Convergence Analysis of Simulation Methods for Hybrid Stochastic Systems
Sprache des Vortragstitels:
RICAM Workshop on Numerical Analysis of Multiscale Problems & Stochastic Modelling
Sprache des Tagungstitel:
Hybrid stochastic systems are widely used to model and numerically study systems in biochemical reaction kinetics and related areas such as, e.g., single neuron models. Hybrid stochastic processes combine continuous time evolution with discontinuous stochastic evolution. For example, these processes arise naturally from microscopic Markov chain models which exhibit a time-scale separation when the fast time scale is approximated by a continuous variable either stochastic (General Stochastic Hybrid Systems) or deterministic (Piecewise Deterministic Processes). We present a class of simulation algorithms based on continuous Runge-Kutta methods and analyze the pathwise convergence behaviour. The methods rely on the fact that the problem of simulating the random, path-dependent jump times can be reformulated into a hitting time problem for a system of ordinary differential equations with random treshold. We find that under suitable conditions on the defining properties of the stochastic processes the deterministic order of convergence is conserved. We further present some preliminary results extending the pathwise convergence to weak convergence which is the appropriate convergence concept for the study of Monte-Carlo algorithms.