Time varying parameter estimation scheme for a linear stochastic differential equation

In this work, an attempt is made to estimate time varying parameters in a linear stochastic differential equation. By defining mk as the local admissible sample/data observation size at time tk, parameters and state at time tk are estimated using past data on interval [tk−mk+1, tk]. We show that the parameter estimates at each time tk converge in probability to the true value of the parameters being estimated. A numerical simulation is presented by applying the local lagged adapted generalized method of moments (LLGMM) method to the stochastic differential models governing prices of energy commodities and stock price processes

Global stability for a 2n + 1 dimensional HIV/AIDS epidemic model with treatments

In this work, we derive and analyze a 2n + 1-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R0, n, Rt, n and Rt,n as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R0, n, Rt, n andRt,n. We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if Rt, n = 1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if Rt, n\u3c1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing Rt,n \u3e 1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation

Finding Positive Solutions of Boundary Value Dynamic Equations on Time Scale

This thesis is on the study of dynamic equations on time scale. Most often, the derivatives and anti-derivatives of functions are taken on the domain of real numbers, which cannot be used to solve some models like insect populations that are continuous while in season and then follow a difference scheme with variable step-size. They die out in winter, while the eggs are incubating or dormant; and then they hatch in a new season, giving rise to a non overlapping population. The general idea of my thesis is to find the conditions for having a positive solution of any boundary value problem for a dynamic equation where the domain of the unknown function is a so called time scale, an arbitrary nonempty closed subset of the reals

Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease

We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (R0) and stochastic (R0) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0 \u3c 1), epidemic can still grow initially (if R0 \u3e 1) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise

Real-Time Nowcasting of Short-Run of the Euro-Dollar Exchange Rate with Economic Fundamentals: Does the measure of Money Supply Matter?

This paper proposes a fundamental-monetary-based econometric model with mixed frequency data to now caste the Euro-Dollar short run exchange rate. We use the exact amount of information that are available to researchers or police makers at the time of prediction of exchange rate. The spot exchange rate information are available at week basis while other macroeconomic data like money supply, in action, industrial production, and interest rate are available at monthly basis. Since the monetary model consists of stable money demand functions (Bianco, 2012), not all the measure of money supply guaranties a stable money demand. Barnett (1978, 1980) has proven that the measure of money supply (simple sum) that central banks published monthly does not provide a stable money demand. Only, the Divisia monetary aggregates provide a stable money demand at any level of aggregation. This paper proposes a multivariate state space model that takes into account not only the asynchronous information inflow, but also the instability of the money demand to predict four weeks spot exchange rate. We obtain great improvement of the model proposed by Bianco (2012) with Divisia monetary aggregates as measure of money supply. The latter performs much better at all forecasting horizons (one, two, three, and four weeks) than all of others models that predict the spot exchange rate with Significant improvements. We estimate our model (dynamic factor model) with the procedure proposed by Barnett et al (2016)

Stochastic Modeling of Energy Commodity Spot Price Processes with Delay in Volatility

Employing basic economic principles, we systematically develop both deterministic and stochastic dynamic models for the log-spot price process of energy commodity. Furthermore, treating a diffusion coefficient parameter in the non-seasonal log-spot price dynamic system as a stochastic volatility functional of log-spot price, an interconnected system of stochastic model for log-spot price, expected log-spot price and hereditary volatility process is developed. By outlining the risk-neutral dynamics and pricing, sufficient conditions are given to guarantee that the risk-neutral dynamic model is equivalent to the developed model. Furthermore, it is shown that the expectation of the square of volatility under the risk-neutral measure is a deterministic continuous-time delay differential equation. The presented oscillatory and non-oscillatory results exhibit the hereditary effects on the mean-square volatility process. Using a numerical scheme, a time-series model is developed to estimate the system parameters by applying the Least Square optimization and Maximum Likelihood techniques. In fact, the developed time-series model includes the extended GARCH model as a special case

Positive Solutions of Boundary Value Dynamic Equations

In this paper, we deal with the existence of a positive solution for 2nd and 3rd order boundary value problem by first defining their respective Green’s function. The Green’s function is used to derive the Green’s function for the 2nth and 3nth order boundary value problem, respectively, where n is a positive integer. The Green’s function is also used to derive conditions for positive solution of the 2nth and 3nth order eigen value differential equation, respectively

Local Lagged Adapted Generalized Method Of Moments And Applications

In this work, an attempt is made for developing the local lagged adapted generalized method of moments (LLGMM). This proposed method is composed of: (1) development of the stochastic model for continuous-time dynamic process, (2) development of the discrete-time interconnected dynamic model for statistic process, (3) utilization of Euler-type discretized scheme for nonlinear and non-stationary system of stochastic differential equations, (4) development of generalized method of moment/observation equations by employing lagged adaptive expectation process, (5) introduction of the conceptual and computational parameter estimation problem, (6) formulation of the conceptual and computational state estimation scheme and (7) definition of the conditional mean square Є -best sub optimal procedure. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, financial, medical, physical and social sciences. The byproducts of LLGMM are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). DTIDMLSMVSP is the generalization of statistic (sample mean and variance) drawn from the static dynamic population problems. Moreover, it is also an alternative approach to the GARCH (1,1) model and its many related variant models (e.g., EGARCH model, GJR GARCH model). It provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equation. Furthermore, application of the LLGMM method to stochastic differential dynamic models for energy commodity price, U. S. Treasury Bill Yield Interest Rate and U. S.-U.K. Foreign Exchange Rate exhibits its unique role and scope

Local Lagged Adapted Generalized Method of Moments: An Innovative Estimation and Forecasting Approach and its Applications.

In this work, an attempt is made to apply the Local Lagged Adapted Generalized Method of Moments (LLGMM) to estimate state and parameters in stochastic differential dynamic models. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, energy commodity markets, financial, medical, military, physical sciences and social sciences. The byproducts of this innovative approach (LLGMM) are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). Moreover, LLGMM is a dynamic non-parametric method. The DTIDMLSMVSP is an alternative approach to the GARCH (1,1) model, and it provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, applications of LLGMM to energy commodities price, U.S. Treasury Bill interest rate and the U.S.–U.K. foreign exchange rate data strongly exhibit its unique role, scope and performance, in particular, in forecasting and confidence-interval problems in applied statistics