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Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates

Moment bounds and geometric ergodicity of diffusions with random switching and unbounded... A diffusion with random switching is a Markov process that consists of a stochastic differential equation part $$X_t$$ X t and a continuous Markov jump process part $$Y_t$$ Y t . Such systems have a wide range of applications, where the transition rates of $$Y_t$$ Y t may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates

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References (43)

Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1186/s40687-016-0089-2
Publisher site
See Article on Publisher Site

Abstract

A diffusion with random switching is a Markov process that consists of a stochastic differential equation part $$X_t$$ X t and a continuous Markov jump process part $$Y_t$$ Y t . Such systems have a wide range of applications, where the transition rates of $$Y_t$$ Y t may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Nov 14, 2016

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