Asymptotic expansion of a nonlinear oscillator with a jump-diffusion process

Abstract

We introduce an asymptotic expansion of the transition density of a nonlinear oscillator involving a jump-diffusion process. We approximate the transition density by expanding the characteristic function of the solution to the nonlinear oscillator with respect to time and present a numerical verification of the asymptotic expansion of the transition density. This study provides us with a new mathematical framework for analyzing the dynamics of stochastic mathematical neuronal models using a jump–diffusion process.