变时滞随机微分方程的指数稳定性

通过构建李雅普偌夫函数和利用半鞅收敛定理,对一类随机变时滞微分方程的全局指数稳定进行了分析,提出了易于判定随机变时滞微分方程几乎必然指数稳定件新的代数判据,推广了现有文献中的主要结论,并给出实例加以验证。     

自治标量随机微分方程混合欧拉格式的收敛性和稳定性

随机微分方程广泛地出现于经济学、生物学、物理学、电子、无线电通讯等领域,所以研究随机微分方程的解是十分必要的。由于随机微分方程的解析解求解困难,其数值方法的研究越来越引起人们的重视。对于求解随机微分方程的数值方法,衡量其有效性的标准是收敛性和稳定性。本文证明混合欧拉格式用于求解自治标量随机微分方程时,在方程的偏移系数和扩散系数均满足线性增长条件和全局Lipschitz条件时的收敛性,并且求出了局部均值收敛阶和均方强收敛阶。接着讨论了两种试验方程混合欧拉格式的稳定性。     

关于倒向随机微分方程解的一点注记

本注记在一定条件下证明了倒向随机微分方程(简记为BSDE)的解满足时齐性,并给出其在金融市场中的解释.     

Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.     

Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps

By introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.     

B空间中无限时滞随机泛函微分方程解的存在唯一性

本文研究抽象空间B中无限时滞随机泛函微分方程解的存在唯一性,在弱化的线性增长条件和一致Lipschitz条件下,得到无限时滞随机泛函微分方程在区间[0,∞)上存在唯一解,进而,得到近似解与精确解之间的误差估计。     

An Effective Numerical Method for the Solution of a Stochastic Coronavirus (2019-nCovid) Pandemic Model

Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model. The comparison of the results of deterministic and stochastic models is also presented. Our proposed efficient computational method well preserves the essential properties of the model. Comparison and convergence analyses of the method are presented.     

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.     

A Reliable Stochastic Numerical Analysis for Typhoid Fever Incorporating With Protection Against Infection

In this paper, a reliable stochastic numerical analysis for typhoid fever incorporating with protection against infection has been considered. We have compared the solutions of stochastic and deterministic typhoid fever model. It has been shown that the stochastic typhoid fever model is more realistic as compared to the deterministic typhoid fever model. The effect of threshold number T* hold in stochastic typhoid fever model. The proposed framework of the stochastic non-standard finite difference scheme (SNSFD) preserves all dynamical properties like positivity, bounded-ness and dynamical consistency defined by Mickens, R. E. The stochastic numerical simulation of the model showed that increase in protection leads to low disease prevalence in a population.     

一类脉冲型随机控制的平稳问题

本文研究一类状态由随机微分方程确定的脉冲型随机控制的平稳问题,证明了最佳脉冲控制的存在性并得以具体刻画。     

股价服从跳-扩过程证券组合的随机微分对策

在服价服从跳—扩过程时，同时考虑流通性这一因素水平，研究两人零和随机微分对策问题，在采用对数效用时分别获得了投资者的最优投资策略。     

基于离散差分进化算法的随机车辆路径问题

针对差分进化算法求解组合优化问题存在的局限性,引入计算机语言中的2种按位运算符,对差分进化算法的变异算子进行重新设计,用来求解不确定需求和旅行时间下同时取货和送货的随机车辆路径问题（SVRPSPD）。通过对车辆路径问题的benchmark问题和SVRPSPD问题进行路径优化,并同差分进化算法和遗传算法的计算结果进行比较,验证了离散差分进化算法的性能。结果表明,离散差分进化算法在解决复杂的SVRPSPD问题时,具有较好的优化性能,不仅能得到更好的优化结果,而且具有更快的收敛速度。     

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

Optimal Production Control in Stochastic Manufacturing Systems with Degenerate Demand

The paper studies the production inventory problem of minimizing the expected discounted present value of production cost control in manufacturing systems with degenerate stochastic demand. We have developed the optimal inventory production control problem by deriving the dynamics of the inventory-demand ratio that evolves according to a stochastic neoclassical differential equation through Ito’s Lemma. We have also established the Riccati based solution of the reduced (one-dimensional) HJB equation corresponding to production inventory control problem through the technique of dynamic programming principle. Finally, the optimal control is shown to exist from the optimality conditions in the HJB equation.     

伊藤型模糊随机微分方程

提出了伊藤(Ito)型模糊随机微分方程的概念,证明了其解的存在唯一性,给出伊藤型线性模糊随机微分方程解的表达式和统计特征方程,并通过例子说明了解法。     

On Optimal Cash Management under a Stochastic Volatility Model

We discuss a mathematical model for optimal cash management. A firm wishes to manage cash to meet demands for daily operations, and to maximize terminal wealth via bank deposits and stock investments that pay dividends and have uncertain capital gains. A Stochastic Volatility (SV) model is adopted for the capital gains rate of a stock, providing a more realistic way to describe its price dynamics. The cash management problem is formulated as a stochastic optimal control problem, and solved numerically using dynamic programming. We analyze the implications of the heteroscedasticity described by the SV model for evaluating risk, by comparing the terminal wealth arising from the SV model to that obtained from a Constant Volatility (CV) model.     

一类耦合Burgers-KdV方程差分解的收敛性与稳定性

针对耦合KdV方程的周期边值问题建立了全离散两层加权中心差分格式,得到了差分解的模估计,证明了差分解的存在性、收敛性和稳定性,并且得到了显格式和弱隐格式对于步长τ和h的限制条件。     

半线性时滞模糊微分方程解的存在性

模糊微分方程常用来模拟不确定条件下的变化过程．其理论被广泛的应用在很多不同的实际问题上．本文主要研究半线性时滞模糊微分方程解的存在性．在广义Hukuhara导数的框架下,运用了一个弱压缩映射的不动点定理,将半线性时滞模糊微分方程解的存在性问题转化为算子不动点的存在性问题,从而建立了方程解的存在性定理．文中例子说明了该理论结果的实用性．     

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations. Received: May 26, 2000; revised version: September 6, 2000