The Application Of Backward Stochastic Differential Equations And Malliavin 's Calculation In Insurance Investment

Recently, optimal investment in financial market for insurers is a very active area in insurance mathematics, because it has very important theoretical and realistic value to combine insurance and finance studies. Due to the particularity of the insurers, they often take reinsurance to transfer a part of the claim risk to the reinsurer, so as to reduce their total risk. The most studied types of reinsurance include proportional insurance and excess of loss reinsurance. There have already been many researchers on optimiza-tion of investment and proportional reinsurance in various aspects. The usually adopted method is HJB equation method. Certainly, there are also some other methods, such as martingale duality method, complete square method especially for quadratic control problems, and so on. This dissertation studies four new problems about investment and proportional reinsurance. The methods used here are very different from existing literature, which are mainly related to backward stochastic differential equations and Malliavin calculus. Detailed works consist of the following items:(1) A problem of optimal investment, consumption and proportional reinsurance for an insurer with option payoff is studied, where the criterion of optimization is max-imizing the utility of terminal wealth. When investment and proportional reinsurance are subjected to general constraint conditions, not confined to convex case, we obtain the expressions for optimal strategies by using techniques of quadratic growth back-ward stochastic differential equation and BMO martingale theory based on martingale optimization principle. In particular, when there is no constraint on investment, and reinsurance proportion takes nonnegative values, we get relatively detailed expressions for optimal strategies by using Malliavin calculus.(2) A problem of optimal investment and proportional reinsurance with hidden Markovian regime switching for an insurer under the criterion of risk minimization is considered. Convex risk measure is adopted as the method of risk valuation. The filtering theory is used to transform the problem into one with complete observations. Then, by using the representation theorem for convex risk measure, we can transform the problem into a two persons zero sum stochastic differential game, where the comparison theorem plays a key role. (3) A problem of optimal investment and proportional reinsurance under partial information about financial assets and claims processes is studied, where maximization of the utility of the terminal wealth is adopted as the criterion. In order to conform to reality, the financial risk assets process and claim process are both assumed to be jump diffusion processes. By using Malliavin calculus for Levy process, we give the characterization of optimal investment and proportional reinsurance and solve it in detail in some particular cases.(4) A problem of optimal investment, consumption and proportional reinsurance under model uncertainty is studied. The optimization criterion is to maximize the utility, where the utility function is assumed to be in a general form, not limited to exponential utility function. First, we can transform the problem into a two persons zero sum forward-backward stochastic differential game. Then we use stochastic maximum principle to solve this problem. In some particular cases, Malliavin calculus is used to get the concrete expressions of the optimal strategies.