非Lipschitz系数McKean-Vlasov随机微分方程的大偏差原理

本文研究有限维框架下一类非Lipschitz系数McKean-Vlasov随机微分方程的Freidlin-Wentzell型大偏差原理,将此类条件下经典随机微分方程的相关结论推广到McKean-Vlasov随机微分方程.在此类McKean-Vlasov随机微分方程解的存在唯一性基础上,采用弱收敛方法得到其大偏差原理.     

一类随机延迟微分方程截断型theta-EM算法的收敛性

本文主要考察随机延迟微分方程截断型theta-EM (Euler-Maruyama)算法的强收敛性问题.将截断型EM算法推广到一般形式,提出截断型theta-EM算法,并讨论随机延迟微分方程在非全局Lipschitz条件下的强收敛率,得到其强收敛阶数.     

Poisson跳的随机延迟微分方程Heun方法的均方收敛性

Heun方法是一类求解随机延迟微分方程的数值方法,本文试图研究Poisson跳的随机延迟微分方程Heun方法的均方收敛性.当Poisson跳的随机延迟微分方程满足一定约束条件时,获得Heun方法求解方程所得的数值解收敛于真解,且均方收敛阶为1的理论结果2.文末数值试验的结果验证了理论结果的正确性.     

随机脉冲随机微分方程解的存在唯一性

提出了随机脉冲随机微分方程模型,其中所谓的随机脉冲是指脉冲幅度由随机变量序列驱动,并且脉冲发生的时间也是一个随机变量序列.因此,随机脉冲随机微分方程是对带跳的随机微分方程模型的推广.利用Gronwall不等式、Lipschtiz条件和随机分析技巧,得到了随机脉冲随机微分方程的解的存在唯一性条件.     

非全局Lipschitz条件下随机延迟微分方程Euler方法的收敛性

大多数随机延迟微分方程数值解的结果是在全局Lipschitz条件下获得的.许多延迟方程不满足全局Lipschitz条件,研究非全局Lipschitz条件下的数值解的性质,具有重要的意义.本文证明了漂移系数满足单边Lipschitz条件和多项式增长条件,扩散系数满足全局Lipschitz条件的一类随机延迟微分方程的Eul...     

非线性中立型随机延迟微分方程随机θ方法的稳定性

本文研究非线性中立型随机延迟微分方程随机θ方法的均方稳定性.在方程解析解均方稳定的条件下,证明了如下结论:当θ∈[0,1/2)时,随机θ方法对于适当小的时间步长是均方稳定的;当θ∈[1/2,1]时,随机θ方法对于任意步长都是均方稳定的.数值结果验证了所获结论的正确性.     

带Lévy跳的中立随机微分方程的EM逼近

本文研究了一类带Lévy跳的中立随机微分方程的Euler近似解的问题.利用Gronwall不等式、H?lder不等式及BDG不等式,在局部Lipschitz和线性增长条件下,本文给出近似解在均方意义下收敛于真实解,推广了带Poisson跳的中立随机微分方程EM逼近结果.     

带跳随机微分方程的Euler-Maruyama方法的几乎处处指数稳定性和矩稳定性

本文首先研究了一维带跳随机微分方程的指数稳定性,并证明Euler-Maruyama(EM)方法保持了解析解的稳定性.其次,研究了多维带跳随机微分方程的稳定性,证明若系数满足全局Lipchitz条件,则EM方法能够很好地保持解析解的几乎处处指数稳定性、均方指数稳定性.最后,给出算例来支持所得结论的正确性.     

带Poisson跳和Markovian调制的中立型随机延迟微分方程的数值解的收敛性

本文研究带Poisson跳和Markovian调制的中立型随机微分方程的数值解的收敛性质.用数值逼近方法求此微分方程的解,并证明了Euler近似解在此线性增长条件和全局Lipschitz条件更弱的条件下仍均方收敛于此方程的解析解.     

随机延迟微分方程平衡方法的均方收敛性与稳定性

本文讨论求解刚性随机延迟微分方程的平衡方法.证明了随机延迟微分方程平衡方法的均方收敛阶为1/2.给出了线性随机延迟微分方程平衡方法均方稳定的条件.     

A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation

In this paper, the Euler–Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations. Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.     

Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

带有弱奇性核的多项分数阶非线性随机微分方程的改进Euler-Maruyama格式

针对一类带有弱奇性核的多项分数阶非线性随机微分方程构造了改进Euler-Maruyama (EM)格式,并证明了该格式的强收敛性.具体地,利用随机积分解的充分条件,将此多项分数阶随机微分方程等价地转化为随机Volterra 积分方程的形式,详细推导出对应的改进EM格式,并对该格式进行了强收敛性分析,其强收敛阶为α_m-α_m-1,其中α_i为分数阶导数的指标,且满足0<α₁<…<α_m-1<α_m<1.最后,通过数值实验验证了理论分析结果的正确性.     

非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂（CEMS）方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.     

Convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay

In this paper, we study the convergence and stability of the stochastic theta method (STM) for a class of index 1 stochastic delay differential algebraic equations. First, in the case of constrained mesh, i.e., the stepsize is a submultiple of the delay, it is proved that the method is strongly consistent and convergent with order 1/2 in the mean-square sense. Then, the result is further extended to the case of non-constrained mesh where we employ linear interpolation to approximate the delay argument. Later, under a sufficient condition for mean-square stability of the analytical solution, it is proved that, when the stepsizes are sufficiently small, the STM approximations reproduce the stability of the analytical solution. Finally, some numerical experiments are presented to illustrate the theoretical findings.     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

线性随机微分方程的离散波形松弛方法的几乎必然收敛性

提出了随机微分方程的离散型波形松弛方法,证明了它是几乎必然收敛的.此外,通过数值实验验证了所得结果.     

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow