Approximation methods for hybrid diffusion systems with state-dependent switching processes : numerical algorithms and existence and uniqueness of solutions

By focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. Different from the existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing stepsize algorithms together with their weak convergence are given. Numerical experiments are also provided for demonstration

Stability And Controls For Stochastic Dynamic Systems

This dissertation focuses on stability analysis and optimal controls for stochastic dynamic systems. It encompasses two parts. One part of our work gives an in-depth study of stability of linear jump diffusion, linear Markovian jump diffusion, multi-dimensional jump diffusion and regime-switching jump diffusion together with the associated numerical solutions. The other part of our work is controls for stochastic dynamic systems, to be specific, we concentrated on mean variance types of control under different formulations. We obtained the nearly optimal mean-variance controls under both two-time-scale and hidden Markov chain formulations and convergence for each case is achieved. In Chapter 2, stability analysis of benchmark linear scalar jump diffusion is studied first. We presented the conditions for exponential p stable and almost surely exponentially stable for SDE and that of numerical solutions. Note that due to the use of Poisson processes, using asymptotic expansions as in the usual approach of treating diffusion processes does not work any longer. Different from the existing treatments of Euler-Maurayama methods for solutions of stochastic differential equations, techniques from stochastic approximation is employed in our work. The similar analysis is carried out for Markov jump diffusion and multi-dimensional jump diffusion. Beside of these, we have a thorough study on regime switching jump diffusion in which asymptotic stability in the large and exponential p-stability are carried out. Connection between almost surely exponential stability and exponential p-stability is exploited. Necessary conditions for exponential p-stability are derived and criteria for asymptotic stability in distribution are provided. In Chapter 3 We work on the famous mean variance problem in which a switching process ( say, market regime) is embedded. We first use a two-time-scale formulation to treat the underlying systems, which is represented by usage of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal. In chapter 4, we revisited the mean variance control problem in which the switching process is a hidden Markov chain. Instead of having full knowledge of switching process, we assume a noisy observation of switching process corrupted by white noise is available, we focus on minimizing the variance subject to a mixed terminal expectation. Using the Wonham filter, we convert the partially observable system to a completely observable one first. Because closed form solutions are virtually impossible to obtain, our main effort is devoted to designing a numerical algorithm. Convergence of the algorithm is obtained

Hybrid Stochastic Systems: Numerical Methods, Limit Results, And Controls

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\u27s 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\u27s 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

Exponential convergence rate of ruin probabilities for level-dependent L\'evy-driven risk processes

We explicitly find the rate of exponential long-term convergence for the ruin probability in a level-dependent L\'evy-driven risk model, as time goes to infinity. Siegmund duality allows to reduce the pro blem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.Comment: 20 pages, 5 figure