In response to the increasing needs for control and optimization of hybrid systems, this work is concerned with such asymptotic properties as recurrence (also known as weak stochastic stability in the literature) and ergodicity of regime-switching diffusions. Using Liapunov functions, necessary and sufficient conditions for positive recurrence are developed. Then, ergodicity of positive recurrent regime-switching diffusions is obtained by constructing cycles using the associated discrete-time Markov chains.

Asymptotic Properties of Hybrid Diffusion Systems

Stabilization and destabilization of hybrid systems of stochastic differential equations

We study a two-time-scale system of jump-diffusion stochastic differential equations. The main goal is to study the convergence rate of the slow components to the effective dynamics. The convergenc...

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Strong Convergence Rate for Two-Time-Scale Jump-Diffusion Stochastic Differential Systems

Focusing on two-component Markov processes having continuous dynamics and discrete events, this work develops asymptotic properties of regime-switching diffusions. First, strong Feller property is established. Next, classifying the underlying processes as recurrence (positive recurrence or null recurrence) or transience, sufficient conditions for such processes are obtained. In addition to providing general criteria for recurrent and transient switching diffusions, for processes that are linearizable with respect to the continuous component, easily verifiable conditions are established. Also provided are conditions for null-recurrent processes. Furthermore, conditions for controlled switching diffusions in which the feedback controls ensure weak stability (or ergodicity) are given. Finally, simple examples are used for demonstration.

On Strong Feller, Recurrence, and Weak Stabilization of Regime-Switching Diffusions

A class of hybrid jump diffusions modulated by a Markov
chain is considered in this work. The motivation stems from
insurance risk models, and emerging applications in production
planning and wireless communications. The models are hybrid in
that they involve both continuous dynamics and discrete events.
Under suitable conditions, asymptotic expansions of the
transition densities for the underlying processes are developed.
The formal expansions are validated and the error bounds
obtained.

Asymptotic Expansions of Transition Densities for Hybrid Jump-diffusions

Asymptotic Expansions of Solutions of Integro- Differential Equations for Transition Densities of Singularly Perturbed Switching Diffusions: Rapid Switchings

A.M Il'in

Papers

Asymptotic Expansions of Solutions of Integro- Differential Equations for Transition Densities of Singularly Perturbed Switching Diffusions: Rapid Switchings