On the stability of stochastic jump kinetics

TL;DR

This paper develops a constructive framework for analyzing stochastic jump models in Systems Biology, providing conditions for existence, uniqueness, and stability of solutions without relying on traditional restrictive assumptions.

Contribution

It introduces explicit, practical conditions for the well-posedness of jump SDEs, challenging standard Lipschitz and growth restrictions, and compares stochastic and deterministic dynamics.

Findings

Established conditions for solution existence and uniqueness.

Derived long-term estimates and perturbation limits.

Contrasted stochastic and deterministic dynamics.

Abstract

Motivated by the lack of a suitable constructive framework for analyzing popular stochastic models of Systems Biology, we devise conditions for existence and uniqueness of solutions to certain jump stochastic differential equations (SDEs). Working from simple examples we find reasonable and explicit assumptions on the driving coefficients for the SDE representation to make sense. By `reasonable' we mean that stronger assumptions generally do not hold for systems of practical interest. In particular, we argue against the traditional use of global Lipschitz conditions and certain common growth restrictions. By `explicit', finally, we like to highlight the fact that the various constants occurring among our assumptions all can be determined once the model is fixed. We show how basic long time estimates and some limit results for perturbations can be derived in this setting such that these can be contrasted with the corresponding estimates from deterministic dynamics. The main complication is that the natural path-wise representation is generated by a counting measure with an intensity that depends nonlinearly on the state.