Диссертация (1137347), страница 16

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There exist c ∈(0, 1], c3 ∈ (0, 1] such that for 0 < ti ≤ T and all (x, y), (x0 , y 0 ) ∈ R2d1−γpc,K (ti , (x, y), (x0 , y 0 ))|H − Hε |(ti , (u, v), (x0 , y 0 ))| ≤ 1 ∨ T 2 c3 ∆dε,γ,∞.(ti )1−γ/2Using the definition of R1h (ti , (x, y), (x0 , y 0 )) and the telescoping sums combined with bounds for thederivatives of the frozen transition density one can get:ZZ 1h,εh0 0−1dη| R1 − R1 (ti , (x, y), (x , y ))| = hdwdzG(w, z)0R2d! (y)h0 0× Dy p̃ ti − h, (xh , yh ) + ηTh (x, y), (x, y), (w, z) , (x , y ) × Th ((x, y), (x, y), (w, x))!0ti − h, (xh , yh ) + ηPh (x0h,i , yh,i), (w, z) , (x0 , y 0 ) ×−Dy p̃h−1ZZ−hdwdzG(w, z)R2d× Thε× Phε1dηDy p̃hεti − h, (xh , yh ) +ηThε0Ph ((x0h,i , yh,i), (w, x))!(y) !0 0(x, y), (x, y), (w, z) , (x , y )0!!(y)0(x, y), (x, y), (w, x)− Dy p̃hε ti − h, (xh , yh ) + ηPhε (x0h,i , yh,i), (w, z) , (x0 , y 0 )!(y) 1−γpc,K (ti , (x, y), (x0 , y 0 ))002c2 ∆dε,γ,∞.(xh,i , yh,i ), (w, x)≤ 1∨T1−γ/2tiAlso the difference |R2h − R2h,ε |(ti , (x, y), (x0 , y 0 )) can be held according to the boundedness ofb(·, ·), bε (·, ·), Hölder properties of a(·, ·), aε (·, ·) and bounds for the derivatives |Dxα p̃h ti −h, (xh , yh ), (x0 , y 0 ) |.65h,εh| R2 − R2 (ti , (x, y), (x0 , y 0 ))| =0 00 0h0 0hbε (x , y ) − b(x , y ), Dx p̃ ti − h, (xh , yh ), (x , y ) i!+hbε (x0 , y 0 ), Dx p̃hε − Dx p̃hti − h, (xh , yh ), (x0 , y 0 ) i(10000− Tra(x, y) − a(xh,i , yh,i ) − aε (x, y) + aε (xh,i , yh,i )2×Dx2 p̃h ti − h, (xh , yh ), (x0 , y 0 )(!)1002 h2 h0 0− Traε (x, y) − aε (xh,i , yh,i )Dx p̃ε − Dx p̃ti − h, (xh , yh ), (x , y )2≤∆dεγ∞ pc,K (ti − h, (xh , yh ), (x0 , y 0 )).(ti − h)1−γ/2Thus, we finally have proved the Lemma.Lemma 4.3.8 (Difference of the iterated kernels).

For all ti , i ∈ (0, j] ,tj ≤ T , (x, y), (x0 , y 0 ) ∈ R2dand r ∈ N:(r)(r)|(p̃h × Hh − p̃hε × Hh,ε )(ti , (x, y), (x0 , y 0 ))|(r+2)γrγ22ttii≤ C r ∆dε,γ,∞+pc,K (ti , (x, y), (x0 , y 0 )). Γ 1 + rγΓ 1 + (r+2)γ 2(4.40)2Proof. Observe that Lemmas 4.3.6 gives (4.40) for r = 0. Let us assume that it holds for a givenr ∈ N∗ and let us prove it for r + 1.Let us denote for all r ≥ 1,(r)ηr (ti , (x, y), (x0 , y 0 )) := |(p̃h ⊗ H (r) − p̃hε ⊗ Hh,ε )(ti , (x, y), (x0 , y 0 ))|.

