Transition probabilities of Lévy‐type processes: Parametrix construction

Abstract

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol q(x,ξ) with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.

Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs

“…(ii) Example 5.2 can be also shown by combining Theorem 2.1 with the heat kernel estimates established in [12], see also [11]; in fact, any continuous negative definite function listed in [12, Table 2] satisfies the assumptions of Theorem 2.1.…”

“…In the first part of this section we establish the existence of a solution to (11), cf. Theorem 4.4.…”

“…The assumption α > 1 is needed to prove that ∇u(t, ·) exists and ∇u(t, ·) ∈ C β b (R d ) whereas (4) is used to show that ∇u(t, ·) ∈ C γ0/2 b (R d ). The fact that ∇u(t, ·) ∈ C β b (R d ) will be needed to construct a solution to (11) using Picard iterations, see Theorem 4.4, and ∇u(t, ·) ∈ C γ0/2 b (R d ) will be used when we apply Itô's formula, see Proposition A.2 and the proof of Theorem 2.1. Note that ∇u(t,…”

Heat kernels of non-symmetric Lévy-type operators

“…The tool used in this paper is the parametrix method, proposed by E. Levi [58] to solve elliptic Cauchy problems. It was successfully applied in the theory of partial differential equations [27], [62], [19], [24], with an overview in the monograph [25], as well as in the theory of pseudo-differential operators [22], [44], [48], [54], [68]. In particular, operators comparable in a sense with the fractional Laplacian were intensively studied [20], [21], [51], [53], [22], also very recently [16], [40], [15], [55].…”

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

“…For the proof of Proposition 6.1 the parametrix construction of the transition density of (X t ) ≥0 in [20] plays a crucial role, see also [23]. In this section, we collect some results from [20] which are needed for our proofs.…”