Optimal Investment Problems in Financial Engineering

Author: Jiacheng Fan

Date: Aug 3, 2020
Department: Financial Engineering
Degree: Doctor of Philosophy

Advisory Committee:
Dr. Zhengyu Cui, Chairman
Dr. Darinka Dentcheva,
Dr. Hamed Ghoddusi,
Dr. Papa Ndiaye,
Dr. Bin Zou

Abstract: This work addresses a fundamental question in mathematical finance: how an individual investor can maximize expected utility from terminal wealth in a continuous-time financial market — the classical Merton Problem. Beyond the standard setting, we extend the problem in several directions, both in terms of objective formulation and model dynamics. Using both primal (stochastic control) and dual (convex duality) methods, we investigate:

Optimal stopping with finite maturity, where we characterize explicit conditions for free-boundary solutions under CRRA, non-HARA, and SAHARA utility functions.

Behavioral extensions, incorporating non-concave utility and subjective probability weighting, and provide explicit conditions under which the duality method yields true solutions to the primal problem.

Random volatility environments, where investors face local volatility, stochastic volatility, and stochastic local volatility with jumps. By applying continuous-time Markov chain approximation (CTMC), we extend solvable cases of the Merton Problem and obtain closed-form (log utility) and semi-closed-form (power utility) solutions.

Together, these contributions broaden the classical Merton framework to more realistic investment settings, offering both theoretical insights and practical implications for optimal portfolio choice under uncertainty.

For full Dissertation, click here.