Split-step Adams–Moulton Milstein methods for systems of stiff stochastic differential equations

In this paper we discuss new split-step methods for solving systems of Itô stochastic differential equations (SDEs). The methods are based on a L-stable (strongly stable) second-order split Adams–Moulton Formula for stiff ordinary differential equations in collusion with Milstein methods for use on SDEs which are stiff in both the deterministic and stochastic components. The L-stability property is particularly useful when the drift components are stiff and contain widely varying decay constants. For SDEs wherein the diffusion is especially stiff, we consider balanced and modified balanced split-step methods which posses larger regions of mean-square stability. Strong order convergence one is established and stability regions are displayed. The methods are tested on problems with one and two noise channels. Numerical results show the effectiveness of the methods in the pathwise approximation of stiff SDEs when compared to some existing split-step methods.     

Five-stage Milstein methods for SDEs

In this paper, we consider the problem of computing numerical solutions for Itô stochastic differential equations (SDEs). The five-stage Milstein (FSM) methods are constructed for solving SDEs driven by an m-dimensional Wiener process. The FSM methods are fully explicit methods. It is proved that the FSM methods are convergent with strong order 1 for SDEs driven by an m-dimensional Wiener process. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the methods proposed in this paper are larger than the Milstein method and three-stage Milstein methods.     

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

This paper deals with the mean-square (MS) stability of the Euler–Maruyama method for stochastic differential delay equations (SDDEs) with jumps. First, the definition of the MS-stability of numerical methods for SDDEs with jumps is established, and then the sufficient condition of the MS-stability of the Euler–Maruyama method for SDDEs with jumps is derived, finally a class scalar test equation is simulated and the numerical experiments verify the results obtained from theory.     

Stability of Stochastic \theta -Methods for Stochastic Delay Hopfield Neural Networks Under Regime Switching

This paper is concerned with the general mean-square (GMS) stability and mean-square (MS) stability of stochastic $\theta $ -methods for stochastic delay Hopfield neural networks under regime switching. The sufficient conditions to guarantee GMS-stability and MS-stability of stochastic $\theta $ -methods are given. Finally, an example is used to illustrate the effectiveness of our result.     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

Stability analysis and stabilization of networked linear systems with random packet losses

This paper is concerned with the stability analysis and stabilization of networked discrete-time and sampled-data linear systems with random packet losses. Asymptotic stability, mean-square stability, and stochastic stability are considered. For networked discrete-time linear systems, the packet loss period is assumed to be a finite-state Markov chain. We establish that the mean-square stability of a related discrete-time system which evolves in random time implies the mean-square stability of the system in deterministic time by using the equivalence of stability properties of Markovian jump linear systems in random time. We also establish the equivalence of asymptotic stability for the systems in deterministic discrete time and in random time. For networked sampled-data systems, a binary Markov chain is used to characterize the packet loss phenomenon of the network. In this case, the packet loss period between two transmission instants is driven by an identically independently distributed sequence assuming any positive values. Two approaches, namely the Markov jump linear system approach and randomly sampled system approach, are introduced. Based on the stability results derived, we present methods for stabilization of networked sampled-data systems in terms of matrix inequalities. Numerical examples are given to illustrate the design methods of stabilizing controllers.     

State feedback control of linear systems in the presence of devices with finite signal-to-noise ratio

This paper provides necessary and sufficient conditions for mean-square state-feedback stabilization of linear systems whose white noise sources have intensities affinely related to the variance of the signal they corrupt. Systems with such noise sources have been called FSN (finite signal-to-noise) models, and the stability results provided in prior work were only sufficient conditions. Upper bounded ?₂ performance is also guaranteed herein by solving a control problem which is non-convex only due to a certain scaling parameter. By fixing this parameter convex programming algorithms provide controllers.     

Stability analysis of explicit multirate methods

If stiff differential systems can be divided into stiff and nonstiff subsystems, each subsystem can be integrated with different step sizes. These methods are called multirate methods, and they have been successfully used in several practical problems, especially in real-time simulations.In this paper a new way to perform the stability analysis of multirate versions of linear multistep methods is presented. This stability analysis makes use of the multirate z-transform method.     

