2 c1 ∆dε,γ,∞(t − u)1−γ/2For q ∈ (4d, +∞) we use the Hölder inequality in the time-space convolution involving the difference of the drifts (last term in (4.32)). SetD(t, (x, y), (x0 , y 0 ))Z t Z:=dup̃ε (u, (x, y), (w, z))h[bε (w, z) − b(w, z)],R2d0Dw p̃(t − u, (w, z)(x0 , y 0 ))idwdz.Denoting by q̄ the conjugate of q, i.e. q, q̄ > 1, q −1 + q̄ −1 = 1, we get from (4.17) and for q > d that:Z tdu0 02kb(., .) − bε (., .)kLq (R2d )|D(t, (x, y), (x , y ))| ≤ c1(t−u)1/20Zno1/q̄×[pc,K (u, (x, y), (w, z))pc,K (t − u, (w, z), (x0 , y 0 ))]q̄ dwdzR2d≤c21 ∆dε,b,qZ0nRR2d×3d/q c2d(2π)2d/q (cq̄)2d/q̄pcq̄,K (u, (x, y), (w, z))pcq̄,K (t − u, (w, z), (x0 , y 0 ))dwdzu2d/q (t√≤tc213ct22π− u)q̄du12 +2d/q!d/qdq̄o1/q̄∆dε,b,q pc,K (t, (x, y), (x0 , y 0 ))Z0Now, the constraint 4d < q < +∞ precisely gives thatis well defined.

We therefore derive:12tduu2d/q (t1− u) 2 +2d/q.+ 2d(1 − 1q̄ ) < 1 so that the last integral|D(t, (x, y), (x0 , y 0 )|11≤ c21 t 2 −2d/q ∆dε,b,q pc,K (t, (x, y), (x0 , y 0 ))B(1 − 2d/q, − 2d/q).2In the case 4d < q < +∞, recalling that α(q) = 12 − 2dq , we eventually get :|p̃ε (s, (x, y), (w, z)) ⊗ H − Hε (t − u, (w, z), (x0 , y 0 ))|1≤ c21 pc,K (t, (x, y), (x0 , y 0 )){∆dε,b,q tα(q) B( + α(q), α(q))2+2∆dε,σ,γ (1 ∨ T (1−γ)/2 )tγ/2 B(1, γ/2)}.59(4.34)The statement now follows in whole generality from (4.30), (4.31), (4.17) for q = ∞ and (4.34)for 4d < q < +∞.The following Lemma associated with Lemmas 4.3.2 and Lemma 4.3.3 allows to complete theproof of Theorem 4.3.1.Lemma 4.3.4 (Difference of the iterated kernels).

For all 0 < t ≤ T, (x, y), (x0 , y 0 ) ∈ (R2d )2 and forall r ∈ N:|(p̃ ⊗ H (r) − p̃ε ⊗ Hε(r) )(t, (x, y), (x0 , y 0 )|(r+2)γrγ22tt≤ C r r∆dε,γ,q+pc,K (t, (x, y), (x0 , y 0 )). Γ 1 + rγΓ 1 + (r+2)γ 2(4.35)2Proof. Observe that Lemmas 4.3.2 and Lemma 4.3.3 respectively give (4.35) for r = 0 and r = 1. Letus assume that it holds for a given r ∈ N∗ and let us prove it for r + 1.Let us denote for all r ≥ 1,(r)ηr (t, (x, y), (x0 , y 0 )) := |(p̃ ⊗ H (r) − p̃ε ⊗ Hε )(t, (x, y), (x0 , y 0 ))|. Writeηr+1 (t, (x, y), (x0 , y 0 )) = (p̃ ⊗ H (r) − p̃ε ⊗ Hε(r) ) ⊗ H(t, (x, y), (x0 , y 0 ))+ p̃ε ⊗ Hε(r) ⊗ (H − Hε )(t, (x, y), (x0 , y 0 ))≤ ηr ⊗ |H| (t, (x, y), (x0 , y 0 )) + p̃ε ⊗ Hε(r) ⊗ |(H − Hε )| (t, (x, y), (x0 , y 0 )).Now, ηr is controlled by the induction hypothesis, |H| - through (4.20), Lemma 4.2.1 provides bounds(r)for the convolution p̃ε ⊗Hε and the difference |(H − Hε )| is controlled in (4.33).

