Диссертация (1137347), страница 17
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Namely, under(ÂD), the exist Cη ≥ 1, c ≤ 1, s.t. for all 0 ≤ i < j ≤ N, (x, y), (x0 , y 0 ) ∈ (R2d )2 :|(p − pε )(tj , (x, y), (x0 , y 0 )| ≤ Cη εγ pc,K (tj , (x, y), (x0 , y 0 )),(4.47)and, similarly, it is established in the Chapter 4 that the same control holds for the scheme (4.23)and its associated perturbation:|(ph − pεh )(tj , (x, y), (x0 , y 0 )| ≤ Cη εγ pc,K (tj , (x, y), (x0 , y 0 )),(4.48)where pc,K has been denoted in (4.15).With these notations and controls at hand we rewrite our initial error as:E 1 (f, (x, y), T, h)h,0,(x,y)h,0,(x,y)0,(x,y)0,(x,y)E[f (XT=h,0,(x,y)h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)E[f (XT, YT)] − E[fε (XT, YT)]ε,0,(x,y)ε,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y), YT)])] − E[fε (XT+E[fε (XT, YTε,0,(x,y)ε,0,(x,y)0,(x,y)0,(x,y)+E[fε (XT, YT)] − E[f (XT, YT)]3X1k,ε=:E, YT)] − E[f (XT=, YT)](f, (x, y), T, h),(4.49)k=1ε,t,(x,y)ε,t,(x,y)where the notation (Xt, Yt)) stands for the solution of the SDE obtained replacing thecoefficients in (4.9) with bε , σε .
The solution exists due to the additional smoothness we assumedw.r.t. (4.9). Let us first control E 11,ε (f, (x, y), T, h) which we again split into two parts:E 11,ε (f, (x, y), T, h)=ε,0,(x,y)E[fε (XTε,0,(x,y), YT0,(x,y)0,(x,y)+E[fε (XT, YT)]=:E111,ε+E112,ε690,(x,y))] − E[fε (XT−0,(x,y), YT0,(x,y)0,(x,y)E[f (XT, YT)](f, (x, y), T, h).)]Now, from the Gaussian upper-bound for the density, deriving from Theorem 4.4.2 above, and similarly to inequality (4.45) (which does not exploit the boundedness of the considered function), weget:Zpc,K (T − t, (x, y), (x0 , y 0 ))|(fε − f )(x0 , y 0 )|dx0 dy 0 ≤ Cεβ .E 112,ε ≤ CR2dOn the other hand, the stability result (4.47) yields:ZE 111,ε ≤ Cη εγpc,K (T − t, (x, y), (x0 , y 0 ))|fε (x0 , y 0 )|dx0 dy 0R2dZ≤ Cηpc,K (T − t, (x, y), (x0 , y 0 ))|fε (x0 , y 0 ) − fε (x0 − x, y 0 − y − x(T − t))|R2d+ pc,K (T − t, (x, y), (x0 , y 0 ))||fε (x0 − x, y 0 − y − x(T − t))|dx0 dy 0xγ≤ Cη ε (1 + |RT −t|β ),ywhere Ru :=Id(T − t)Id0d, exploiting as well the Young inequality for the last control.
ThisIdfinally gives:E 11,ε −→ 0.ε→0(4.50)The sensitivity result (4.48) for the scheme similarly yields:E 13,ε −→ 0.ε→0(4.51)We now focus on the contribution E 12,ε for which we can rely on a PDE type analysis technique.Such an approach had first been used in the context of non-degenerate Hölder continuous Eulerschemes by Mikulevičius and Platen [MP91] through Schauder estimates.
This in particularly requiredthe final test function to be smooth (specifically f ∈ C 2+γ for γ-Hölder continuous coefficients b, σ).This approach was extended in [KM17] using direct control bounds on the heat-kernel allowing thatway to consider only β-Hölder continuous test functions β ∈ (0, 1].The point here is that, through the regularization we are able to use pointwise bounds of thederivatives of the functionZε,t,(x,y)ε,t,(x,y)vε (t, x, y) := E[fε (XT, YT)] =pε (T − t, (x, y), (x0 , y 0 ))fε (x0 , y 0 )dx0 dy 0R2dand to control as well pointwise and uniformly in ε, for ε small enough, under (ÂD), the spatialderivatives of vε w.r.t. the non-degenerate component.Observe that, since fε , bε , σε are smooth, it is readily seen, from the smoothness of vε and the Markovproperty (see e.g.
