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A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. This chapter is devoted to the mathematical(More)
We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then(More)
1 Semimartingales Let W denote a standard Wiener process with W 0 = 0. For a variety of reasons, it is desirable to have a notion of an integral 1 0 H s dW s , where H is a stochastic process; or more generally an indefinite integral t 0 H s dW s , 0 ≤ t < ∞. If H is a process with continuous paths, an obvious way to define a stochastic integral is by a(More)
A stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have(More)
This paper is concerned with the numerical approximation of the expected value IE(g(X t)), where g is a suitable test function and X is the solution of a stochastic differential equation driven by a Lévy process Y. More precisely we consider an Euler scheme or an " approximate " Euler scheme with stepsize 1/n, giving rise to a simulable variable X n t , and(More)