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Hybrid systems, and Piecewise Deterministic Markov Processes in particular,are widely used to model and numerically study systems exhibiting multiple timescales in biochemical reaction kinetics and related areas. In this paper analmost sure convergence analysis for numerical simulation algorithms forPiecewise Deterministic Markov Processes is presented. The discussed numericalmethods arise through discretisina a constructive method defining theseprocesses. The stochastic problem of simulating the random, path-dependent jumptimes of such processes is reformulated as a hitting time problem for a systemof ordinary differential equations with random threshold. Then deterministiccontinuous methods (methods with dense output) are serially employed to solvethese problems numerically. We show that the almost sure asymptotic convergencerate of the stochastic algorithm is identical to the order of the embeddeddeterministic method. We illustrate our theoretical findings by numericalexamples from mathematical neuroscience were Piecewise Deterministic MarkovProcesses are used as biophysically accurate stochastic models of neuronalmembranes.

Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes