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

Abstract

This work is concerned with the stability of regime-switching processes under the perturbation of the transition rate matrices. From the viewpoint of application, two kinds of perturbations are studied: the size of the transition rate matrix is fixed, and only the values of entries are perturbed; the values of entries and the size of the transition matrix are all perturbed. Moreover, both regular and irregular coefficients of the underlying system are investigated, which clarifies the impact of the regularity of the coefficients on the stability of the underlying system.

Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes

“…In Section 3, we also first provide the construction of the coupling process ( t , ˜ t ) in (1.1), the key point of which is the construction of intervals used in Skorokhod's representation theorem. Then we provide an estimate of (1.1) which improves the one given in [23].…”

“…Moreover, it was also shown in [23] that the quantity (t, , ˜ ) plays a crucial role in study of the stability of RSPs under perturbation of the Q-matrix. However, the estimates in [23,24] do not work for switching processes in an infinite state space, which needs to be generalized from the viewpoint of applications. Also, the estimation of (t, , ˜ ) develops the classical perturbation theory for Markov chains (cf.…”

“…Our improvements in the previous two concerned problems are all based on a new observation on Skorokhod's representation theorem for jumping processes. The approach of using Skorokhod's representation theorem to study RSPs has been widely used in the literature; see, e.g., [8,17,18,23,28], etc. The basic idea of Skorokhod's representation theorem is to represent the switching process in terms of an integral with respect to a Poisson random measure based on a sequence of constructed intervals on the real line.…”

Positive recurrence of a solution of an SDE with variable switching intensities

“…We are looking for sufficient conditions for positive recurrence of the strong Markov process (X t , Z t ). Such a problem was considered in [2] for the exponentially recurrent case; for other references see [1], [5], [8], [11], and the references therein. Under weak ergodic and transient conditions the setting was earlier investigated in [15] for the case of the constant intensities λ 0 , λ 1 (i.e., not depending on x).…”

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

“…The contemporary state of affairs in this field and new applied problems related to the link between convergence rate and perturbation bounds in the 'uniform' case were described in [34]. In some recent papers uniform perturbation bounds of homogeneous Markov chains were studied by the techniques of stochastic differential equations, see for instance [42] and the references therein.…”