Degree Type

Dissertation

Date of Award

2015

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

L. Steven Hou

Second Advisor

Huaiqing Wu

Abstract

This dissertation studies stochastic delay differential equations (SDDEs), applies them to real market data, and compares them with classic models. In Chapter 2, we study the mathematical properties of stochastic differential equations with or without delay, and introduce the linear SDDE for several specific financial market behaviors we are interested in. Since it is hard to find an explicit solution of an SDDE, we introduce a numerical technique and use it to analyze the SDDE. In Chapter 3, we use the Euler-Maruyama method to discretize a continues-time stochastic system and show the convergence in different senses of the numerical scheme to the true solution of the linear SDDE. Furthermore, it is crucial to understand the quantitative behavior of the parameters for the stochastic system and the impact of introducing the delay term, but these parameters are unknown and hard to estimate. In Chapter 4, we propose a blocking method to group the price points and use the Bayesian methods to estimate the parameters in the linear SDDE. We then apply the model to real stock price data, estimate and calibrate all the parameters in the stochastic system, and compare them with the parameters obtained from the classic geometric Brownian motion model.

DOI

https://doi.org/10.31274/etd-180810-3984

Copyright Owner

Yun Zheng

Language

en

File Format

application/pdf

File Size

73 pages

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