Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching

Abstract

This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with regime-switching, which is applied to study the long term returns of interest rates. Then under the existence of the stationary distribution, we provide a criterion to justify whether its stationary distribution is heavy-tailed or not.

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

“…By the arbitrariness of γ and λ, letting first λ → +∞ then γ ↓ 0 in (2.14), we obtain that 15) and further that sup t>0 E|Z(t)| 2 < ∞ due to the positiveness of η α .…”

“…The monographs [19] and [37] provide good summaries of the recent progress in the study of state-independent and state-dependent RSDPs respectively. As shown in [11], [6] for the Ornstein-Uhlenbeck process with Markovian switching, and in [15] for the Cox-Ingeroll-Ross process with Markovian switching, the stationary distributions of the corresponding processes with switching could be heavy-tailed, but the stationary distributions of the processes without switching must be light-tailed. Therefore, the heavy-tailed empirical evidence promotes the application of models with regime-switching.…”

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

“…Then there are many new difficulties and phenomena appeared in the study of regime-switching processes. See, for instance, [3,12,16,24,31,32] and references therein on the stability of such system; [4,7,18,24,19,20] on the recurrence of such system; [8,2,10] on the heavy or light tail behavior of the invariant probability measure of such system. Besides, there are some literature on the regime-switching processes driven by Lévy processes [5,27,30,31].…”

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

“…Moreover, recent works have found more and more special characteristics of these models compared with those models without regime-switching. For instance, the invariant probability measures of Ornstein-Uhlenback processes and Cox-Ingersoll-Ross processes with regime-switching may be heavy tailed, whereas without regime-switching, their invariant probability measures must be light tailed; see, [2,9] and [11].…”