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Abstract
The stochastic optimal decision-making problem concerns the process of dynamically deciding actions to optimize pre-specified criteria based on specific stochastic models. It is, however, common that a decision-maker is unable to obtain complete information to formulate fully reliable models and faces the issue of model uncertainty. Existing empirical studies have shown that ignoring model uncertainty leads to improper decisions and causes losses in the financial market. Thus, it is important to incorporate model uncertainty into decision-making. To our best knowledge, no existing works on dividend optimization have taken model uncertainty into consideration. This thesis is an early attempt to fill such a gap in the actuarial literature.
This thesis studies three popular optimization problems in the framework of model uncertainty, which involve different models with multiple control variables and various assumptions. It consists of three projects. The first project investigates an optimal risk exposure-dividend control problem under a diffusion model with model uncertainty. Due to the concerns about model uncertainty, the ambiguity averse insurer aims at finding the robust strategies such that a penalized reward function is maximized in the worst-case scenario. The problem is formulated as a zero-sum stochastic differential game between the insurer and the market. Explicit expressions for the value functions are obtained and the optimal dividend strategies are identified as barrier strategies. The second project incorporates model uncertainty into a dividend optimization problem of a singular type under the classical risk model with general assumptions on the claim size distribution. Using the standard stochastic control techniques, we characterize the value function as the smallest viscosity supersolution to the existing Hamilton-Jacobi-Bellman equation and show that the optimal strategies are of band type. The third project extends the second project by incorporating fixed and proportional transaction costs on dividend payments. The problem is an impulse control problem and the optimal dividend strategies are shown to be n-level lump sum strategies. Numerical studies are provided for each project and the economic implications of model uncertainty on insurer’s decision-making are discussed. It is shown that the insurer who is more averse to ambiguity tends to be more conservative in the optimal strategies.
