On the cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces

Weak Euler Approximation for Itô Diffusion and Jump Processes

“…For β ∈ (0, 1), the results on Kolmogorov equations in Hölder classes have been proved [31,32]. They can be extended to the case β > 1 in a standard analytic way.…”

“…For α ∈ (0, 2] and β ∈ (0, 1), given f ∈ C β (H), there exists a unique solution u ∈ C α+β (H) to the Kolmogorov equation (14) and |u| α+β ≤ C|f | β [31].…”

On $$L_{p}$$ - theory for stochastic parabolic integro-differential equations

“…The estimate (ii) follows by Theorem 2.1 in [11]. The estimate (iii) is proved in [15] (we apply Corollary 1 and Proposition 2 with V = L 2 (U, U, )).…”

“…The assertion (iii) and embedding theorem imply that u ∈ D p (E). Using Lemma 3.2 in [11], we get that there is a constant C such that for every υ ∈ H β+α p (E)…”

“…According to Lemma 12 in [11] or inequality (36) and Lemma 16 in [15], there are positive constants C and c such that for all s < t, j ≥ 1,…”