Parameter estimation

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.

This dissertation is a study about applied dynamical systems on two concentrations. First, on the basis of the growing association between opioid addiction and HIV infection, a compartmental model is developed to study dynamics and optimal control of two epidemics; opioid addiction and HIV infection. We show that the disease-free-equilibrium is locally asymptotically stable when the basic reproduction number R0 = max(Ru0; Rv0) < 1; here Rv0 is the reproduction number of the HIV infection, and Ru0 is the reproduction number of the opioid addiction. The addiction-only boundary equilibrium exists when Ru0 > 1 and it is locally asymptotically stable when the invasion number of the opioid addiction is Ruinv < 1: Similarly, HIV-only boundary equilibrium exists when Rv0 > 1 and it is locally asymptotically stable when the invasion number of the HIV infection is Rvinv < 1. We study structural identifiability of the parameters, estimate parameters employing yearly reported data from Central for Disease Control and Prevention (CDC), and study practical identifiability of estimated parameters. We observe the basic reproduction number R0 using the parameters. Next, we introduce four distinct controls in the model for the sake of control approach, including treatment for addictions, health care education about not sharing syringes, highly active anti-retroviral therapy (HAART), and rehab treatment for opiate addicts who are HIV infected. US population using CDC data, first applying a single control in the model and observing the results, we better understand the influence of individual control. After completing each of the four applications, we apply them together at the same time in the model and compare the outcomes using different control bounds and state variable weights. We conclude the results by presenting several graphs.

The purpose of online parameter learning and modeling is to validate and restore the properties of a structure based on legitimate observations. Online parameter learning assists in determining the unidentified characteristics of a structure by offering enhanced predictions of the vibration responses of the system. From the utilization of modeling, the predicted outcomes can be produced with a minimal amount of given measurements, which can be compared to the true response of the system. In this simulation study, the Kalman filter technique is used to produce sets of predictions and to infer the stiffness parameter based on noisy measurement. From this, the performance of online parameter identification can be tested with respect to different noise levels. This research is based on simulation work showcasing how effective the Kalman filtering techniques are in dealing with analytical uncertainties of data.

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.