Browsing by Subject "Stochastic differential equations"
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Item A deterministsic numerical method for solutions of first-passage time problems(Texas Tech University, 1996-12) Sharp, Wyatt D.In this research, a new deterministic numerical procedure for first passage time problems is introduced, analyzed, and numerically tested. In this procedure, the Green's function solution to the forward Kolmogovorov equation is approximated for a small time step. The reliability function is then approximated by recursively and numerically solving an iterated integral whose integrand involves the approximate Green's function. The reliability function, which solves the backward Kolmogorov equation, yields the probability distribution of first passage times, and hence the expected exit time. The error analysis is shown for the one dimensional case and can be modified for higher dimensions. Three numerical examples are given. Two of the examples are two dimensional problems which have been given special attention by other investigators [2, 10] because of their computational difficulty.Item A stochastic age-structured population model(Texas Tech University, 1998-05) Chowdhury, MarufulAge-structured models for biological species have received much study. In this thesis the age-structured population model is extended to a stochastic agestructured model. The model is solved numerically for two age-structured population examples. The calculational results are compared with Monte Carlo calculations. The results indicate that the two calculational methods are in agreement. Persistence times for two age-structured populations are then studied computationally. It is shown that persistence times increase as the number of age groups increases. Also, persistence times are higher for a stohastic age-structured population than for a nonstochastic (or deterministic) age-structured population.Item A stochastic differential equation model for charged-particle straggling(Texas Tech University, 2001-08) Thompson, Carolyn M.Considered in the present investigation is the energy loss due to charged particles that pass through an absorber. It is assumed that the rest mass of the charged particle is much greater than the rest mass of an electron and the energy of the charged particle is sufficiently high so that the particles do not capture electrons. With these assumptions, the charged particles lose energy by excitation and ionization with the absorber atoms. The mechanism of the energy loss is primarily the interaction of the Coulomb fields of the particles with those of the bound electrons in the absorber. If a charged particle passes through a target of given thickness, the particle will interact with the target's electrons. The charged particle will lose a certain amount of kinetic energy while passing through the target or stopping material. The amount of energy lost by this charged particle in the stopping material can be measured as the stopping power. A stochastic differential equation model is used to determine the amount of energy lost with respect to the distance traveled by the charged particle.Item Algorithms and analysis for next generation biosensing and sequencing systems(2012-08) Shamaiah, Manohar; Vikalo, Haris; Hassibi, Arjang; Vishwanath, Sriram; Tewfik, Ahmed; Yoon, Byung-JunRecent advancements in massively parallel biosensing and sequencing technologies have revolutionized the field of molecular biology and paved the way to novel and exciting innovations in medicine, biology, and environmental monitoring. Among them, biosensor arrays (e.g., DNA and protein microarrays) have gained a lot of attention. DNA microarrays are parallel affinity biosensors that can detect the presence and quantify the amounts of nucleic acid molecules of interest. They rely on chemical attraction between target nucleic acid sequences and their Watson-Crick complements that serve as probes and capture the targets. The molecular binding between the probes and targets is a stochastic process and hence the number of captured targets at any time is a random variable. Detection in conventional DNA microarrays is based on a single measurement taken in the steady state of the binding process. Recently developed real-time DNA microarrays, on the other hand, acquire multiple temporal measurements which allow more precise characterization of the reaction and enable faster detection based on the early dynamics of the binding process. In this thesis, I study target estimation and limits of performance of real time affinity biosensors. Target estimation is mapped to the problem of estimating parameters of discretely observed nonlinear diffusion processes. Performance of the estimators is characterized analytically via Cramer-Rao lower bound on the mean-square error. The proposed algorithms are verified on both simulated and experimental data, demonstrating significant gains over state-of-the-art techniques. In addition to biosensor arrays, in this thesis I present studies of the signal processing aspects of next-generation sequencing systems. Novel sequencing technologies will provide significant improvements in many aspects of human condition, ultimately leading towards the understanding, diagnosis, treatment and prevention of diseases. Reliable decision-making in such downstream applications is predicated upon accurate base-calling, i.e., identification of the order of nucleotides from noisy sequencing data. Base-calling error rates are nonuniform and typically deteriorate with the length of the reads. I have studied performance limits of base-calling, characterizing it by means of an upper bound on the error rates. Moreover, in the context of shotgun sequencing, I analyzed how accuracy of an assembled sequence depends on coverage, i.e., on the average number of times each base in a target sequence is represented in different reads. These analytical results are verified using experimental data. Among many downstream applications of high-throughput biosensing and sequencing technologies, reconstruction of gene regulatory networks is of particular importance. In this thesis, I consider the gene network inference problem and propose a probabilistic graphical approach for solving it. Specifically, I develop