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Date of Award
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.
Jin, Zhuo, "Numerical methods for problems arising in risk management and insurance" (2011). Wayne State University Dissertations. Paper 281.