Author: Jiacheng Fan
Advisor: Dr. Zhenyu Cui
Date: Aug 3, 2020
Department: School of Busisness
Degree: Doctor of Philosophy
Advisory Committee:
Dr. Zhengyu Cui, Chairman
Dr. Darinka Dentcheva,
Dr. Hamed Ghoddusi,
Dr. Papa Ndiaye,
Dr. Bin Zou
Abstract: A fundamental problem in mathematical finance and financial economics is the problem of an individual investor who invests in a financial market so as to maximize the expected utility from his terminal wealth. In the framework of a continuous time model, the economic agent is allowed to rebalance his positions of his portfolio consisting of both risky assets and risk-free bond in a continuous way while being sub ject to the budget and miscellaneous investment constraints (i.e. Merton Problem). The standard methods to solve this problem in the literature are stochastic optimal control approach based on the dynamic programming principle (primal method) and the convex duality method based on the semimartingale structure of the underlying assets (dual method). This thesis includes a series of original works that generalize the original Merton Problem from both the objective formulations and the model setting of the underlying dynamics. We have investigated different variations of the problem by applying both duality method and primal method which incorporates (i) a discretionary stopping in the investment cycle in a finite maturity setting; (ii) behavioral extensions including the non-concave utility function and the subjective probability weighting on the distribution of the underlying assets; (iii) stochastic lo cal volatility models with jumps for the underlying dynamics. Corresponding to the above variations, we have shown in this thesis that (i) how the individual investor make the stopping selection mixed with his optimal stocks trading. By dual transfor mation, the conditions on the existence of the free boundaries have been characterized in explicit form for CRRA, non-HARA and SAHARA utility functions; (ii) how the nonconcavity and subjective probability distortions influence the individual investors’ iv decision making. The conditions on when the Lagrange duality method provides the true solutions to the primal problem have been fully interpreted explicitly; (iii) how the individual investors react to different random volatility processes. By continuous time Markov chain approximation (CTMC), the solvable cases for the Merton problem have be extended to general Markov process including local volatility model, stochastic volatility model and stochastic local volatility model with jumps in the underlying assets, the closed form solutions have been obtained for the log utility and the semi-closed form solution was obtained for power utility function.
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