On the stability of stochastic jump kinetics

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.

💡 Research Summary

The paper addresses a fundamental gap in the mathematical analysis of stochastic reaction‑network models that are widely used in systems biology. While such models are often represented as jump stochastic differential equations (SDEs) driven by counting measures, the standard theoretical toolbox relies on global Lipschitz continuity and linear growth conditions that are rarely satisfied by realistic biochemical networks. The authors propose a new constructive framework that replaces these restrictive assumptions with conditions that are both “reasonable” (i.e., they hold for models of practical interest) and “explicit” (all constants can be computed directly from the model parameters).

The authors begin by formulating the dynamics of a reaction network as a jump SDE
( dX_t = \sum_{k=1}^m \nu_k , dN_k(t) ),
where each (N_k) is a state‑dependent counting process with intensity ( \lambda_k(X_{t-}) ). The intensity functions are allowed to be nonlinear (e.g., polynomial or log‑linear) and may grow faster than linearly with the state. By imposing a polynomial‑type bound on the product ( \lambda_k(x),|\nu_k| ) – namely, the existence of constants (C>0) and (p\ge 0) such that ( \lambda_k(x),|\nu_k| \le C(1+|x|^p) ) for all (x) – the authors guarantee that the total jump intensity remains controllable. Importantly, these constants are directly computable from the reaction rates and stoichiometries of the given network.

To prove existence and uniqueness, the paper departs from the classical Picard iteration and instead uses the compensated Poisson random measure representation. By defining a state‑dependent compensator ( \mu(dt,dz)=\Lambda(X_{t-}),dt,\delta_{\nu(z)}(dz) ) with ( \Lambda(x)=\sum_k \lambda_k(x) ), the SDE can be rewritten as a stochastic integral with respect to the compensated measure ( \tilde{\mu} ). Under the polynomial growth bound, the authors establish an (L^2) contraction property for the integral operator, allowing the application of Banach’s fixed‑point theorem to obtain a unique càdlàg solution that exists almost surely for all finite times.

Long‑time behavior is investigated through Lyapunov‑type functions of the form ( V(x)=1+|x|^q ) (with ( q\ge 2 )). By deriving a differential inequality for the expectation of (V), they obtain
( \frac{d}{dt}\mathbb{E}