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Kurtz, T.G. (1982) Representation and Approximation of Counting Processes. In: Fleming, W.H. and Gorostiza, L.G., Eds., Advances in Filtering and Optimal Stochastic Control, Vol. 42, Lecture Notes in Control and Information Sciences, Springer, Berlin, 177-191.

has been cited by the following article:

• TITLE: On the Stability of Stochastic Jump Kinetics

  AUTHORS: Stefan Engblom

  KEYWORDS: Nonlinear Stability, Perturbation, Continuous-Time Markov Chain, Jump Process, Uncertainty, Rate Equation

  JOURNAL NAME: Applied Mathematics, Vol.5 No.19, November 18, 2014

  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.