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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)

Models of populations in which a type or location, represented by a point in a metric space E, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measure-valued processes are obtained as infinite population limits for a large… (More)

- David F Anderson, Gheorghe Craciun, Thomas G Kurtz
- Bulletin of mathematical biology
- 2010

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)

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)

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)

We perform an error analysis for numerical approximation methods of continuous time Markov chain models commonly found in the chemistry and biochemistry literature. The motivation for the analysis is to be able to compare the accuracy of different approximation methods and, specifically, Euler tau-leaping and midpoint tau-leaping. We perform our analysis… (More)

- Jin Feng, Thomas G. Kurtz
- 2000

- Thomas G. Kurtz
- 1998

Let X be a Markov process with generator A and let Y (t) = γ(X(t)). The conditional distribution π t of X(t) given σ(Y (s) : s ≤ t) is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of… (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)

We give complete proofs of the theorem of convergence of types and the Kesten-Stigum theorem for multi-type branching processes. Very little analysis is used beyond the strong law of large numbers and some basic measure theory. Consider a multi-type Galton-Watson branching process with J types. Let L (i,j) be a random variable representing the number of… (More)