[TT90]), that vε satisfies the PDE((∂t vε + Lε vε )(t, x, y) = 0,vε (T, x, y) = fε (x, y), (x, y) ∈ R2d ,where Lε stands for the generator associated with SDE obtained replacing the coefficients in (4.9)with bε , σε , i.e. for all ϕ ∈ C02 (R2d , R), (x, y) ∈ R2d ,1Lε ϕ(x, y) = bε (x, y) · ∇x ϕ(x, y) + x∇y ϕ(x, y) + Tr(aε (x, y)Dx2 ϕ(x, y)).270For the further analysis we have to apply the Ito formula directly to the scheme (4.23) viewed asan Ito process exploiting Hölder continuity of coefficients.E 12,εε,h,0,(x,y)= E[fε (XT=N−1Xε,h,0,(x,y), YTε,h,0,(x,y)E[vε (ti+1 , Xti+1ε,0,(x,y))] − E[fε (XTε,h,0,(x,y), Yti+1ε,0,(x,y), YT)]ε,h,0,(x,y)) − vε (ti , Xtiε,0,h,(x,y), Yti)]i=0=N−1XhZEti+1n∂s vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) ) + ∇x vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )tii=0ε,h,0,(x,y)ε,h,0,(x,y)bε (Xti, Yti)+ Xsε,h,0,(x,y) ∇y vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )o i1ε,h,0,(x,y)ε,h,0,(x,y)Tr(Dx2 vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )a(Xti, Yti)) ds+2N−1 h Z ti+1 noiX=E∂s vε + Lε vε (Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )ds×tii=0hZ+ Eti+1n∇x vε (Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )tiε,h,0,(x,y)ε,h,0,(x,y)× (bε (Xti, Yti) − bε (Xsε,h,0,(x,y) , Ysε,h,0,x,y ))1ε,h,0,(x,y)+Tr(Dx2 vε (Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )(a(Xti, Ytε,h,0,x,y)i2o i− a(Xsε,h,0,(x,y) , Ysε,h,0,(x,y) ))) ds=N−1XhZEti+1n∇x vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )tii=0×ε,h,0,(x,y)ε,h,0,(x,y)(bε (Xti, Yti)+1Tr(Dx2 vε (s, Xsε,h,0,(x,y) , Ysε,h,0,(x,y) )2×(a(Xtiε,h,0,(x,y)ε,h,0,(x,y), Yti− bε (Xsε,h,0,(x,y) , Ysε,h,0,(x,y) ))o i) − a(Xsε,h,0,(x,y) , Ysε,h,0,(x,y) ))) ds ,(4.52)exploiting the PDE satisfied by vε for the last equality.To finish the analysis we need to control ∇x vε (s, x, y) and Dx2 vε (s, x, y) uniformly over ε ∈(0, 1].
Moreover, we will also use the Hölder properties of bε and aε to control the differencesε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)ε,h,0,(x,y)bε (Xti, Yti)−bε (Xs, Ys) and aε (Xti, Yti)−aε (Xs, Ys).To achieve bounds for the heat kernel derivatives we refer the reader to Theorem 4.4.2 below.Theorem 4.4.2.
Under (ÂD), for any t ∈ [0, T ] and |α| ≤ 2 there exist C ≥ 1, c ∈ (0, 1] such that,for ε ∈ [0, ε0 ], for ε0 > 0 small enough,|Dxα pε (t, (x, y), (x0 , y 0 ))| ≤Cpc,K (t, (x, y), (xt|α|/20, y 0 )).Let us postpone the proof to Appendix, Section 4.6.Remark 4.4.3. The result of the Theorem 4.4.2 is of interest by itself. Up to the best of our knowledge,these are the first pointwise bounds obtained on the derivatives w.r.t. the non-degenerate variablesunder the sole Hölder continuity assumption for the coefficients in (4.9).
They extend to Kolmogorov71diffusions the well-known controls derived by Il’in et al. [IKO62]. Investigating the quantitativebehaviour of the derivatives w.r.t. the degenerate under minimal smoothness assumptions remains avery interesting and open problem. Provided the coefficients are Lipschitz, the Konakov and Mammentrick should apply to get the expected control, namely the normalized Kolmogorov density multipliedby an additional singularity of the characteristic order in time, here (T − t)−3/2 .
