Explicit Approximations Of Nonlinear Stochastic Functional Differential Equations

Time delays are omnipresent and entrenched in real systems.In order to describe the world more accurately,functional differential equations have a wide range of emerging and existing applications in the fields of physics,biology,medical sciences,automatic control.It is unavoidable that systems in nature are often perturbed by random interference from inside or outside.Stochastic functional differential equations as mathematical models are thus more realistic.Stochastic functional differential equations valued in the infinite-dimensional space can hardly be solved explicitly.Therefore,it is extremely important in theories and applications to construct reliable and easily implementable numerical algorithms for stochastic functional differential equations.This dissertation mainly investigates explicit approximations of nonlinear stochastic functional differential equations driven by a Brownian motion,including the construction of numerical schemes,convergence,convergence rate,and how the numerical solutions preserve the long-time behaviors of exact ones.The whole dissertation consists of the following chapters:Chapter 1 briefly introduces some applications of stochastic functional differential equations,research progress of the corresponding numerical methods,and lists the outline and contributions of this dissertation.Chapter 2 focuses on the explicit approximation for nonlinear stochastic delay differential equations.Utilizing the inherent characteristic and borrowing the truncated idea,the explicit truncated Euler-Maruyama(EM)scheme is proposed,and the boundedness and convergence in the pth moment are obtained.Furthermore,the 1/2 order optimal convergence rate is proved.Chapter 3 further studies the explicit numerical methods for approximating the long-time asymptotic behaviors of nonlinear stochastic delay differential equations.Taking advantage of the truncated idea we construct the more precise explicit truncated EM scheme,and prove the exponential stability in mean square and with probability 1 of numerical solutions.Furthermore,borrowing the linear interpolation we construct a continuous function-valued explicit truncated EM linear interpolation segment process.Making use of the uniform boundedness and the attraction of the numerical segment processes in probability,we go a further step to prove the existence and uniqueness of the numerical invariant measure.Furthermore,we show that the numerical invariant measure converges to the exact one in the FortetMourier distance as the step size tends to zero.Chapter 4 investigates the explicit numerical method for approximating nonlinear stochastic functional differential equations.Precisely,the explicit truncated EM scheme suitable for nonlinear stochastic functional differential equations is constructed.The pth moment boundedness and convergence of numerical solutions are obtained(p≥2).Moreover,the 1/2 order optimal convergence rate is proved.The global Lipschitz restriction on the diffusion coefficient is released.Compared to solving nonlinear implicit equations in the infinite-dimensional space,the computational cost of this explicit truncated EM scheme is low.Chapter 5 uses the numerical solutions constructed by Chapter 4 to approximate the exponential stability in Lp and with probability 1 of exact ones(p≥2)for nonlinear stochastic functional differential equations.