Open Access Dissertation
Date of Award
Recently, a class of dynamic systems called ``hybrid systems" containing both continuous dynamics and discrete events has been adapted to treat a wide variety of situations arising in many real-world situations. Motivated by such development, this dissertation is devoted to the study of dynamical systems involving a Markov chain as the randomly switching process. The systems studied include hybrid competitive Lotka-Volterra ecosystems and non-zero-sum stochastic differential games between two insurance companies with regime-switching.
The first part is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective is to reduce the computational complexity by using the two-time-scale formulation. Because the existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this work that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex original system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods.
The second part develops an approximation procedure for a class of non-zero-sum stochastic differential games for investment and reinsurance between two insurance companies. Both proportional reinsurance and excess-of-loss reinsurance policies are considered. We develop numerical algorithms to obtain the approximation to the Nash equilibrium by adopting the Markov chain approximation methodology. We establish the convergence of the approximation sequences and the approximation to the value functions. Numerical examples are presented to illustrate the applicability of the algorithms.
Bui, Trang Thi-Huyen, "Switching Diffusions: Applications To Ecological Models, And Numerical Methods For Games In Insurance" (2019). Wayne State University Dissertations. 2235.