graphical models and design message passing algorithms which are then verified using experimental data provided by the Dialogue for Reverse Engineering Assessment and Methods (DREAM) initiative.Item Cumulants of an IQF via differential equations(Texas Tech University, 1973-12) Hartwig, Ronald CraigNot availableItem Deterministic and random particle methods applied to Vlasov-Poisson-Fokker-Planck kinetic equations(Texas Tech University, 1996-05) Havlak, KarlWe devise and study two different particle methods for approximating Vlasov-Poisson-Fokker-Planck systems. We first consider a random particle method. Such a proposed scheme takes into account the fact that the trajectories of a particle, undergoing Brownian motion due to collisions with the medium or background particles, can be obtained as the solutions of stochastic differential equations, i.e., the Langevin equations. These equations are the precise analogs of the (deterministic) Hamiltonian system in the collisionless model. The particle approximation, in particular, simulates the action of viscosity by the use of independent Wiener processes (Brownian motions). The analysis relies heavily on the machinery developed by K. Ganguly and H.D. Victory, Jr. [SIAM J. Numer. Anal., 26 (1989), pp. 249-288] to treat the sampling errors due to random motions of the particles. For example, the key idea in the consistency error analysis is to separate the moment and discretization errors - accounting for the deterministic portions of the error - from the sampling errors introduced by the random motion of the particles. The latter errors constitute the dominant component of the overall consistency error in terms of order and are gauged by applying Bennett's Inequality utilized to estimate tail probabilities for standardized sums of independent random variables with zero means. Moreover, the stability estimates for the particle approximations to the collisionless model are extended to the Vlasov-Poisson-Fokker-Planck setting by means of this inequality. We then consider a deterministic method. Such a proposed scheme is a splitting method, whereby particle methods are used to treat the convective part and the diffusion is simulated by convolving the particle approximation with the fieldfree Fokker-Planck kernel. The states of the particles are not affected by the diffusion per se, but the charge or mass on the particles in their previous states is redistributed via the diffusion. Because of this redistribution of mass or charge, it is necessary to monitor the growth in time of the velocity moments of the approximate distribution. Convergence of the errors in both the density and the fields is shown to be first order in time with respect to both the uniform and L^- senses. This treatment is the first application of the velocity moment analysis by P.L. Lions and B. Perthame [Invent. Math., 105 (1991), 415- 30] in a numerical analysis of Vlasov-type kinetic equations. Our study is made feasible by some formulas by F. Bouchut [J. Fund Anal, 111 (1993), 239-258] concerning the field-free fundamental solution and recent extensions to the periodic setting. The splitting procedure we employ is related to the viscous splitting or fractional step procedure of G.H. Cottet and S. Mas-Gallic [Numer. Math., 57 (1990), 805-827] for treating Navier-Stokes equations modeling viscous, incompressible flow.Item Deterministic and stochastic discrete-time epidemic models with applications to amphibians(Texas Tech University, 2004-08) Emmert, Keith EricA discrete-time model is formulated for spread of disease in a structured host population. The host population is sub-divided into three developmental stages, larval, juvenile, and adult, and each stage can be infected by the pathogen. Recovery from the disease is possible with this model. We investigate conditions on the parameters where either the host population does not survive or the host population survives and is free from disease. The analysis assumes parameters of the model are constants. Several different submodels of the full structured epidemic model are studied and conditions are derived for global stability of the extinction equilibrium and local stability of the disease-free equilibrium. Numerical examples are presented to illustrate the dynamics of the model when the disease-free equilibrium is not stable. The motivation for this model is the spread of a fungal pathogen in an amphibian population. A second discrete-time deterministic and stochastic epidemic model is formulated for spread of disease in a structured host population. This model differs from the previous model because the parameters of this model are periodic. The host population is again subdivided, but this time into two developmental stages, juvenile and adult. Each stage can be infected by the pathogen, but there is no recovery from the disease. Several submodels of the full model are studied and conditions for global extinction as well as local stability of the disease-free solutions are given. Stochastic and deterministic examples illustrating the dynamics of the model are presented. The motivation for this model is the spread of a fungal pathogen in amphibian populations which are explosive breeders.Item Deterministic and stochastic epidemic models with multiple pathogens(Texas Tech University, 2003-08) Kirupaharan, NadarajahCompetitive exclusion and coexistence of multiple pathogens in deterministic and stochastic epidemic models are investigated in this dissertation which consists of three parts. In the first part, the persistence and extinction dynamics of multiple pathogen strains for discrete-time SIS epidemic model in a single patch and in two patches are studied. It is shown for the single patch model that the basic reproduction number determines which strain dominates and persists. However, in the two-patch epidemic model, both the dispersal probabilities and the basic reproduction numbers for each strain determine whether a strain persists. With two patches, there is a greater chance that more than one starin will co-exist. In the second part, the stochastic spatial epidemic models with multiple pathogen strains for the above deterministic models are formulated as discrete-time Markov chain models and analyzed for coexistence and comptetitive