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

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