Thus, the inductionhypothesis we get the result.Theorem 4.3.1 now simply follows from the controls of Lemma 4.3.4, the parametrix expansions(4.9) and (4.27) of the densities p, pε and the asymptotic of the Gamma function.Stability for perturbed Euler schemesLet us describe precisely the analogue of the scheme (4.23) with perturbed coefficients as in (4.27)which approximates the process (4.27) with perturbed coefficients bε , σε :(RtRtε,hε,hε,hε,hXtε,h = x + 0 bε (Xφ(s), Yφ(s))ds + 0 σε (Xφ(s), Yφ(s))dWs ,R(4.36)tYtε,h = y + 0 Xsε,h ds.for t ∈ [0, tj ), 0 < j ≤ N , where φ(t) = ti ∀t ∈ [ti , ti+1 ).Recall first from the Section 4.2.2 that we have the following representations for the densities phand phε :ph (tj , (x, y), (x0 , y 0 )) =jX(r)p̃h ⊗h Hh (tj , (x, y), (x0 , y 0 )),r=0phε (tj , (x, y), (x0 , y 0 )) =jX(r)p̃hε ⊗h Hε,h (tj , (x, y), (x0 , y 0 )),r=060ε,(r)(r)where Hhis defined analogously to Hhperturbed counterparts in ε.in (4.25) with Lh , L̃h , p̃h changed respectively by theirTheorem 4.3.5.

Fix T > 0 and let us define a time-grid Λh := {(ti )i∈[[1,N ]] }, N ∈ N∗ . Under (AD),there exists C ≥ 1, c ∈ (0, 1] s.t. for all 0 < tj ≤ T, ((x, y), (x0 , y 0 )) ∈ (R2d )2 , q ∈ (4d, +∞]:|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 (4.23), (4.36) starting from (x, y) at time 0.The closeness of "main" parts p̃h and p̃hε in the above expansions can be derived analogously toLemma 1 [KKM17] as the difference between two Gaussian densities with the small differences inmeans and covariances. The only point we would like to emphasize - the Kolmogorov-like densitypc,K which stands in the bounds due to the control for the scheme transition density in the degeneratecase.

The complete proof could be found in [LM10], Theorem 2.1, (b).Lemma 4.3.6 (Control and Comparison of the densities and their derivatives). There exist c1 ≥1, c ∈ (0, 1] s.t. for all 0 < tj ≤ T, (x, y), (x0 , y 0 ) ∈ R2d and all multi-index α, |α| ≤ 4,|Dxα p̃h (tj , (x, y), (x0 , y 0 )) − Dxα p̃hε (tj , (x, y), (x0 , y 0 ))| ≤c1 ∆dε,σ,γ pc,K (tj , (x, y), (x0 , y 0 ))|α|/2.tjwhere the last inequality holds for all η ∈ (0, γ) due to the mollification procedure.Proof. According to the definitionp̃hε (tj , (x, y), (x0 , y 0 )) =1G (Vjε )−1/2 ε )1/2t2ddet(Vjj0x√−xtj0y −y−xtj ,(4.37)3/2tjwhereVjε = 1tjR tj1t2j 0R tjãεφ(s) ds0ãεφ(s) (tj − s)dsR tj ε1ãφ(s) (tj − s)dst2j 0Rtj1ãεφ(s) (tj − s)2 dst3j 0and ∀z ∈ R2d , G(z)= exp(−|z|2 /2)(2π)−d stands for the density of the standard Gaussian vector ofR2d .

We emphasize that, in (4.37) we introduced the matrix Vjε which is non-degenerate and hasorder one, i.e. there exists c := c(AD) ≥ 1 s.t. c−1 I2d ≤ Vj ≤ cI2d . The matrix Vjε then acts on the x0 −x √tjcomponents renormalized at their intrinsic scales, namely  y0 −y−xtj .3/2tjTaking the result from Lemma 4.3.6, the control for the difference |p̃h − p̃hε |(tj , (x, y), (x0 , y 0 )) comesfrom the closeness of two Gaussian densities with the same mean and slightly different covariancematrices Vj and Vjε , recalling that by the definition Vj0 is equal to Vj .As |detVj − detVjε | ≤ C(T, d)∆dε,σ,γ for any j ≤ N , where C(T, d) stands for the constant whichdepends only on the fixed time T and the dimension d.

Also due to the definition of detVj , it hasthe first order in time.61Thus,|p̃h − p̃hε |(tj , (x, y), (x0 , y 0 ))1111≤− exp − hVj−1 1/22det(Vjε )1/2 (2π)d t2dj det(Vj ) x0 −x  √11tj− hVj−1  y0 −y−xt,exp+jε )1/2 2det(V(2π)d t2d3/2jjtj x0 −x   x0 −x   √√1tjtjε −1 i 00− exp − h(Vj ),y −y−xtjy −y−xtj23/23/2tj0x√−xtjy 0 −y−xtj ,3/2tj0x√−xtjy 0 −y−xtj0x√−xtjy 0 −y−xtji3/2tji3/2tjtj00≤ C(T, d)∆ε,σ,γ pc,K (tj , (x, y), (x , y )),where the difference between two exponents of scalar products can be controlled as usual - using thefirst order Taylor expansion.

