On the Stability of Stochastic Jump Kinetics

Stefan Engblom

February 2012

Abstract:

Motivated by the lack of a suitable framework for analyzing popular stochastic models of Systems Biology, we devise conditions for existence and uniqueness of solutions to certain jump stochastic differential equations (jump 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 perturbation results can be derived in this setting such that these can readily be compared 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.

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