Risk management in finance and insurance via stochastic optimization

Abstract

This thesis is concerned with the study of the risk-constrained portfolio selection problem arising from an ordinary investor and the insurer being an investor. We first consider the problem for an insurer who can invest her surplus into financial market. With value at risk (VaR) imposed as the dynamic risk constraint, the portfolio selection problem is considered with two objectives: the ruin probability minimization and wealth utility maximization. A closed-form solution is found by solving the associated Hamilton-Jacob-Bellman (HJB) equation for the first problem. By using the exponential utility function, we solve the second problem by transforming this stochastic optimal control problem into a deterministic optimal control one and using control parametrization method. Second, we consider the risk-constrained utility maximizing problem with a jump diffusion model and a regime switching model for an ordinary investor. Conditional value at risk (CVaR) and maximal value at risk (MVaR) are used as the risk constraint in the two models, respectively. The associated HJB equations are treated with numerical techniques.