Invariant Measures and Euler--Maruyama's Approximations of State-Dependent Regime-Switching Diffusions

Abstract

Regime-switching processes contain two components: continuous component and discrete component, which can be used to describe a continuous dynamical system in a random environment. Such processes have many different properties than general diffusion processes, and much more difficulties are needed to be overcome due to the intensive interaction between continuous and discrete component. We give conditions for the existence and uniqueness of invariant measures for state-dependent regime-switching diffusion processes by constructing a new Markov chain to control the evolution of the state-dependent switching process. We also establish the strong convergence in the L 1 -norm of the Euler-Maruyama's approximation and estimate the order of error. A refined application of Skorokhod's representation of jumping processes plays a substantial role in this work.

Stability of regime-switching processes under perturbation of transition rate matrices

“…However, we do not find an explicit condition in terms of the difference between Q and Q, for example, Q− Q ℓ 1 used in this work at current stage. Our result shows once again the significant effect of Skorkhod's representation of continuous-time Markov chains which has been applied in [22] to deal with state-dependent regime-switching processes.…”

Stabilization of Regime-Switching Processes by Feedback Control Based on Discrete Time Observations II: State-Dependent Case

“…Next, employing the idea of Shao [23], we go to construct two auxiliary continuous-time Markov chains (Λ(t)) and (Λ * (t)) such that Λ * (t) ≤ Λ(t) ≤Λ(t), t ≥ 0, a.s., under some appropriate conditions. Our stochastic comparisons are based on Skorokhod's representation of (Λ(t)) in terms of the Poisson random measure by following the line of [26, Chapter II-2.1] or [32].…”

Euler-Maruyama Method for Regime Switching Stochastic Differential Equations with Hölder Coefficients

“…Numerical schemes for regime switching SDEs therefore have become an active area since the pioneer work by Yuan and Mao [31] with numerous results on various aspects [11,12,15,16,19,22,23,24,25,27,28,32,33]. See, for instance, [31] for Euler-Maruyama method, [24] for weak Euler-Maruyama method, [22] for tamed-Euler method, [23] for Milstein-type algorithm, [11,32] for stability of numerical approximations, [33] for stabilization of numerical solutions, [15] for approximation of invariant measures, [25,27] for numerical scheme for state-dependent switching systems, [28] for scheme for hybrid systems with jumps, [16] for approximation of delayed hybrid systems (see also [12]). However, most of the aforementioned works (except [19], to the best of our knowledge, which focuses on somewhat specific models) require the global or local Lipschitz conditions for the drift and diffusion coefficients despite of a vital fact that many models in reality violate these conditions.…”