Stochastic optimization problems in insurance

Stochastic modeling of the reserve surplus of an insurance business plays a critical role in the foundation of actuarial mathematics. With the recent development in insurance products, the insurance company often faces both the market risk coming from the investments in the equity market and the intrinsic risk coming from the claim liabilities. This Thesis establishes a general framework in which the risk reserve process is formulated as the solution of a stochastic differential equations with jumps. Using the martingale theory and Beyesian analysis, it gives the budget constraint in this new framework. Following von Neumann-Morgenstern preference structure and the duality method in finance, it reveals the equivalence between the solution of the utility optimization problem and the solution of one new type of FBSDEs. Using the approximation method and the viscosity method in the PDE theory, it solves one type of non-linear BSDEs with jumps and the associated PDIE arising from the utility optimization problem. Utilizing the embedding method and LQ control theory in the optimization theory, it addresses one type mean-variance optimization problem and describes the optimal solutions in analytic form with the help of one stochastic Riccati equation. In the end, this Thesis includes some counterexamples and one numerical illustration for the extended mean-variance problem