Conditions for the existence and smoothness of the distribution density of the Ornstein–Uhlenbeck process with Lévy noise

Abstract

Abstract. Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with Lévy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.

Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with lévy noise

“…The present paper is a sequel of [1]. In these two papers, we study local properties of the distributions of Ornstein-Uhlenbeck processes with Lévy noise, i.e., of the solutions of linear stochastic differential equations of the form…”

“…Structurally, these equations belong to the simplest class of stochastic differential equations with Lévy noise. The main aim of our investigation is to establish necessary and sufficient conditions for the existence and/or smoothness of the density of transition probability for the Markov processes specified by stochastic differential equations with Lévy noise by using the indicated important class of processes as an example (for details, see [1]). One-dimensional Ornstein-Uhlenbeck processes, i.e., the solutions of (1) with m D 1; are completely investigated in [1].…”

“…The main aim of our investigation is to establish necessary and sufficient conditions for the existence and/or smoothness of the density of transition probability for the Markov processes specified by stochastic differential equations with Lévy noise by using the indicated important class of processes as an example (for details, see [1]). One-dimensional Ornstein-Uhlenbeck processes, i.e., the solutions of (1) with m D 1; are completely investigated in [1]. The following two facts of principal importance for subsequent investigations were established: First, it was shown that, in the presence of a nontrivial transfer coefficient A; we observe an effect of "regularization" which can be described as follows:…”

“…There are several known sufficient conditions, such as the Sato or Kallenberg conditions (assertions 1 and 2 of Proposition 1 in [1], respectively). However, at present, there are no general criteria establishing the existence and/or smoothness of density in terms of efficiently verified conditions imposed on the measure of the Lévy process.…”

“…However, at present, there are no general criteria establishing the existence and/or smoothness of density in terms of efficiently verified conditions imposed on the measure of the Lévy process. At the same time, for A 6 D 0; these criteria are available, at least in the one-dimensional case (see [1], Proposition 2 and Theorem 1). Second, the problems of existence of density and its regularity are essentially different and should be investigated separately.…”