Write(r)(r)ηr+1 (ti , (x, y), (x0 , y 0 )) ≤ (p̃h ⊗ Hh − p̃hε ⊗ Hh,ε ) ⊗ Hh (ti , (x, y), (x0 , y 0 ))(r)+ p̃hε ⊗ Hh,ε ⊗ (Hh − Hh,ε )(ti , (x, y), (x0 , y 0 ))(r) ≤ ηr ⊗ |Hh | (ti , (x, y), (x0 , y 0 )) + p̃hε ⊗ Hh,ε ⊗ |(Hh − Hh,ε )| (ti , (x, y), (x0 , y 0 )).Thus,from the induction hypothesis, similarly to Lemma 4.3.4, we get the result.Through the Lemma 4.3.8 one can prove the Lemma 4.3.5.4.4Weak errorIn the same manner as in the article [KM17] we would like to consider the analogue to the differencebetween the degenerate diffusion and it’s Euler scheme in the case of non-smooth coefficients.66Remark 4.4.1.

We would like to emphasize that for our error controls, we need to consider γ/2for the Hölder index of the degenerate second variable. According to the existing literature, see e.g.Lunardi [Lun97] or Priola [Pri09], concerning Schauder estimates for PDEs associated with generatorsderiving from (4.9), one could expect this regularity to be γ/3 which corresponds to the homogeneityindex of the degenerate variable (see again the above references or Bramanti et al.

[BCLP10] or[Men18] for some related applications to harmonic analysis). The current index appears through ouranalysis because of some specific properties of the model, namely the increment over time step ofthe degenerate component needs to be handled (unbounded coefficient).

This precisely leads to theindicated restriction (see Theorem 4.5.1 and its proof).There are two kinds of quantities we would be interested in while studying approximations of theSDE’s solution. First, we can focus on the analogue to (1.4):Ew (f, (x, y), T, h) := E(x,y) [f (XTh , YTh )] − E(x,y) [f (XT , YT )],(4.41)where f is a test function that lies in a suitable functional space. The second quantity we will beinterested in concerns directly the difference of the densities. We have indicated above that the Eulerscheme (4.23) has a density enjoying Gaussian bounds.

We can refer to the Chapter 4 to justify that,under the current Assumptions (AD), the diffusion in (4.9) itself has a density. The existence of thedensity also follows from the well-posedness of the martingale problem associated with the generatorof (4.9) and the estimates in Theorem 4.4.2.

We will try to quantify, for a given time t ∈ {(ti )i∈[[0,N ]] },in terms of h the differenceEd ((x, y), (x0 , y 0 ), t, h) := (p − ph )(t, (x, y), (x0 , y 0 )),(4.42)where p(t, (x, y), (x0 , y 0 )) (resp. ph (t, (x, y), (x0 , y 0 )) denotes the density of the unique weak solution ofthe SDE (4.9), at time t and point (x0 , y 0 ) when the starting point at time 0 is (x, y) (resp. X h givenby the Euler scheme (4.23) at time t and point (x0 , y 0 ) when the starting point at time 0 is (x, y)).To perform the further analysis we have to assume more about Hölder properties of coefficientsas it has been already mentioned in Remark 4.4.1. Namely, instead of (AD3), we assume for someγ ∈ (0, 1] , κ,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.Remark 4.4.2. Due to the boundedness of coefficient (ÂD3) is included in (AD3), meaning that allprevious results, achieved under (AD3) still hold under (ÂD3).Our first main result, which we have already mentioned in Chapter 1, is the following theorem.Theorem 4.4.1. Assume (ÂD) holds and fix T > 0. 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],there exists C > 0, such that for E 1 as in (4.41):|Ew (f, (x, y), T, h)| ≤ Chγ/2 (1 + |x|γ/2 ).Rt,(x,y)t,(x,y)Proof.

