Some optimal control problems in financial and actuarial mathematics

Ng, Kenneth

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https://hdl.handle.net/2142/124658 Copy

Description

Title

Some optimal control problems in financial and actuarial mathematics

Author(s)

Ng, Kenneth

Issue Date

2024-04-20

Director of Research (if dissertation) or Advisor (if thesis)

Chong, Wing Fung

Doctoral Committee Chair(s)

Wei, Wei

Committee Member(s)

• Song, Renming
• Jing, Xiaochen

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Stochastic Control
• Actuarial Science
• Mean Field Games
• Variable Annuities, Forward Utility Preferences
• Optimal Stopping

Abstract

This thesis encompasses four stochastic control problems in actuarial science and financial mathematics. Chapter 1 provides an overview of the thesis and relevant topics that would be covered in the subsequent chapters. Chapters 2, 3, and 4 are concerned with three individual actuarial problems, which respectively study the fee structures in variable annuities, pension management, and the optimal insurance strategies for policyholders of a mutual insurance company (MIC). In the last chapter, we present an optimal stopping problem which concerns with finding the optimal time to sell or purchase a risky asset in the financial market. The main content for each chapter is elaborated below. In Chapter 2, we formulate a mathematical model to study the role of different fee structures on Variable Annuities with a ratchet feature and provide analytical solutions of the associated optimal investment-withdrawal problem. We consider a performance-based fee structure and manifests its advantages over a conventional constant fee structure in both insurer’s and policyholder’s perspectives. Specifically, we demonstrate that policyholders tend to invest less aggressively into risky assets under the performance fee. Consequently, the risk-neutral insurer receives an increase in expected profit and a reduction in tail risk. Mathematically, we also contribute by establishing the well-posedness of the associated free-boundary value problems and the verification theorems for the underlying control problems, which involve non-canonical analysis and estimations due to the ratchet feature and the guaranteed protection. In Chapter 3, we revisit the problem of optimal portfolio selection of an individual worker during the accumulation phase in the defined contribution pension scheme, which has been well studied in the literature. Most of them adopted the classical backward model and approach, but any pre-specifications of retirement time, preferences, and market environment models do not often hold in such a prolonged horizon of the pension scheme. Pre-commitment to ensure the time-consistency of an optimal investment strategy derived from the backward model and approach leads the supposedly optimal strategy to be sub-optimal in the actual realizations. To address these shortcomings, we consider the problem via the forward preferences, in which an environment-adapting strategy is able to hold optimality and time-consistency together. Stochastic partial differential equation representation for the worker’s forward preferences is illustrated. We construct two of the forward utility preferences and solves the corresponding optimal investment strategies, in the cases of initial power and exponential utility functions. In Chapter 4, we consider the optimal insurance problem for policyholders who purchase policies from an MIC. Unlike shareholder-owned insurance companies, an MIC is owned by its policyholders, who are entitled to receive a portion of the operational surplus of the MIC in the form of dividends ex post. The wealth process and the optimal insurance strategy of a policyholder thus depend on those of the other policyholders within the MIC. To tackle the complex interactions between policyholders emerged from the surplus-sharing mechanism, herein, we employ a linear-quadratic extended mean field game framework to model the insurance network asymptotically. In a generic scenario with insurance constraints being imposed, we characterize the Nash equilibrium strategies using a McKean-Vlasov forward backward stochastic differential equation (FBSDE). The local well-posedness of the FBSDE is addressed. When no insurance constraint is imposed, we obtain the closed-form optimal strategies in terms of simpler Riccati equations, whose well-posedness shall be established globally. Numerical studies on the effect of the equilibrium strategies with respect to different model parameters are also provided. In Chapter 5, we consider an open problem in optimal prediction, where an investor wishes to sell at a time in order to maximize the probability that the selling price is greater than a given portion p of the ultimate maximum price over a finite-time planning horizon. The problem was first formulated in the manuscript by Pedersen (2009), where the solution to a special case can be dated back to Pedersen (2003). The optimal strategy is essentially determined by the roots of a theta-type function, which is the difference of solutions to two partial differential equations (PDEs). In Pedersen (2009), only the case of unique existence of root was considered. Yet, the general number of roots, which appears to be a highly non-canonical PDE problem, was not discussed. Herein, we show that depending on a goodness index $\lambda$ of the stock, the function can admit either one, two or no roots. Our work is devoted to providing a complete solution to the problem by a comprehensive study of the number of roots of the aforementioned function using a hybrid of hard analysis and a computer-aided proof. Simulations and backtesting results show that this trading strategy could provide considerable returns even in bearish markets, and outperform some well-recognized strategies, whence confirming the financial importance of this criterion.

Graduation Semester

2024-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2024 Kenneth Ng

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