Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients

TLDR: In this paper, the authors used the backward parametrix method to prove the existence and regularity of the transition density associated to the solution process of a stable-like driven stochastic differential equation with Holder continuous coefficients.

Abstract: In this article, we use the backward parametrix method in order to prove the existence and regularity of the the transition density associated to the solution process of a stable-like driven stochastic differential equation (SDE) with Holder continuous coefficients. The method of proof uses the parametrix method on the Gaussian component of a subordinated Brownian motion. This analysis which can be generalized also provides a stochastic representation of the density which is potentially useful for other applications.Abbrevations: B: Brownian motion; V: α-stable-like subordinator independent of B; μ: Levy measure of the subordinator V; m(·): positive concave increasing function; ; δy(dx): Dirac measure with unit mass at ; ψ: Levy exponent of Z; q(M, x): Gaussian density with covariance matrix M and ; ϕ: a regular varying function; b: drift coefficient of X; σ: coefficient of associated with the driving Levy process Z ≔ BV; ζ: coefficient associated with the diffusion (if X is a jump diffusion proce...