exclusion. When infected individuals disperse between two patches, coexistence may occur in the stochastic model. However, in the stochastic model, eventually disease extinction occurs but it will take a long time. An estimate for the probability of disease extinction is obtained for the stochastic model. The distribution conditioned on non-extiction is compared to the solution of the deterministic model. In the third part, the dynamics of continuous- time stochastic SIS and SIR epidemic models with multiple pathogen strains and density-dependent mortality are studied using stochastic differential equation models. The dynamics of these stochastic models are then compared to the analogous deterministic models. In the deterministic model, there can be competitive exclusion, where only one strain, the dominant one, persists or there can be coexistence, persistence of more than one strain. In the stochastic model, all strains will eventually be eliminated because the disease-free state is an absorbing state. Generally, it will take a long time until all strains are eliminated. Numerical examples show that coexistence cases predicted in the deterministic models may not occur in the stochastic models.Item Deterministic and stochastic models of virus dynamics(Texas Tech University, 2003-12) Perera, Niranjala CA variety of mathematical models ranging from very simple ones to complicated ones have been developed and analyzed in order to capture different phenomena associated with the spread of diseases. Even though none of these models behave exactly according to the observed clinical data, major features of disease dynamics can be captured merely by means of a simple model. The model introduced by Nowak and May [12] is such simple deterministic model of which a stability analysis has not been done. Our objectives in this endeavor are two-fold. The first objective of this thesis is to carry out a thorough analysis of the aforementioned deterministic model of virus dynamics while obtaining the related system of Ito stochastic differential equations which has not been obtained to date. The motivation for obtaining the related stochastic model is also two-fold. The first reason is the capability of stochastic models to capture the randomness associated with the disease dynamics. The second reason is while a deterministic model predicts a single outcome for a given set of parameter values, a stochastic model predicts an infinite set of possible outcomes weighed by their likelihoods and probabilities. Any mathematical model which describes virus dynamics, is not complete until it describes the immune response. With analogy to a predator-prey model, immune cells play the role of the predator while the virus plays the role of the prey. The immune response is triggered by encountering a foreign antigen. The role of the immune system is to fight off invasion by foreign pathogens. In this endeavor, our interest is a special kind of T cell, namely cytotoxic T lymphocyte (CTL) which can also identify and eliminate infected cells. Then the immune response is incorporated with the aforementioned simple model of virus dynamics. This is done under three different assumptions on the CTL proliferation rate. This evidently results in three different models. The second objective of this thesis is to carry out a thorough stability analysis of the three deterministic models of virus dynamics with the CTL response while obtaining respective systems of Ito stochastic differential equations.Item Development and implementation of stochastic neutron transport equations and development and analysis of finite difference and Galerkin methods for approximate solution to Volterra's population equation with diffusion and noise(Texas Tech University, 1999-05) Sharp, Wyatt D.Many systems in this world are influenced by stochastic (random) processes either from within the system or from external agents. When modeling these systems, these processes and their derivatives arise naturally in a field of study called stochastic differential equations (SDEs). SDEs find application in diverse areas of engineering, chemistry, physics, economics and finance, population dynamics, pharmacology and medicine, and social sciences, to name a few. This research is divided into two parts, the common thread being SDEs. In the second chapter, a new system of SDEs for modeling the random behavior of neutron travel is derived. Numerical methods are developed to solve this system and shown to be accurate when compared with the Monte Carlo method. In the third chapter, two independent numerical methods are developed to solve Volterra's population equation with diffusion and noise. Error analyses are performed on the two methods which prove convergence of the approximations to the exact solution. Three numerical examples are given which confirm the results of the error analyses.Item Extrapolation of difference methods in option valuation, rounding error in numerical solution of stochastic differential equations, and shooting methods for stochastic boundary-value problems(Texas Tech University, 2003-08) Arciniega, ArmandoMy dissertation involves three different projects. The first project involves numerical solution of option prices. In particular, the fully implicit and Crank-Nicolson difference schemes for solving option prices are analyzed. It is proved that the error expansions for the difference methods have the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. The difference methods are applied to European, American, and down-and-out knock-out call options. Computational results indicate that Richardson extrapolation significantly decreases the amount of computational work (by as much as a factor of 16) in estimation of option prices. The second project involves an analysis of rounding errors in numerical solution of stochastic differential equations. A statistical rounding error analysis of Euler's method for numerically solving stochastic differential equations is performed. In particular, rounding errors associated with the mean square error and for functional expectations of the solutions are investigated. It is shown that rounding error is inversely proportional to the square root of the step size. An extrapolation technique provides second-order accuracy, and