Access Type
Open Access Dissertation
Date of Award
January 2017
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
George Yin
Abstract
This dissertation is concerned with the so-called stochastic hybrid systems, which are
featured by the coexistence of continuous dynamics and discrete events and their interactions. Such systems have drawn much needed attentions in recent years. One of the main reasons is that such systems can be used to better reflect the reality for a wide range of applications in networked systems, communication systems, economic systems, cyber-physical systems, and biological and ecological systems, among others. Our main interest is centered around one class of such hybrid systems known as switching diffusions. In such a system, in addition to the driving force of a Brownian motion as in a stochastic system represented by a stochastic differential equation (SDE), there is an additional continuous-time switching process that models the environmental changes due to random events.
In the first part, we develops numerical schemes for stochastic differential equations with
Markovian switching (Markovian switching SDEs). By utilizing a special form of It^o's formula for switching SDEs and special structural of the jumps of the switching component we derived a new scheme to simulate switching SDEs in the spirit of Milstein's scheme for purely SDEs. We also develop a new approach to establish the convergence of the proposed algorithm that incorporates martingale methods, quadratic variations, and Markovian stopping times. Detailed and delicate analysis is carried out. Under suitable conditions which are natural extensions of the classical ones, the convergence of the algorithms is established. The rate of convergence is also ascertained.
The second part is concerned with a limit theorem for general stochastic differential
equations with Markovian regime switching. Given a sequence of stochastic regime switching systems where the discrete switching processes are independent of the state of the systems. The continuous-state component of these systems are governed by stochastic differential equations with driving processes that are continuous increasing processes and square integrable martingales. We establish the convergence of the sequence of systems to the one described by a state independent regime-switching diffusion process when the two driving processes converge to the usual time process and the Brownian motion in suitable sense.
The third part is concerned with controlled hybrid systems that are good approximations
to controlled switching diffusion processes. In lieu of a Brownian motion noise, we use a
wide-band noise formulation, which facilitates the treatment of non-Markovian models. The wide-band noise is one whose spectrum has band width wide enough. We work with a basic stationary mixing type process. On top of this wide-band noise process, we allow the system to be subject to random discrete event influence. The discrete event process is a continuous time Markov chain with a finite state space. Although the state space is finite, we assume that the state space is rather large and the Markov chain is irreducible. Using a two-time-scale formulation and assuming the Markov chain also subjects to fast variations, using weak convergence and singular perturbation test function method we first proved that the when controlled by nearly optimal and equilibrium controls, the state and the corresponding costs of the original systems would "converge" to those of controlled diffusions systems. Using the limit controlled dynamic system as a guidance, we construct controls for the original problem and show that the controls so constructed are near optimal and nearly equilibrium.
Recommended Citation
Hoang, Tuan A., "Hybrid Stochastic Systems: Numerical Methods, Limit Results, And Controls" (2017). Wayne State University Dissertations. 1708.
https://digitalcommons.wayne.edu/oa_dissertations/1708