This thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk... Show moreThis thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk management is made via developing dynamic risk measures and dynamic acceptability indices that can be characterized by distortion functions. In particular, we proved a representation theorem illustrating that the class of dynamic coherent risk measures generated by distortion functions coincides with a specific type of dynamic risk measures, the dynamic WV@R. We also investigate thoroughly various types of time consistencies for dynamic risk measures and dynamic acceptability indices in terms of distortion functions.
Another contribution to risk management is proving strong consistency and asymptotic normality of two estimators of dynamic WV@R. In contrast to the exist- ing literature, our results do not rely on the assumptions of distribution of random variables. Instead, we investigate the asymptotic normality of estimators in terms of the generating distortion functions. Last but not least, we give counterexample to show that a sufficient condition of asymptotic normality is not necessary.
The contribution to optimization is twofold. On the one hand, we formulate the (scalar) diversification optimization problem as a vector optimization problem (VOP), and show that a set-valued Bellman principle is satisfied by this VOP. On the other hand, we derive explicit policy gradient formula and implement the deep neural network to solve diversification optimization problem numerically. This deep learning technique allows to overcome computation difficulty caused by the non-convexity of VOP. Show less
