Stochastic processes

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.

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.

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.

For over a decade, researchers at Harbor Branch Oceanographic Institute (HBOI)
have conducted surveys of the bottlenose dolphin (Tursiops truncatus) population of
Indian River Lagoon (IRL) in Florida. I have constructed a 4-stage population model
using the statistical program R. The model is used to conduct a viability analysis by
analyzing the relationship between birth, calf and adult survival rates. The power
analysis compares survey frequency to expected confidence intervals in estimating
abundance. The sensitivity analysis shows that the population is most sensitive to
changes in adult survival, followed by birth rate and calf survival. The model shows a
strong chance of viability over a 50 year time span. The population is vulnerable to long
periods of decline if birth, calf or adult survival rates fall below certain thresholds.
Overall, the model simulates the future impacts of demographic change, providing a tool
for conservation efforts.

Survey is time-consuming and expensive. Therefore, it needs to be both effective and efficient. Some archaeologists have argued that current survey techniques are not effective (Shott 1985, 1989), but most archaeologists continue to employ these methods and therefore must believe they are effective. If our survey techniques are effective, why do simulations suggest otherwise? If they are ineffective, can we improve them? The answers to these practical questions depend on the topological characteristics of archaeological site distributions. In this study I analyze archaeological site distributions in the Valley of Oaxaca, Mexico, using lacunarity and fractal dimension. Fractal dimension is a parameter of fractal patterns, which are complex, space-filling designs exhibiting self-similarity and power-law scaling. Lacunarity is a statistical measure that describes the texture of a spatial dispersion. It is useful in understanding how archaeological tests should be spaced during surveys. Between these two measures, I accurately describe the regional topology and suggest new considerations for archaeological survey design.

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.

In this study we developed a supply chain contract model for multiple scheduling period with dynamic demand patterns of stochastic nature, and with elastic price structures. The model presented here combined and enhanced several supply chain contract models developed previously. It is unique in that it considered multiple periods, dynamic, stochastic and price-elastic demand patterns, and flexible order quantities. Using a linear demand price-elastic relation and normal distribution pattern, optimal solutions for minimum cost, maximum profit, price structure, and order policies for the entire supply chain were derived. Sensitivity analyses performed in this study gave a better understanding of relative importance of various system variables.

Alopex is a biologically influenced computation paradigm that uses a stochastic procedure to find the global optimum of linear and nonlinear functions. It maps to a hierarchical SIMD (Single-Instruction-Multiple-Data) architecture with simple neuronal processing elements (PE's), therefore the large amount of interconnects in other types of neural networks are not required and more efficient utilization of chip level and board level "real estate" is realized. In this study, verifications were performed on the use of a simplified Alopex algorithm in handwritten digit recognition with the intent that the verified algorithm be digitally implementable. The inputs to the simulated Alopex hardware are a set of 32 features extracted from the input characters. Although the goal of verifying the algorithm was not achieved, a firm direction for future studies has been established and a flexible software model for these future studies is available.

This Thesis is concerned with the application of stochastical mixture considerations in the analytical modeling of certain classes of electromagnetic composites. It refers to the elucidation of the electromagnetic properties of such composite materials when used in engineering applications. The analytical studies refer to the extension of the existing stochastical mixture permittivity formulations to characterize magnetic mixture materials as well as chiralic mixture media. In both cases the mixture medium is presumed to consist of a host (receptacle) and dispersed particulates (inclusions). The effects of particulate shape in both chiralic and achiralic systems are also considered. Further, the concept of particulate polarization in deciding the permittivity and/or permeability characteristics of orderly-textured mixture media is addressed so as to determine the electromagnetic properties of such orderly-textured media. Application potentials of the present studies in the design of electromagnetic composites are indicated and the scope for the future research is portrayed.

This dissertation deals with the non-perturbative finite element methods for stochastic structures and conditional simulation techniques for random fields. Three different non-perturbative finite element schemes have been proposed to compute the first and second moments of displacement responses of stochastic structures. These three methods are based, respectively, on (i) the exact inverse of the global stiffness matrix for simple stochastic structures; (ii) the variational principles for statically-determinate beams; and (iii) the element-level flexibility for general stochastic statically indeterminate structures. The non-perturbative finite element method for stochastic structures possesses several advantages over the conventional perturbation-based finite element method for stochastic structures, including (i) applicability to large values of the coefficient of variation of random parameters; (ii) convergence to exact solutions when the finite element mesh is refined; (iii) requirement of less statistical information than that demanded by the high-order perturbation methods. Conditional simulation of random fields has been an extremely important research field in most recent years due to its application in urban earthquake monitoring systems. This study generalizes the available simulation technique for one-variate Gaussian random fields, conditioned by realizations of the fields, to multi-variate vector random field, conditioned by the realizations of the fields themselves as well as the realizations of the fields derivatives. Furthermore, a conditional simulation for non-Gaussian random fields is also proposed in this study by combining the unconditional simulation technique of non-Gaussian fields and the conditional simulation technique of Gaussian fields. Finally, the dissertation incorporates the simulation technique of random field into the non-perturbation finite element method for stochastic structures, to handle the cases where only one-dimensional probability density function and the correlation function of the random parameters are available, the demanded two-dimensional probability density function is unavailable. Simulation technique is applied to generate the samples of random fields which are used to estimate the correlation between flexibilities over elements. The estimated correlation of flexibility is then used in finite element analysis for stochastic structures. For each proposed approach, numerous examples and numerical results have been implemented.