Dealing with α : |α| > 0 brings us additional polynomials multipliedwith each exponents - the same as for the frozen densities for the diffusions.Lemma 4.3.7 (Control of the One-Step Convolution for the Chain). For all β ∈ (0, γ), 0 < ti ≤ T,(x, y), (x0 , y 0 ) ∈ R2d there exists Cβ such that:|Hh (ti , (u, v), (x0 , y 0 )) − Hh,ε (ti , (u, v), (x0 , y 0 ))| ≤Cβ ∆dε,γ,∞1−γ/2tipc,K (ti , (u, v), (x0 , y 0 ))Proof.

1. One step transition.Note that if ti = h, the transition probability p̃h (ti − h, (·, ·), (x0 , y 0 )) is the Dirac measure δx0 ,y0 sothatHh (h, (x, y), (x0 , y 0 )) =−1hhhhhhhhhE[δx0 ,y0 (Xh , Yh )|X0 = x, Y0 = y] − E[δx0 ,y0 (X̃h , Ỹh )|X̃0 = x, Ỹ0 = y] ,−1h0 0h0 0=hp (h, (x, y), (x , y )) − p̃ (h, (x, y), (x , y )) .As ph (h, (x, y), (x0 , y 0 )), p̃h (h, (x, y), (x0 , y 0 )) are Gaussian densities, one can get(h1/2 σ(x, y))−1 (x0 − x − b(x0 , y 0 )h)√ 3/20G√2 3(h σ(x, y))−1 (y 0 − y − x+x2 h)Hh (h, (x, y), (x0 , y 0 )) = h−1 (2 3)d ×h2d (det(a(x, y)))1/200(h1/2 σ(xh , y h ))−1 (x0 − x)√G0002 3(h3/2 σ(xh , y h ))−1 (y 0 − y − x+x2 h)−,h2d (det(a(xh0 , y h0 )))1/200where (xh , y h ) := (x0 , y 0 − x0 h).62Applying the same technique for the perturbed version of the kernel function we get(h1/2 σε (x, y))−1 (x0 − x − bε (x, y)h)√ 3/20G√2 3(h σε (x, y))−1 (y 0 − y − x+x2 h)Hh,ε (h, (x, y), (x0 , y 0 )) = h−1 (2 3)d ×2d1/2h det(aε (x, y))001/2(h σε (xh , y h ))−1 (x0 − x)√ 3/2G0002 3(h σε (xh , y h ))−1 (y 0 − y − x+x2 h).−h2d (det(aε (xh0 , y h0 )))1/2As a result, the difference between kernel functions in the case of one-step transition could beestimated as the differencebetween Gaussian h0 densities with close coefficients as in the Chapter 4.

xx ≤ |x0 − x|(1 + h ) + |y 0 − y − x+x0 h| it follows thatAlso due to the fact that −0h22yy∃c > 0, C ≥ 1 s.t|Hh (h, (x, y), (x0 , y 0 )) − Hh,ε (h, (x, y), (x0 , y 0 )| ≤ Ch−1+γ/2 ∆dε,γ,∞ pc,K (h, (x, y), (x0 , y 0 )).Case ti > h. Recall that for all 0 < i ≤ NHh (ti , (u, v), (x0 , y 0 )) =Z h−1ph − p̃h (h, (u, v), (w, z))p̃h (ti − h, (w, z), (x0 , y 0 ))dwdz.(4.38)R2d(4.39)0Set (xh , yh ) := (x, y + hx), (x0h,i , yh,i) := (x0 , y 0 − x0 ti ).

