Numerical Simulation of Stochastic Reaction Networks modelled by Piecewise Deterministic Markov Processes
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Hybrid systems, and Piecewise Deterministic Markov Processes in particular, are widely used to model and numerically study multiscale systems in biochemical reaction kinetics and related areas where I put an emphasise on models arising in mathematical neuroscience. I will present an almost sure convergence analysis for numerical simulation algorithms for Piecewise Deterministic Markov Processes. These algorithm are built by appropriately discretising constructive methods defining these processes. The stochastic problem of simulating the random, path-dependent jump times is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold which is solved using continuous approximation methods. In particular show that the almost sure asymptotic convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. Finally, we present an extension of our results to more general Piecewise Deterministic Processes.