Explicit computations for some Markov modulated counting processes

In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the `state of the economy', to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies. Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor

Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation

This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain with two components. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of the numerical methods.Comment: Key words: Singular control, dividend policy, Markov chain approximation, numerical method, reinsurance, regime switchin

Numerical methods for problems arising in risk management and insurance

In this dissertation we investigate numerical methods for problems annuity purchasing and dividend optimization arising in risk management and insurance. We consider the models with Markov regime-switching process. The regime-switching model contains both continuous and discrete components in their evolution and is referred to as a hybrid system. The discrete events are used to model the random factors that cannot formulated by differential equations. The switching process between regimes is modulated as a finite state Markov chain. As is widely recognized, this regime-switching model appears to be more versatile and more realistic. However, because of the regime switching and the nonlinearity, it is virtually impossible to obtain closed-form or analytic solutions for our problems. Thus we are seeking numerical solutions by using Markov chain approximation methods. Focusing on numerical solutions of the regime-switching models in the area of actuarial science, and based on the theory of weak convergence of probability measures, the convergence of the approximating sequences is obtained. In fact, under very broad conditions, we prove that the sequences of approximating Markov chain, the cost functions, and the value functions all converge to that of the underlying original processes. The proofs are purely probabilistic. It need not appeal to regularity properties of or even explicitly use the Bellman equation. Moreover, the feasibility of regime-switching model and Markov chain approximation method are illustrated by the examples