Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Abstract

We study a real-valued Lévy-type process X, which is locally α-stable in the sense that its jump kernel is a combination of a 'principal' (state dependent) α-stable part with a 'residual' lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate X in law by a non-linear regression Xwith a deterministic regressor term f t (x) and α-stable innovation term U x t , and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case α < 1, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.

Construction and heat kernel estimates of generalstable-like Markov processes

“…The case where the matrix A is non-diagonal is particularly interesting, since the structure of the transition density estimate is completely different; in particular, it is impossible to obtain an estimate of the type (2.5). Similar effects have also been observed in [59] in the non-symmetric scalar setting (N = µ plus a further perturbation); see Chapter 4 for a detailed discussion.…”

“…a strongly continuous, positivity preserving and contractive semigroup-whose generator is an extension of L; we also have to verify the uniqueness stated in Theorem 3.1. For that, we adapt the strategy developed in [48,59]: in particular, we show that p t (x, y), which was constructed by means of the parametrix approach in Chapter 5 as a candidate for the fundamental solution, is an approximate fundamental solution (in the sense of Section 7.1). For the readers' convenience and in order to have a self-contained presentation, we give full proofs below.…”

“…Following [59,Definition 5.1] we say that a function h(t, x) is approximate harmonic for the operator ∂ t − L if there exists a family {h…”

“…We show that the martingale problem for such stable-like models with microstructural perturbations is well-posed, and that its solution is a Lévy-type (or Feller) process which has a transition probability density p t (x, y); this answers the first of the two general questions formulated above. In order to approach the second question, we adapt the framework used in the paper [59], where a simpler one-dimensional model with constant α(x) ≡ α is considered. In the present paper, we represent the transition density p t (x, y) as a sum of an explicitly given 'principal' part and a 'residual' part which has explicit bounds (see (3.13), (3.14) and (3.20)).…”

Non-symmetric Lévy-type operators

“…A classical method of proving (CoJ1) -(CoJ4) is to associate P t f with an operator L satisfying (L1) -(L3) by showing that it solves the equation ∂ t u − Lu = 0, in other words, that it is harmonic for ∂ t − L. In general, a typical problem is that we do not know whether P t f ∈ D(L), which is usually a question of sufficient regularity. In a series of papers [34], [45], [35] this problem was resolved for certain operators by introducing the notion of approximate harmonicity and by specifying the so called approximate fundamental solution. Namely, for t, ε > 0, t + ε 1 and f ∈ C 0 (R d ) we let (in the sense of Bochner, thus…”