Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential
equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance
have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To
modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem
are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce
to free-boundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the
solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a
fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs.
We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility
of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a
numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference
method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also
satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013). / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the
degree of Doctor of Philosophy. / Fall Semester 2017. / October 30, 2017. / backward stochastic differential equations, Hamilton-Jacobi-Bellman equation, Monte Carlo approximation, portfolio
optimization, stochastic control, transaction costs / Includes bibliographical references. / Arash Fahim, Professor Directing Dissertation; Jen Atkins, University Representative; Giray Ökten,
Committee Member; Lingjiong Zhu, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_605026 |
| Contributors | Tsai, Wan-Yu (author), Fahim, Arash (professor directing dissertation), Atkins, Jennifer (university representative), Zhu, Lingjiong (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Mathematics (degree granting departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, doctoral thesis |
| Format | 1 online resource (81 pages), computer, application/pdf |
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