A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions

TL;DR

This paper introduces a new way to characterize the jump rate of piecewise-deterministic Markov processes with discrete transitions and proposes a nonparametric estimator with proven uniform convergence.

Contribution

It provides a novel characterization of the jump rate and a nonparametric estimation method with convergence guarantees for these processes.

Findings

New characterization of jump rate

Nonparametric estimator with uniform convergence

Illustration on a numerical example

Abstract

Piecewise-deterministic Markov processes form a general class of non-diffusion stochastic models that involve both deterministic trajectories and random jumps at random times. In this paper, we state a new characterization of the jump rate of such a process with discrete transitions. We deduce from this result a nonparametric technique for estimating this feature of interest. We state the uniform convergence in probability of the estimator. The methodology is illustrated on a numerical example.