Date of Award
Doctor of Philosophy
L. Steven Hou
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.
Zheng, Yun, "Asset pricing based on stochastic delay differential equations" (2015). Graduate Theses and Dissertations. 14433.