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for all 0 < tj ≤ T, ((x, y), (x0 , y 0 )) ∈(R2d )2 :|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 of equations (14), (17) starting from (x, y) at time 0.These two theorems will be restated and discussed in Section 4.3.1.The sensitivity analysis will then be applied, in the flavour of [KM17] to investigate the weak error associated to a specific Euler scheme which had already beenconsidered in [LM10] for equations of type (13). However, to perform the analysiswe need to change assumptions (AD) slightly. Precisely, we have to assume moreabout Hölder properties of coefficients than in (AD).Instead of (AD3), we assume for some γ ∈ (0, 1] , 0 < κ < ∞ it holds: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 whenconditions (AD1),(AD2), (ÂD3) are in force.Theorem. Fix T > 0. Under assumptions (ÂD) for any test function f ∈C β,β/2 (R2d ) (β−Hölder in the first variable and β/2−Hölder in the second variablefunctions) for β ∈ (0, 1], there exists C > 0, such that:|E(x,y) [f (XTh , YTh )] − E(x,y) [f (XT , YT )]| ≤ Chγ/2 (1 + |x|γ/2 ).where γ ∈ (0, 1] stands for the Hölder index of γ, γ/2 (γ for the variable x, γ/2 fory) Hölder continuous time-homogeneous functions b, σ.The theorem will be restated in Section 4.4.We also would like to present our control for the direct difference of transitiondensities p(t, (x, y), (x0 , y 0 )) and ph (t, (x, y), (x0 , y 0 )).
The result below is in clearcontrast with the one of Theorem 4.4.1 for the weak error, i.e. when additionally oneconsiders an integration of a Hölder function w.r.t. the final (or forward variable).We finally can reach a global error of order hβ , β < γ − 1/2 which is close to theexpected one in hγ/2 when γ goes to 1.10Theorem (4.5.1).
Fix a final time horizon T > 0 and a time step h = T /N, N ∈ N∗for the Euler scheme. Under assumptions (ÂD), for γ ∈ (1/2, 1] and β ∈ (0, γ− 21 ),for all t in the time grid Λh := {(ti )i∈[[1,N ]] } and (x, y), (x0 , y 0 ) ∈ R2d there existC := (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 )),(18)s∈[t−h,t]where pc,K (s, (x, y), (x0 , y 0 )) stands for the Kolmogorov-type Gaussian density (15)at time s.The theorem will be discussed in Section 4.5.References[Aro59]D. G Aronson. The fundamental solution of a linear parabolic equationcontaining a small parameter. Illinois Journal of Mathematics, 3:580–619, 1959.[Bai17]N.T.G. Bailey.
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