Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients. Issue 6 (1st November 2016)

• Record Type: Journal Article
• Authors: Kohatsu-Higa, Arturo; Li, Libo
• Is Part Of: Stochastic analysis and applications. Volume 34:Issue 6(2016)
• Page Start: 979
• Page End: 1024
• Publication Date: 2016-11-01
• Subjects: SDEs with jumps -- Lévy process -- parametrix method -- stochastic representation; Primary: 60J35 -- Secondary: 60J75 -- 60J22; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2016.1198706
• Languages: English
• ISSNs: 0736-2994

Abstract

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 Hölder 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: Lévy measure of the subordinator V ; m (·): positive concave increasing function; ; δ y ( dx ): Dirac measure with unit mass at ; ψ: Lévy 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 Lévy process Z ≔ B V ; ζ: coefficient associated with the diffusion (if X is a jump diffusion process); : generator of process X, solution of the stochastic differential equation (1 ); p t ( x, y ): density of the process X ; : generator of the parametrix or frozen process ; : density of the parametrix ; k : Hölder exponential of a ≔ σσ T ; and : uniform upper and lower bounds for the eigenvalues of a (·); k ′: Hölder exponential of e ≔ ζζ T ; η and (η i ) i : auxiliary Bernoulli used to link the drift part and the jump part; and : auxiliary Bernoulli used to link the small and large jumps; : ordered jump times of an Poisson process;, where w ∈ (α, 1∧(α + λ)) and λ > 0; ; ; x + ≔ x ∨0 for ; ; ; I n t ( y, x ) ≔ ∫ t 0 …∫ t n − 1 0 dtn … dt 1 K ( tn, …, t 1, t ; x, y ); ; ; ; ; ; ; ;