Stochastic control problems and related free boundaries in mathematical finance and insurance

Abstract

This thesis deals with free boundary problems arising from stochastic control models in the context of investment portfolios and insurance. It consists of three parts. The first part is devoted to an optimal dynamic reinsurance and dividend-payout problem for an insurer who has to manage her risk exposure in order to maintain the solvency capital requirement and business viability. We aim to determine the optimal dividend and reinsurance policies with the objective of maximizing the expected cumulative discounted dividend payout until either bankruptcy or a given maturity time comes. Mathematically, it is a combined classical-singular control issue. The corresponding Hamilton-Jacobi-Bellman equation is a variational inequality with a fully nonlinear operator and a gradient constraint. Using a standard penalty approximation method and the comparison principle for its gradient function, we can prove the existence and uniqueness of a C2,1 solution to the variational inequality. We find that a risk and time dependent reinsurance barrier and a time dependent dividend-payout barrier can partition the surplus-time space into three non-overlapping zones. The localities of the regions are also explicitly estimated. The second part is concerned with an infinite-time optimal stopping problem where a mutual fund manager wants to maximize the expected utility of her capital with option compensation at the stopping time. We formally derive an explicit solution of the value function using the Legendre transformation approach and determine the optimal investing strategy as well as the ideal selling price. Furthermore, we give numerical examples to illustrate our results. The third part investigates an exit strategy problem in a finite-time horizon. We assume that a portfolio manager invests on behalf of small investors with an expected utility maximization investment schedule, while each investor has the option of redeeming her position before maturity. We seek to identify an appropriate exit point for an investor so as to minimize the expected relative error between her redeeming worth and the high-water mark over a given period. The problem can be also formulated as a stopping problem with a variational inequality. We employ partial differential equation techniques to solve it and show that an investor will either reject the manager's wealth management program from the start or hold the position until the end.