Density estimates for jump diffusion processes

TL;DR

This paper derives upper and lower bounds for the density of a jump diffusion process driven by a Poisson process, with specific focus on Gaussian and Laplacian jump amplitudes, extending results to multiple dimensions.

Contribution

It provides new density estimates for jump diffusion processes with Gaussian or Laplacian jumps, including multidimensional cases, using convolution and Malliavin calculus techniques.

Findings

Established upper and lower density bounds for jump diffusion processes.

Extended density bounds to multidimensional jump processes.

Applied Malliavin calculus for tail estimates of the process.

Abstract

We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.