Approximation Methods for Hybrid Diffusion Systems with State-Dependent Switching Processes: Numerical Algorithms and Existence and Uniqueness of Solutions

Focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state-dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. In contrast to existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing-stepsize algorithms together with their weak convergence are given. Numerical experiments are also provided for demonstration.