[Paper Review] Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions

Jinghai Shao|arXiv (Cornell University)|Oct 25, 2017

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper establishes conditions for the existence and uniqueness of invariant measures in state-dependent regime-switching diffusions by constructing a controlling Markov chain, and proves the strong $L^1$-norm convergence of the Euler-Maruyama approximation with an error order of $O(\delta^{1/2})$. The analysis relies on refined Skorokhod representation and coupling techniques to handle the complex interaction between continuous and discrete components in the switching process.

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.

Motivation & Objective

• To establish sufficient conditions for the existence and uniqueness of invariant measures in state-dependent regime-switching diffusion processes (RSDPs).
• To analyze the convergence properties of the Euler-Maruyama scheme for RSDPs with state-dependent switching.
• To overcome the challenges posed by the intense interaction between the continuous diffusion component and the discrete Markovian switching component.
• To extend existing results on invariant measures and numerical approximations from state-independent to state-dependent RSDPs.

Proposed method

• Constructs a new Markov chain to control the evolution of the state-dependent switching process and analyze its long-term behavior.
• Applies Skorokhod’s representation of jumping processes to handle the discontinuous dynamics of the regime-switching component.
• Uses successful coupling techniques to compare paths of the true process and its Euler-Maruyama approximation.
• Employs Gronwall’s inequality in conjunction with moment estimates to bound the $L^1$-norm difference between the true and approximate processes.
• Imposes conditions (Q1)-(Q3), (A1)-(A2), (H1)-(H2) on drift, diffusion, and switching rates to ensure regularity and stability.
• Derives error bounds by decomposing the difference into components involving state, regime, and time-discretization errors.

Experimental results

Research questions

• RQ1 Under what conditions does a state-dependent regime-switching diffusion process admit a unique invariant measure?
• RQ2 What is the rate of convergence of the Euler-Maruyama scheme for such processes in the $L^1$-norm?
• RQ3 How can the interaction between the continuous diffusion and discrete switching components be effectively controlled to ensure ergodicity and numerical stability?
• RQ4 Can the coupling method be refined to handle pathwise differences in processes with state-dependent switching rates?
• RQ5 What role does Skorokhod’s representation play in analyzing the convergence of the approximation scheme?

Key findings

• The existence and uniqueness of an invariant measure for state-dependent RSDPs are established under suitable Foster-Lyapunov type conditions and the constructed controlling Markov chain.
• The Euler-Maruyama approximation converges to the true process in the $L^1$-norm with an error bound of order $O(\delta^{1/2})$, where $\delta$ is the time step.
• The convergence rate is derived via a pathwise coupling argument and moment estimates, leveraging the boundedness of drift and diffusion coefficients.
• The analysis shows that the probability of regime mismatch between the true and approximate processes decays as $O(\delta^{1/2})$ over finite time horizons.
• The method successfully handles the non-Markovian nature of the switching process by controlling the switching intensity through the state-dependent transition rates.
• The results extend the theoretical foundation of numerical methods for RSDPs beyond the state-independent case, enabling broader application in finance, biology, and engineering.