This work focuses on a class of jump-diffusions with state-dependent switching. First, compared with the existing results in the literature, in our model, the characteristic measure is allowed to be σ-finite. The existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method. Then, the Feller property is proved by means of introducing auxiliary processes and by making use of Radon-Nikodym derivatives. Furthermore, the irreducibility and all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable x (the state variable corresponding to the jump-diffusion component) in a neighborhood of the infinity, and some examples are presented to illustrate the results.
Adaptive filtering algorithms are investigated when system models are subject to model structure errors and regressor signal perturbations. System models for practical applications are often approximations of high-order or nonlinear systems, introducing model structure uncertainties. Measurement and actuation errors cause signal perturbations, which in turn lead to uncertainties in regressors of adaptive filtering algorithms. Employing ordinary differential equation (ODE) methodologies, we show that convergence properties and estimation bias can be characterized by certain differential inclusions. Conditions to ensure algorithm convergence and bounds on estimation bias are derived. These findings yield better understanding of the robustness of adaptive algorithms against structural and signal uncertainties.
This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin.Compound Poisson processes with regime switching are used to model the surplus and the switching(a continuous-time controlled Markov chain) represents random environment and other economic conditions.Assuming the switching to be fast varying together with suitable conditions,it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain.Under simple conditions,the optimal policy of the limit dividend strategy is a threshold policy.Using the optimal policy of the limit system as a guide,feedback control for the original surplus is then developed.It is demonstrated that the constructed dividend policy is asymptotically optimal.