Denote, using Markovian notations, v(t, x, y) := E[f (XT, YT)] = R2d p(T −t, (x, y), (x0 , y 0 ))f (x0 , y 0 )dx0 dy 0 .Now, well posedness of the martingale problem yields that v is actually a weak solution of the PDE:((∂t v + Lv)(t, x, y) = 0,(4.43)v(T, x, y) = f (x, y), (x, y) ∈ R2d ,67where L stands for the generator of (4.9) at time t, i.e. for all ϕ ∈ C02 (R2d , R), (x, y) ∈ R2d ,1Lϕ(x, y) = b(x, y) · ∇x ϕ(x, y) + x∇y ϕ(x, y) + Tr(a(x, y)Dx2 ϕ(x, y)).2Pay attention that, even though we have good controls on the spatial gradients for the non-degeneratevariables, see again Theorem 4.4.2 below, in the current degenerate setting this does not yield thatv is a classical solution to (4.43). Indeed, it does not seem to be an easy task to directly controlpointwise, under our mild Hölder assumption (ÂD) 1 , the derivatives w.r.t. degenerate variable of thedensity p expressed as a convergent parametrix sum (see once more the proof of Theorem 4.4.2 for theparametrix expansion of the density).

We also mention that similar features appear in the papers whohandle Schauder estimates for PDEs related with (4.9). In [Lun97] and [Pri09] the derivatives w.r.t.to the non-degenerate variable are controlled up to order 2, whereas for the degenerate variable(s)the bounds obtained are for Hölder moduli of continuity of v (w.r.t. to those variables).To circumvent this difficulty we need to introduce a smoothing procedure of the coefficients.Mollification procedure.Let us specify the mollification procedure. Namely, for a small parameter ε, we smooth suitably thecoefficients and the function f introducing:Rbε (x, y) := b ? ρε (x, y) = R2d b(u, v)ρε (x − u, y − v)dudv,Rσε (x, y) := σ ? ρε (x, y) = R2d σ(u, v)ρε (x − u, y − v)dudv,Rfε (x, y) := f ? ρε (x, y) = R2d f (u, v)ρε (x − u, y − v)dudv,(4.44)where ? stands for the spatial convolution and ρε is a spatial mollifier, i.e.Zρε (x, y) = ε−3d ρ(x/ε, y/ε2 ), ρ ∈ C ∞ (R2d ),ρ(x, y)dxdy = 1, |supp(ρ)| ⊂ K,R2d2dfor some compact set K ⊂ R .According to the notations and the Hölder and boundness properties of the coefficients one canprove:Z2(b(x, y) − b(x − uε, y − vε )ρ(u, v)dudv .|b − bε | ≤ R2dFrom the Hölder continuity of b:sup|(b − bε )(x, y)| ≤ Cρ εγ , Cρ := κ(x,y)∈R2dZ(|u|γ + |v|γ/2 )ρ(u, v)dudv.R2dThe same analysis can be performed for σε and fε so that σε and fε satisfies Hölder conditions.This gives|b − bε | + |σ − σε |≤ Cεγ ,|f − fε |≤ Cεβ .(4.45)As the result we get the following controls for closeness of coefficients:1 Observe that if the coefficients were smooth, the Konakov and Mammen trick would also give the pointwise controlson the derivatives w.r.t.

y.68|b(x, y) − bε (x, y)| ≤ Cεγ ,sup(x,y)∈R2d|f (x, y) − fε (x, y)| ≤ Cεβ ,sup(x,y)∈R2d∀η ∈ (0, γ),sup|σ(x, y) − σε (x, y)| + |(σ − σε )|η(x,y)∈R2d≤ Cη (εγ + εγ−η ).Let us introduce a suitable distance:∀(x, y), (x0 , y 0 ) ∈ (Rd )2 , d̂ (x, y), (x0 , y 0 ) := |x − x0 | + |y 0 − y|1/2 .(4.46)γNamely, a measurable function f is in Cb,d(Rd , Rd ⊗ Rd ) if|f |d̂,γ := sup |f (x)| + [f ]d̂,γ , [f ]d̂,γ :=x∈Rdsup(x,y)6=(x0 ,y 0 )∈R2d|f (x, y) − f (x0 , y 0 )|γ < +∞.d̂ (x, y), (x0 , y 0 )Following the arguments of Chapter 4 since we can control the closeness between the transitiondensities corresponding to the SDEs with mollified and non-mollified coefficients, it is then possibleto control the difference between transition densities of the corresponding diffusions.

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