Finding out theminimal assumption yielding such a bound is rather challenging.Coming back to the proof of Theorem 4.4.1, we derive from the control in Theorem 4.4.2 that for Zxαα0 00 0Dx vε (t, x, y) =Dx pε (T − t, (x, y), (x , y ))[fε (x , y ) − fε (RT −t)]dx0 dy 0 ,ydRthe following inequality holds:Z[fε ]C β,β/2 pc,K (T − t, (x, y), (x0 , y 0 ))|Dxα vε (t, x, y)| ≤ C(T − t)α/2R2don× |x0 − x|β + |y 0 − (y + (T − t)x|β/2 dx0 dy 0 ,(4.53)00(x ,y )|where [f ]C β,β/2 := sup(x,y)6=(x0 ,y0 ) |fdβ(x,y)−f, dβ ((x, y), (x0 , y 0 )) := |x − x0 |β + |y − y 0 |β/2 .((x,y),(x0 ,y 0 ))To have the same scale as in the exponent in (4.15) let us rewrite (4.53) taking into account that:β|x0 − x|(T − t)1/2!β/2(x−x0 )(T −t)(x+x0 )(T −t)0−||y−y−22+(T − t)β/2(T − t)!β/2 β(x+x0 )(T −t)00|y−y−||x−x|2≤ C(T − t)β/2 +(T − t)1/2(T − t)3/2|x0 − x|β + |y 0 − y − x(T − t)|β/2 = (T − t)β/2From (4.15) and (4.54) one can get, up to a modification of the constant c to c̄:ZC[fε ]C β,β/2 pc̄,K (T − t, (x, y), (x0 , y 0 )) 0 0|Dxα vε (t, x, y)| ≤dx dy(T − t)α/2−β/2R2dC[fε ]C β,β/2≤.(T − t)|α|/2−β/2(4.54)(4.55)Plugging (4.55) into (4.52) we get:(ZT|E 12,ε |≤C[fε ]C β,β/20×+hiε,h,0,x,yε,h,0,x,yds E |bε (Xsε,h,0,x,y , Ysε,h,0,x,y ) − bε (Xφ(s), Yφ(s))|1(T − s)1/2−β/2hiε,h,0,x,yε,h,0,x,yE |aε (Xsε,h,0,x,y , Ysε,h,0,x,y ) − aε (Xφ(s), Yφ(s))|where φ(s) = ti for s ∈ [ti , ti+1 ), i = 0, .
. . , N − 1.72)1,(T − s)1−β/2(4.56)Let us denote by Ψε (x, y) any of two functions bε (x, y) or aε (x, y) since they both satisfy the sameHölder continuity assumptions. Following the introduced notations, we are tempted to boundhiε,h,0,x,yε,h,0,x,yE |Ψε (Xsε,h,0,x,y , Ysε,h,0,x,y 0) − Ψε (Xφ(s), Yφ(s))|ihion hε,h,0,x,y γ/2ε,h,0,x,y γ|| + E |Ysε,h,0,x,y − Yφ(s)≤ [Ψε ]C γ,γ/2 E |(Xsε,h,0,x,y − Xφ(s)From the definition (4.23) of the approximation scheme:ihε,h,0,x,y γE |(Xsε,h,0,x,y − Xφ(s)|γ ihε,h,0,x,yε,h,0,x,yε,h,0,x,yε,h,0,x,y)(Ws − Wφ(s) ), Yφ(s))(s − φ(s)) + σ(Xφ(s), Yφ(s)= E bε (Xφ(s)≤ C(|bε |∞ ∨ |σε |∞ )hγ/2 .(4.57)To control the error in the second component we cannot compensate the transport so we have tokeep the dependency on the starting point in the final bound:γ/2 Z shiε,h,0,x,yE |(Ysε,h,0,x,y − Yφ(s)|γ/2 = E Xuε,h,0,x,y du φ(s)hγ/2 i γ/2≤ sup E Xsε,h,0,x,y h≤ Chγ/2 |x|γ/2 .(4.58)s∈[0,T ]Applying controls in (4.57) and (4.58) for (4.56) we have a final control for the E 12,ε :|E 12,ε | ≤ C(a, b, fε , T )hγ/2 (1 + |x|γ/2 ).(4.59)which proves the statement of the Theorem 4.4.2.Remark 4.4.4.
The rate hγ/2 (1 + |x|γ/2 ) holds even for tests functions f ∈ C β1 ,β2 (R2d ), (β1 , β2 ) ∈(0, 1]2 .Our second result, already mentioned in the Introduction (Chapter 1), provides bounds on thedifference between the densities Ed .4.5Global errorTheorem 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 − 21 ), 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 )|≤ Chβ (1 + (|x| ∧ |x0 |))1+γ )suppc,K (s, (x, y), (x0 , y 0 )),(4.60)s∈[t−h,t]where as in (4.15) pc,K (s, (x, y), (x0 , y 0 )) stands for the Kolmogorov-type gaussian density at time s.The proof is given below in Section 4.5.1.
The above result is in clear contrast with the oneof Theorem 4.4.1 for the weak error, i.e. when additionally consider an integration of a Hölderfunction w.r.t. the final (or forward variable). The point is that such an integration allows to73exploit directly the spatial bounds of Theorem 4.4.2 on the underlying heat-kernel (with possiblymollified coefficients). When handling directly the difference of the densities we cannot avoid tocontrol sensitivities of the kernels w.r.t. to the degenerate variable.
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