Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching

In this paper, we consider the ergodicity and transience of the Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the theory of [Formula: see text]-matrices, we give some necessary and sufficient conditions for ergodicity of the CIR interest rate model with Markov switching. Besides, we show that the transition semigroup converges to the stationary distribution at an exponential rate in the Wasserstein distance. Finally, two examples are presented to illustrate our theory.

Exponential ergodicity for non-Lipschitz multivalued stochastic differential equations with Lévy jumps

We study the exponential ergodicity of diffusions generated by a multivalued stochastic differential equation with Lévy jumps when the coefficients are non-Lipschitz continuous by proving that the transition semigroup is strongly Feller and irreducible, and that it admits a unique invariant measure. This is obtained through an [Formula: see text]-convergence result, Girsanov’s theorem, coupling method combined and a stopping argument.

Exponential ergodicity of CIR interest rate model with random switching

In this paper, we consider ergodicity of Cox–Ingersoll–Ross (CIR) interest rate model with random switching. First, we show that the CIR model with switching has a unique stationary distribution. Next, we prove that the transition semigroup for the CIR model with switching converges to the stationary distribution at an exponential rate in the Wasserstein distance. Moreover, under two particular cases, the explicit expressions for stationary distributions are presented. Finally, the central limit theorem for the CIR model with random switching is established.

ON THE TRANSITION SEMIGROUPS OF CENTRALLY LABELED RAUZY GRAPHS

Rauzy graphs of subshifts are endowed with an automaton structure. For Sturmian subshifts, it is shown that its transition semigroup is the syntactic semigroup of the language recognized by the automaton. An inverse limit of the partial semigroups of nonzero regular elements of their transition semigroups is described. If the subshift is minimal, then this inverse limit is isomorphic, as a partial semigroup, to the [Formula: see text]-class associated to it in the free pro-aperiodic semigroup.

HARNACK INEQUALITIES AND APPLICATIONS FOR MULTIVALUED STOCHASTIC EVOLUTION EQUATIONS

By the method of coupling and Girsanov transformation, Harnack inequalities (F.-Y. Wang, 1997) and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand triple. The concentration property of the invariant measure for the semigroup is investigated. As applications of Harnack inequalities, explicit upper bounds of the Lp-norm of the density, contractivity, compactness and entropy-cost inequality for the semigroup are also presented.

ANALYSIS OF THE STOCHASTIC FITZHUGH–NAGUMO SYSTEM

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup Pt; further, we identify its infinitesimal generator in the space L2 (H; ν).