is one way to increase accuracy while avoiding rounding error. Several computational results are given, which support the theoretical results. The third project is a development of shooting methods for numerical solution of Stratonovich boundary-value problems. In particular, shooting methods are examined for numerically solving systems of Stratonovich boundary-value problems. It is proved that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is performed. Computational simulations are given.Item Extrapolation of implicit numerical methods for stochastic differential equations and stochastic models for multiple assets with application to options(2006-08) Koskodan, Rachel C.; Allen, Edward J.; Allen, Linda J. S.; Victory, Harold D.; Williams, G. BrockThis dissertation involves two different projects. The first project involves improving the accuracy of approximate solutions to stochastic differential equations. The stochastic theta numerical method forms a family of implicit Euler methods for approximating solutions to Ito stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation. In the second project, consistent Monte Carlo, discrete stochastic, and stochastic differential equation models are constructed from first principles for a mutual fund with multiple assets. The different stochastic models are shown to be consistent in the estimation of mutual fund values. The models are applied to the calculation of European call option prices. It is shown that option prices are insensitive to the form of the stochastic model's diffusion term. In addition, a general n-dimensional Black-Scholes partial differential equation is derived for option prices. Computational examples illustrate that the Black-Scholes partial differential equation and the stochastic differential equation models are consistent in estimating option prices.Item Initial Studies of Riccati Equations Arising in Stochastic Linear System Theory(Texas Tech University, 1977-12) Yao, Mong LingNot Available.Item Mathematical models with antibody and cytotoxic T lymphocyte responses due to hantavirus infections in rodents and humans(2011-08) Lewis, Chelsea J; Allen, Linda J. S.; Roeger, Lih-Ing W.Viral-hemorrhagic fevers are a suite of diseases that pose a threat to public health. These viruses induce an overwhelming response of cytotoxic T-cells (CTLs) which eliminates the virus by either killing or damaging infected cells. This response may lead to a considerable loss of functioning cells that are critical to the appropriate execution of certain processes. For hantavirus, a zoonotic disease carried by rodents, many of the target cells are the specialized endothelial cells that regulate solute movements, and are hence essential components of osmotic regulation and metabolic waste disposal of particular solutes in the blood. Because of the intimate relationship these endothelial cells have with the vascular system, an infection from the hantavirus may lead to either hantavirus pulmonary syndrome (HPS) or hemorrhagic fever with renal syndrome (HFRS). A crucial loss of too many functioning cells results in pulmonary edema, hemorrhage and even death. Rodents are the natural reservoirs for hantavirus that spread the virus to humans. Mice have been exposed to this virus much longer than humans and may have co-evolved with the virus. Because of this, rodents have developed a mechanism of down-regulation of their own CTL response. This means that hantavirus infection does not result in fatality in rodents; this is good for the virus as well as the rodents. Humans, having had limited exposure to the virus in evolutionary time, have yet to develop a means of co-existing with hantavirus. When there is no previous exposure to, or memory for, the virus, it is the natural response of our immune system to attack an infection via the CTL response, resulting in many of the symptoms associated with HPS or HFRS. We use a mathematical model for the virus and the immune response, including antibodies and the CTLs, to describe the cellular dynamics of a viral infection. The original model is an extension of a model originally formulated by Wodarz \cite{wodarz}, a system of ordinary differential equations (ODEs) for healthy target cells, infected target cells, virions, antibodies, and CTLs. Our extension includes T-helper cells which are important in CTL activation and antibody production. In addition, we extend this ODE model to a stochastic differential equation model. It is the aim of these models to predict the effects of the immunological mechanisms on the host both in human and in rodent reservoirs. Understanding the mechanisms by which rodents are able to successfully down-regulate the immunological response to hantavirus will help in finding appropriate treatments for this disease in humans.Item Runge-Kutta and recursive distribution numerical methods for approximate solution of stochastic differential equations(Texas Tech University, 1995-05) Abukhaled, Marwan IbrahimThis research concerns numerical solution of stochastic differential equations and is divided into two different and independent approaches. In the first approach, a class of Runge-Kutta methods is developed, analyzed and numerically tested. It is shown that these methods are of second-order accuracy in the weak sense for estimating expectations of functions of the solution for scalar as well as for systems of stochastic differential equations. It is also shown that in these methods, variance reduction techniques can be applied to reduce the stochastic error involved in estimating the expectations of functions of the solution. These second-order explicit methods are unique in the sense that they do not involve derivatives of the drift and diffusion coefficients and they can be easily programmed and implemented. In the second approach, it is shown that probability distributions of approximate sample paths of the solution satisfy a recursive integral equation. These probability distributions can then be approximated by numerically solving the integral equation. The advantage of this approach is the avoidance of computing thousands of sample paths as is generally the case in most standard numerical methods. This approach is shown to be useful for numerical solution of first-passage time problems.