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

TL;DR: In this work, decreasing-stepsize algorithms together with their weak convergence are given and the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained.

Abstract: 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 demonstr...