Wellposedness Of Stochastic Differential Equations And Bayesian Estimation Of Parameters In The Ordinary Differential Equations

This thesis consists of two parts. In the first part, we study the blow-up and global existence of solutions to stochastic Schrodinger equations. In the second part, we provide a Bayesian penalized B-spline method to estimate parameters (constant and/or time-varying) in several differential equation models. Chapter 1 is devoted to the background and the motivation of these two problems.Mathematical theories for differential equations play an important role in Physics, Medical science, Biology, Finance and so on. Nonlinear Schrodinger equation, a type of differential equations, is very important in atom, molecule, solid state physics, nuclear physics, chemistry. However, in the real world, everything is not deterministic and is dis-turbed by random factors. In Chapter 2, we study the dynamics of a stochastic nonlinear Schrodinger equation with both a quadratic potential and an additive noise. We show that in both cases of repulsive potential and attractive one, any initial data with finite energy gives birth to a solution that blows up in arbitrarily small time. This is different from the deterministic case when the potential is repulsive, where strong potentials could prevent the solutions from blowing up. Our result hence indicates that the additive noise rather than the potential dominates the dynamical behaviors of the solutions to the stochastic nonlinear Schrodinger equations.Corresponding to the blow-up property, in Chapter 3 we study the global wellposed-ness of energy solution to a stochastic Schrodinger-Poisson system under the effect of multiplicative noise of Stratonovich type. Different from the deterministic case, we es-tablish communicator estimates in the stochastic version, based on which we obtain the existence and uniqueness of global solution.During the study of wellposedness of stochastic Schrodinger equation, we notice that parameters play an essential role in characterizing the dynamics of the equation. Different parameters could result in completely different dynamical behaviors. This mo-tivates us to explore statistical method with observed data to determine the parameters of the model, before applying mathematical notions to the applied sciences. Therefore, in the second part of this thesis, we develop a statistical method-Bayesian penalized B-spline method to estimate parameters in several differential equation models. In par-ticular, we allow the parameters to be constant and even to be time-varying.In Chapter 4, after introducing the general theory of Bayesian penalized B-spline method, we apply this method to a 2 x 2 linear model and a nonlinear Lotka-Volterra model, under the assumption that all state variables are observed. Moreover, the simula-tion results indicate that this method is efficient.In Chapter 5, we consider the estimation of parameters in the epidemic model. The difference between this kind of model and the models in Chapter 4 is that, only part of the state variables or even only one state variable can be observed in the epidemic mod-els. In the simulation part of this chapter, we estimate the parameters of the Kermack-Mckendrick model with only one state variable observed. Furthermore, we compare the Bayesian penalized B-spline method with the least square method in the simulation. It is shown that the former one performs better than the latter one. Also, we illustrate the pro-posed method through a real data analysis, which is the monthly number of HCV from Jan 2004 to Dec 2014 of mainland China. We use the data to estimate the parameters of the model which is provide by Zhang and Zhou in 2012. Results show that this method is still efficient when we have observed data of only one state variables.In Chapter 6, we further adopt the Bayesian penalized B-spline method to estimate time-varying parameters. The key ingredient of applying this method is that, we first ex-pand the time-varying parameters with B-spline, through this procedure, the estimation of time-varying parameters is transformed into the estimation of constant parameters. In the simulation, we estimate the time-varying parameters in two differential equation models:one is HIV model, and the other is Hong and Lian's model. Our simulation results show that this method is also efficient. Moreover, we compare the Bayesian penalized B-spline method with the two-stage local polynomial method in estimating time-varying parameters in the model of Hong and Lian, and the comparison shows that the former method has smaller RASE than the latter one does.