均方差图象匹配系统的局部精度

本文用均方可微的二维齐次高斯随机场模型描述图象信号和噪声.在信号和噪声相互独 立且噪声的数学期望为零等条件下,证明了均方差图象匹配系统的三个重要性质,导出了均方 差匹配系统局部精度的解析表达式,为系统的精度分析和计算提供了理论和应用基础.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.     

Parallel Rosenbrock methods for chemical systems

Parallel Rosenbrock methods are developed for systems with stiff chemical reactions. Unlike classical Runge-Kutta methods, these linearly implicit schemes avoid the necessity to iterate at each time step. Parallelism across the method allows for the solution of the linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. In addition to possessing excellent stability properties, these methods are computationally efficient while preserving positivity of the solutions. Numerical results confirm these characteristics when applied to problems involving stiff chemistry, and enzyme kinetics.     

Convergence and stability of the balanced methods for stochastic differential equations with jumps

This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.     

Approximation of jump diffusions in finance and economics

In finance and economics the key dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps. Strong schemes provide pathwise approximations and therefore can be employed in scenario analysis, filtering or hedge simulation. Weak schemes are appropriate for problems such as derivative pricing or the evaluation of risk measures and expected utilities. Here only an approximation of the probability distribution of the jump-diffusion process is needed. As a framework for applications of these methods in finance and economics we use the benchmark approach. Strong approximation methods are illustrated by scenario simulations. Numerical results on the pricing of options on an index are presented using weak approximation methods.     

The stability of a proportional rate extremum regulator

When the signal component of the input to a linear unity feedback sampled-data system is additively contaminated by a relatively wide-band noise process, it has been shown previously that the mean-square values of the true and apparent error quantities are minimized by essentially the same value of loop gain. This paper presents an investigation into the stability of a proportional-rate extremum regulator, which utilizes this fact to establish the optimum gain setting from finite estimates of the measurable mean-square apparent error. Assuming a parabolic operating characteristic, the stability and dynamic behavior of the adaptive loop is studied using path tangent curves in the incremental phase plane. The ability of the regulator to establish the optimum gain setting for a given system is verified experimentally for a number of different operating conditions. In addition, the experimental results show that the transient disturbances generated by the corrections to the gain parameter may adversely affect the regulator stability if they exist over a sufficient fraction of the measurement period.     

Practical estimation of high dimensional stochastic differential mixed-effects models

Stochastic differential equations (SDEs) are established tools for modeling physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE, intrinsic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects modelling allows for the distinction of between and within experiment variability. A framework for modeling dynamics within a population using SDEs is proposed, representing simultaneously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics, using SDEs. These stochastic differential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models.     

Monotone Finite Volume Schemes for Diffusion Equation with Imperfect Interface on Distorted Meshes

In this paper we consider the numerical solution of stiff problems in which the eigenvalues are separated into two clusters, one containing the “stiff”, or fast, components and one containing the slow components, that is, there is a gap in their eigenvalue spectrum. By using exponential fitting techniques we develop a class of explicit Runge–Kutta methods, that we call stability fitted methods, for which the stability domain has two regions, one close to the origin and the other one fitting the large eigenvalues. We obtain the size of their stability regions as a function of the order and the fitting conditions. We also obtain conditions that the coefficients of these methods must satisfy to have a given stiff order for the Prothero–Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments.     

Stability of nonequidistant variable order multistep methods for stiff systems

The order, stability and convergence of the nonequidistant variable order multistep methods (NVOMMs) for stiff systems are discussed. The stability properties of certain class of variable step multistep methods (VSMMs) depending on one free parameter β^* for some order will be discussed, some theorems and lemmas are proved. New effective techniques restricted by larger interval of β^* to get strongly stable methods are determined     

(ρ, σ)-方法关于刚性延迟微分代数系统的非线性稳定性

本文涉及（ρ,σ）－方法应用于1－0指标的非线性刚性延迟微分代数系统的稳定性,证明了求常微分方程（ODEs）的（ρ,σ）－方法的强（G）（c,p,q）－代数稳定性导致相应延迟微分代数系统方法的（渐近）整体稳定性。     

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler–Taylor expansion and stochastic Runge–Kutta type.     

Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ∈ [0, 1]. Since θ-method covers the commonly used Euler–Maruyama (EM) method and the backward Euler–Maruyama (BEM) method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations (SDDEs) with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.