Explicit Approximations Of Nonlinear Stochastic Differential Equations And Its Applications

With the continuous development of science and technology,nonlinear stochas-tic differential equations(SDEs)are widely used in control engineering,wireless communications,financial engineering,ecology and other fields to describe the world more accurately.In general,the analytical expressions of nonlinear SDEs are diffi-cult to obtain.Therefore,constructing simple and easily implementable numerical algorithms or appropriate approximation techniques,which can reproduce the dy-namical behavior of nonlinear SDEs by numerical solutions,has great theoretical significance and wide application value.This dissertation investigates the problems of explicit approximations for two types of nonlinear SDEs;the strong convergence and rate between the numerical solutions and exact solutions in finite horizons are analysed,moreover,the approximations of long-time dynamical properties for con-tinuous systems including the moment boundedness,stability and ergodicity are studied.Furthermore,the applications of nonlinear switching diffusion systems(S-DSs)on the Gilpin-Ayala model are studied.It examines the impact of stochastic factors(white noises and color noises)on long-term dynamical behaviors.The whole dissertation consists of the following chapters:Chapter 1 briefly introduces the research status of the numerical methods for nonlinear SDEs and research progress of the applications on the Gilpin-Ayala model.The outline and contribution of this dissertation are also listed in this chapter.Chapter 2 studies the explicit approximations of nonlinear SDEs for a wider range of Lyapunov functions.Using the idea of truncation and the Lyapunov func-tions,the growth rate of drift coefficient and diffusion coefficient is estimated,and the truncation mapping is constructed to modify the value of Euler-Maruyama(EM)numerical solutions at each grid point.Thus,the excessive deviation between the nonlinear term and white noise term driven by Brown motion is corrected,so that the Khasminski’s theorem can be applied to the discrete numerical solution-s.And the strong convergence and rate in finite horizons are established.Based on this,using the nonnegative semi-martingale convergence theorem,the discrete stochastic LaSalle-type theorem is obtained,which makes numerical solutions repro-duce almost sure stability of the original system.Some examples and simulations are provided to support the theoretical results and demonstrate the validity of the approach.Chapter 3 studies the explicit approximations for nonlinear SDEs modulated by a Markov chain.Truncated EM schemes depending on the Markov state and easily implementing are constructed,the strong convergence between the numeri-cal solutions and exact solutions in finite horizons is provided,and the(1/2)-order rate of convergence is obtained.On these bases,for long-time dynamical prop-erties of nonlinear SDSs,the value of EM numerical solutions at each grid point is further modified,so that numerical methods are more suitable for the analysis of long-time dynamical properties,not only the asymptotic moment boundedness,moment exponential stability and almost sure stability of numerical solutions,but also the existence and uniqueness of a numerical invariant measure are proved,and the convergence between the invariant measure and underlying invariant measure is obtained.In the end,some examples and simulations are provided to support the theoretical results and demonstrate the validity of the approach.Chapter 4 mainly considers the application of nonlinear SDEs modulated by a Markov chain in ecosystem,the dynamical properties of the stochastic Gilpin-Ayala model with Markov switching are studied.Using the Fredholm alternative theorem,the Lyapunov functions depending on the Markov state are constructed,and asymptotic properties are analysed.We give necessary and sufficient condi-tions for stochastic permanence or extinction of the stochastic Gilpin-Ayala model with Markov switching,and simultaneously obtain ergodicity of the model.These results explain the phenomena of recurrence for the population and reveal the ran-dom switching can suppress the impermanence in ecosystem.Furthermore,several examples and simulations are given to illustrate our main results.