Prescription-induced jump distributions in multiplicative Poisson processes

TL;DR

This paper investigates how different stochastic calculus prescriptions arise in multiplicative Poisson noise models, especially in ecological applications, and proposes a method to determine the most suitable prescription based on data.

Contribution

It introduces a framework linking inertial effects to prescription choices in multiplicative Poisson processes and applies it to soil salinization modeling.

Findings

Inertial terms induce natural emergence of Ito and Stratonovich prescriptions.

Linear multiplicative noise allows transformation between prescriptions.

A data-driven method to select the appropriate stochastic prescription is proposed.

Abstract

Generalized Langevin equations (GLE) with multiplicative white Poisson noise pose the usual prescription dilemma leading to different evolution equations (master equations) for the probability distribution. Contrary to the case of multiplicative gaussian white noise, the Stratonovich prescription does not correspond to the well known mid-point (or any other intermediate) prescription. By introducing an inertial term in the GLE we show that the Ito and Stratonovich prescriptions naturally arise depending on two time scales, the one induced by the inertial term and the other determined by the jump event. We also show that when the multiplicative noise is linear in the random variable one prescription can be made equivalent to the other by a suitable transformation in the jump probability distribution. We apply these results to a recently proposed stochastic model describing the dynamics of primary soil salinization, in which the salt mass balance within the soil root zone requires the analysis of different prescriptions arising from the resulting stochastic differential equation forced by multiplicative white Poisson noise whose features are tailored to the characters of the daily precipitation. A method is finally suggested to infer the most appropriate prescription from the data.