Strong Error Analysis of the $Θ$-Method for Stochastic Hybrid Systems

We discuss numerical approximation methods for Random Time Change equations which possess a deterministic drift part and jump with state-dependent rates. It is first established that solutions to such equations are versions of certain Piecewise Deterministic Markov Processes. Then we present a convergence theorem establishing strong convergence (convergence in the mean) for semi-implicit Maruyama-type one step methods based on a local error analysis. The family of $\Theta$--Maruyama methods is analysed in detail where the local error is analysed in terms of It{\^o}-Taylor expansions of the exact solution and the approximation process. The study is concluded with numerical experiments that illustrate the theoretical findings.