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

Abstract

In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise $\mathit{L}\mathrm{(}\mathit{t}\mathrm{)}={\mathit{L}}_{\mathit{t}}$ , $\mathit{t}\mathrm{>}0$ . The transition probability density ${\mathit{p}}_{\mathit{t}}$ , $\mathit{t}\mathrm{>}0$ of the semigroup associated to the solution ${\mathit{u}}_{\mathit{t}}$ , $\mathit{t}⩾0$ of the SDE is given by a power series expansion. The series expansion of ${\mathit{p}}_{\mathit{t}}$ can be re-expressed in terms of Feynman graphs and rules. We will also prove that ${\mathit{p}}_{\mathit{t}}$ , $\mathit{t}\mathrm{>}0$ has an asymptotic expansion in power of a parameter $\mathit{\beta }\mathrm{>}0$ , and it can be given by a convergent integral.

A remark on some applications will be given in this work.