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

2013/10/01 by Martin Riedler, Riedler, Martin G., Girolama Notarangelo +1
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Numerical Analysis (math.NA) #Probability (math.PR) #Simulation Techniques and Applications #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1310.0392

openalex publication_date 2013/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Θ--Maruyama methods is analysed in detail where the local error is analysed in terms of Itô-Taylor expansions of the exact solution and the approximation process. The study is concluded with numerical experiments that illustrate the theoretical findings.

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