Marcus versus Stratonovich for Systems with Jump Noise

TL;DR

This paper compares Itô, Stratonovich, and Marcus stochastic calculus prescriptions for systems driven by jump noise, highlighting their differences and suitability for various physical models involving Poisson and Levy processes.

Contribution

It provides an in-depth comparison of the three stochastic integral prescriptions for jump noise systems using concrete examples, clarifying their differences and applications.

Findings

Marcus calculus preserves key properties for jump processes.

Solutions differ significantly between prescriptions for the same system.

The paper clarifies when to use each stochastic calculus in physical models.

Abstract

The famous It\^o-Stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise. In physics literature, this dilemma is often resolved in favour of the Stratonovich prescription because of its two characteristic properties valid for systems driven by Brownian motion: (i) it allows physicists to treat stochastic integrals in the same way as conventional integrals, and (ii) it appears naturally as a result of a small correlation time limit procedure. On the other hand, the Marcus prescription [IEEE Trans. Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to retain (i) and (ii) for systems driven by a Poisson process, L\'evy flights or more general jump processes. In present communication we present an in-depth comparison of the It\^o, Stratonovich, and Marcus equations for systems with multiplicative jump noise. By the examples of areal-valued linear system and a complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we compare solutions obtained with the three prescriptions.