Long, Hongwei

Since the population growth systems may suffer impulsive environmental disturbances such as earthquakes, epidemics, tsunamis, hurricanes, and so on, stochastic differential equations(SDEs) that are driven not only by Brownian motion but also by α-stable Lévy noises are more appropriate to model such statistical behavior of non-Gaussian processes with heavy-tailed distribution, having infinite variance and in some cases infinite first moment. In this dissertation, we study stochastic processes defined as solutions to stochastic logistic differential equations driven by multiplicative α-stable Lévy noise. We mainly focus on one-dimensional stochastic logistic jump-diffusion processes driven by Brownian motion and α-stable Lévy motion. First, we present the stability analysis of the solution of a stochastic logistic growth model with multiplicative α-stable Lévy. We establish the existence of a unique global positive solution of this model under certain conditions. Then, we find the sufficient conditions for the almost sure exponential stability of the trivial solution of the model. Next, we provide parameter estimation for the proposed model. In parameter estimation, we use statistical methods to get an optimal and applicable estimator. We also investigate the consistency and asymptotics of the proposed estimator. We assess the validity of the estimators with a simulation study.

We consider a portfolio optimization problem in stochastic volatility jump-diffusion model. The model is a mispriced Lévy market that contains informed and uninformed investors. Contrarily to the uninformed investor, the informed investor knows that a mispricing exists in the market. The stock price follows a jump-diffusion process, the mispricing and volatility are modelled by Ornstein-Uhlenbeck (O-U) process and Cox-Ingersoll-Ross (CIR) process, respectively. We only present results for the informed investor whose goal is to maximize utility from terminal wealth over a finite investment horizon under the power utility function. We employ methods of stochastic calculus namely Hamilton-Jacobi-Bellman equation, instantaneous centralized moments of returns and three-level Crank-Nicolson method. We solve numerically the partial differential equation associated with the optimal portfolio. Under the power utility function, analogous results to those obtain in the jump-diffusion model under logarithmic utility function and deterministic volatility are obtained.

The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for Ornstein-Uhlenbeck processes driven by _-stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters.

In finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alpha-stable Levy motion to the process related to dynamics of log-returns in the Black-Scholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter estimation for discretely observed stochastic differential equations driven by Levy noises. We also discuss the consistency and asymptotic properties of the proposed estimators. Simulation results of the model are also presented to show the validity of the estimators. We then propose a new model where the volatility is not a constant. We consider generalized alpha-stable geometric Levy processes where the stochastic volatility follows the Cox-Ingersoll-Ross (CIR) model in Cox et al. [9]. A number of methods have been proposed for estimating parameters for stable laws. However, a complication arises in estimation of the parameters in our model because of the presence of the unobservable stochastic volatility. To combat this complication we use the sample characteristic function method proposed by Press [48] and the conditional least squares method as mentioned in Overbeck and Ryden [47] to estimate all the parameters. We then discuss the consistency and asymptotic properties of the proposed estimators and establish a Central Limit Theorem. We perform simulations to assess the validity of the estimators. We also present several tables to show the comparison of estimators using different choices of arguments ui's. We conclude that all the estimators converge as expected regardless of the choice of ui's.

This thesis is composed of three main parts. Each chapter is concerned with
characterizing some properties of a random ensemble or stochastic process. The
properties of interest and the methods for investigating them di er between chapters.
We begin by establishing some asymptotic results regarding zeros of random
harmonic mappings, a topic of much interest to mathematicians and astrophysicists
alike. We introduce a new model of harmonic polynomials based on the so-called
"Weyl ensemble" of random analytic polynomials. Building on the work of Li and
Wei [28] we obtain precise asymptotics for the average number of zeros of this model.
The primary tools used in this section are the famous Kac-Rice formula as well as
classical methods in the asymptotic analysis of integrals such as the Laplace method.
Continuing, we characterize several topological properties of this model of
harmonic polynomials. In chapter 3 we obtain experimental results concerning the
number of connected components of the orientation-reversing region as well as the geometry
of the distribution of zeros. The tools used in this section are primarily Monte
Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology
software known as Perseus. While the results in this chapter are empirical rather
than formal proofs, they lead to several enticing conjectures and open problems.
Finally, in chapter 4 we address an industry problem in applied mathematics
and machine learning. The analysis in this chapter implements similar techniques to
those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or
Control Area Network) is a network for allowing micro-controllers inside of vehicles
to communicate with each other. We propose and demonstrate the e ectiveness of an
algorithm for detecting malicious tra c using an approach that discovers and exploits
the natural geometry of the CAN surface and its relationship to random walk Markov
chains.

The purpose of this thesis is to compare the effectiveness of several interest
rate models in fitting the true value of interest rates. Up until 1990, the universally
accepted models were the equilibrium models, namely the Rendleman-Bartter model,
the Vasicek model, and the Cox-Ingersoll-Ross (CIR) model. While these models
were probably considered relatively accurate around the time of their discovery, they
do not provide a good fit to the initial term structure of interest rates, making them
substandard for use by traders in pricing interest rate options. The fourth model
we consider is the Hull-White one-factor model, which does provide this fit. After
calibrating, simulating, and comparing these four models, we find that the Hull-White
model gives the best fit to our data sets.

The main objective of this thesis is to simulate, evaluate and discuss several
methods for pricing European-style options. The Black-Scholes model has long been
considered the standard method for pricing options. One of the downfalls of the
Black-Scholes model is that it is strictly continuous and does not incorporate discrete
jumps. This thesis will consider two alternate Levy models that include discretized
jumps; The Merton Jump Diffusion and Kou's Double Exponential Jump Diffusion.
We will use each of the three models to price real world stock data through software
simulations and explore the results.Keywords: Levy Processes, Brownian motion, Option pricing, Simulation, Black-Scholes, Merton Jump Diffusion, Kou, Kou's Double Exponential Jump Diffusion.