On some stochastic control models and related free-boundary problems in insurance mathematics
Creators: name: Zhu, Shihao
Publisher: name: Universität Bielefeld
Year: 2024
Subjects: subject: 510

description: This thesis investigates stochastic control models arising in insurance mathematics. From a mathematical point of view, the models presented in Chapters 2-4 can be formulated as stochastic control-stopping problems, while the model considered in Chapter 5 is a singular stochastic control problem. One of the primary challenges lies in characterizing the free boundaries when solving above problems.<br><br>

Specifically, Chapter 2 proposes a dynamic framework for the joint determination of optimal consumption, portfolio choice and irreversible investment in healthcare, which extends the classical Merton problem by incorporating irreversible healthcare investment. Chapter 3 examines the optimal retirement time decision, optimal investment, and consumption strategies, in particular under an age-dependent mortality force. Chapter 4 explores consumption and investment decisions, when the agent is faced with an uncertain lifespan and stochastic labor income within a Black-Scholes market framework. A new aspect is that the time at which purchasing life insurance for bequest purposes can be chosen. Chapter 5 analyzes an optimal dividend problem in which a two-state regime-switching environment affects the dynamics of the company’s cash surplus and, as a novel feature, also the level of bankruptcy.<br><br>

To solve the considered stochastic control-stopping problems, a combination of the martingale duality method and of the free-boundary approach is employed. This involves transforming the original stochastic control-stopping problems into pure stopping problems, which can be then studied via optimal stopping theory. However, the optimal stopping problems exhibit distinct features that vary from Chapter 2 to Chapter 4. Specifically, Chapter 2 concerns a finite time-horizon, two-dimensional degenerate optimal stopping problem with interconnected dynamics and non-monotonic stopping boundary. On the other hand, Chapter 3 deals with a three-dimensional degenerate optimal stopping problem with finite time-horizon and interconnected dynamics, which however exhibits a monotonic stopping boundary. The main mathematical contribution of Chapters 2 and 3 is the complete characterization of the optimal control-stopping strategy, as well as the regularity of the problems’ value function and optimal stopping boundary. The latter is indeed shown to be Lipschitz continuous via employing purely probabilistic methods. Chapter 4 also addresses a two-dimensional optimal stopping problem, which, however, can be reduced to a one-dimensional optimal stopping problem, and then explicitly solved by using the classical “guess and verify” approach. Chapter 5 is devoted to the study of an intricate singular stochastic control problem with regime-switching. Via a “guess and verify” approach, the explicit solutions for the value function, the optimal control, and the corresponding free boundaries are provided. In addition, detailed numerical studies are performed in each chapter to explore the economic and actuarial implications of the models considered in this thesis.