Define ∀(u, v) ∈ R2d , B h (u, v) :=h1/2 σ(u, v)0b(u, v)hh√.,Σ(u,v)=b(u, v)h2 /2h3/2 σ(u, v)/2 h3/2 σ(u, v)/(2 3) 0x0 02d 30 0hIntroducing for all (x, y), (w, z), (x , y ) ∈ (R ) transitions: Ph (w, z), (x , y ) := Σ (w, z) 0 , Th (x, y), (w, z), (xyB h (x, y) + Ph (w, z), (x0 , y 0 ) , we can rewrite (4.39)("h×p̃ti − h,"− p̃hti − h,xhyhxhyhHh (ti , (x, y), (x0 , y 0 )) = h−1!+ Th (x, y), (x, y), (u, v) , (x0 , y 0 )+ Ph− p̃h ti − h,!0(x0h,i , yh,i), (u, v)00, (x , y )63Zh− p̃ti − h,xhyhdudvG(u, v)!#R2dxhyh, (x0 , y 0 )!#)00, (x , y ).According to the Taylor expansion at order one:× Dx p̃Hh (ti , (x, y), (x0 , y 0 )) = h−1! ti − h, (xh , yh ) + ηTh (x, y), (x, y), (w, z) , (x0 , y 0 )h−Dx p̃hti − h, (xh , yh ) + ηPh0(x0h,i , yh,i), (w, z)!00, (x , y )× ThZZR2d(x)(x, y), (x, y), (w, z)h−1ZZti − h, (xh , yh ) + ηTh× Dy p̃hDy p̃ti − h, (xh , yh ) + ηPh0(x0h,i , yh,i!00, (w, z)), (x , y )dη0(y)! 0 0(x, y), (x, y), (w, z) , (x , y ) × Th ((x, y), (x, y), (w, x))1dwdzG(w, z)R2dhdη0(x) 0× Ph ((x0h,i , yh,i), (w, z))+1dwdzG(w, z)− (y) 0× Ph (x0h,i , yh,i), (w, x):= (M1h + R1h )(ti , (x, y), (x0 , y 0 )),where Dx , Dy denote respectively the differentiation w.r.t.

the d first components x = (x1 , . . . , xd ) ∈Rd and the d second components y = (y1 , . . . , yd ) ∈ Rd of the point (x, y) in R2d .As we are interested in the difference |Hh − Hh,ε |(ti , (u, v), (x0 , y 0 )) it is enough to estimate thecloseness of R1h , R1h,ε and M1h , M1h,ε .We need to recall two following controls which has been mentioned before in (4.17). Let µ =(µ1 , .

. . , µd ) ∈ Nd , ν = (ν1 , . . . , νd ) ∈ Nd be multi-indices. We have, ∃c > 0, C ≥ 1, ∀(µ, ν), |µ| ≤3, |ν| ≤ 4, ∀0 < i ≤ N, (x, y), (x0 , y 0 ) ∈ R2d ,−(|Dxν Dyµ p̃h (ti , (x, y), (x0 , y 0 ))| ≤ Cti|ν|32 + 2 |µ|)pc,K (ti , (x, y), (x0 , y 0 )).Observe as well that there exists C > 0 s.t.(x)(x)000|Th (x, y), (x, y), (w, z)− Ph (xh,i , yh,i ), (w, z)| ≤ C(h + |(x, y) − (x0h,i , yh,i)|γ h1/2 |(w, z)|),(y)(y)00|Th (x, y), (x, y), (w, z)− Ph (x0h,i , yh,i), (w, z)| ≤ C(h2 + |(x, y) − (x0h,i , yh,i)|γ h3/2 |(w, z)|)0since b(·, ·) is bounded and the difference between Σ(x, y) and Σ(x0h,i , yh,i) can be controlled due tothe Hölder continuity.!h0 0As in the article [LM10] expanding terms Dx p̃ ti , (xh , yh ) + ηTh (x, y), (x, y), (w, z) , (x , y )!and Dy p̃h ti , (xh , yh ) + ηTh (x, y), (x, y), (w, z) , (x0 , y 0 )at order 2 around (xh , yh ) in M1h one canget:Hh (ti , (x, y), (x0 , y 0 )) = H(ti , (x, y), (x0 , y 0 )) + (R1h + R2h )(ti , (x, y), (x0 , y 0 ))where, we have denoted, with a slight abuse of notation H(ti , (x, y), (x0 , y 0 )) = (L−L̃)p̃h (ti , (x, y), (x0 , y 0 ))whereas from the continuous case H(ti , (x, y), (x0 , y 0 )) = (L − L̃)p̃(ti , (x, y), (x0 , y 0 )).

Pay attentionthat a priori, p̃h (ti , (x, y), (x0 , y 0 )) 6= p̃(ti , (x, y), (x0 , y 0 )). The only difference between those two objects is in the covariance matrices for which the backward transport of the final point is taken in64continuous time in p̃ and in discrete time in δph .R2h (ti , (x, y), (x0 , y 0 )) := h−b(x0 , y 0 ), Dx p̃h ti − h, (xh , yh ), (x0 , y 0 ) i +()1002 h0 0Tra(x, y) − a(xh,i , yh,i ) Dx p̃ ti − h, (xh , yh ), (x , y )2The difference between |H − Hε |(ti , (x, y), (x0 , y 0 )) can be controlled as in (4.33).

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