Wang, Zhen
(2024)
Optimal Investment Strategies for Investment of Wealth and Avoiding Ruin.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
The goal of this dissertation is to provide a mathematically rigorous treatment of the problem of minimizing the probability of ruin while simultaneously maximizing terminal wealth. This analysis will be carried out in the classic context of a wealth process which consists of a risky asset (stock, option or a portfolio of such) and a risk-free asset (bond, money market account). We will consider two distinct types of optimal investment strategies: one in which the strategy is fixed-mix (constant) and one in which the strategy is time-dependent and random through dependence on the past history of the wealth process.
The first case is the situation in which the investor only controls the investment strategy at the beginning of the investment. In this case, we begin with the probability density function of a Brownian motion that never hits a given lower ruin boundary before the terminal time in order to solve the avoidance of ruin problem. We then extend this method to maximizing the terminal wealth problem while avoiding ruin by considering the outcome at ruin to be equal to the ruin value. We compute the analytical solution to this problem in term of a utility function.
The second case is closer to the real market, where the investor can adjust the allocation among the different assets at any time. Since the time of ruin is a random time, we cannot directly use well-known results from stochastic control theory. We address this issue by extending these results to our random stopping time if it occurs before the constant terminal time. We consider several approaches to solving the resulting Hamilton-Jacobi-Bellman (HJB) PDE. A new approach outlined in the thesis is to transform the HJB PDE into a heat equation with a free-boundary condition. We then use numerical results developed for the Stefan problem to solve our original optimal control problem. We show that the results from this new approach compare favorably with finite difference and finite element methods constructed to solve the original optimization problem.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
27 August 2024 |
| Date Type: |
Publication |
| Defense Date: |
6 June 2024 |
| Approval Date: |
27 August 2024 |
| Submission Date: |
11 June 2024 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
93 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Merton's problem, reflection principle, stochastic control, Hamilton-Jacobi-Bellman equation |
| Date Deposited: |
27 Aug 2024 14:14 |
| Last Modified: |
27 Aug 2024 14:14 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/46579 |
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