Stochastic Control in Optimal Insurance and Investment with Regime Switching

Abstract

Motivated by the financial crisis of 2007-2009 and the increasing demand for portfolio and risk management, we study optimal insurance and investment problems with regime switching in this thesis. We incorporate an insurable risk into the classical consumption and investment framework and consider an investor who wants to select optimal consumption, investment and insurance policies in a regime switching economy. We allow not only the financial market but also the insurable risk to depend on the regime of the economy. The objective of the investor is to maximize his/her expected total discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absolute risk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimal consumption, investment and insurance problems when there is regime switching. Next we consider an insurer who wants to maximize his/her expected utility of terminal wealth by selecting optimal investment and risk control policies. The insurer’s risk is modeled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market. In the case of no regime switching in the economy, we apply the martingale approach to obtain optimal policies for HARA utility functions, constant absolute risk aversion (CARA) utility functions, and quadratic utility functions. When there is regime switching in the economy, we apply dynamic programming to derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Optimal investment and risk control policies are then obtained in explicit forms by solving the HJB equation. ii We provide economic analyses for all optimal control problems considered in this thesis. We study how optimal policies are affected by the economic conditions, the financial and insurance markets, and investor’s risk preference.