Asymptotic expansion of the transition density of the semigroup associated to a SDE driven by Lévy noise

Article type: Research Article

Authors: Smii, Boubaker^{; *}

Affiliations: King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, KFUPM Box 82, Dhahran 31261, Saudi Arabia. E-mail: boubaker@kfupm.edu.sa

Correspondence: [*] Corresponding author. E-mail: boubaker@kfupm.edu.sa.

Abstract: In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L(t)=Lt, t>0. The transition probability density pt, t>0 of the semigroup associated to the solution ut, t⩾0 of the SDE is given by a power series expansion. The series expansion of pt can be re-expressed in terms of Feynman graphs and rules. We will also prove that pt, t>0 has an asymptotic expansion in power of a parameter β>0, and it can be given by a convergent integral. A remark on some applications will be given in this work.

Keywords: Stochastic differential equations, transition probability densities, Lévy processes, Feynman graphs and rules, Borel summability, neural networks

DOI: 10.3233/ASY-201640

Journal: Asymptotic Analysis, vol. 124, no. 1-2, pp. 51-68, 2021