Nonlinear Stochastic Differential Delay Equations Research Articles

Escaping the Curse of Dimensionality in Estimating Multivariate Transfer Entropy

Multivariate transfer entropy (TE) is a model-free approach to detect causalities in multivariate time series. It is able to distinguish direct from indirect causality and common drivers without assuming any underlying model. But despite these advantages it has mostly been applied in a bivariate setting as it is hard to estimate reliably in high dimensions since its definition involves infinite vectors. To overcome this limitation, we propose to embed TE into the framework of graphical models and present a formula that decomposes TE into a sum of finite-dimensional contributions that we call decomposed transfer entropy. Graphical models further provide a richer picture because they also yield the causal coupling delays. To estimate the graphical model we suggest an iterative algorithm, a modified version of the PC-algorithm with a very low estimation dimension. We present an appropriate significance test and demonstrate the method's performance using examples of nonlinear stochastic delay-differential equations and observational climate data (sea level pressure).

Delay-dependent exponential stability of the backward Euler method for nonlinear stochastic delay differential equations

Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results.

Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean‐square exponential stability of the exact solution is derived. Then, mean‐square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean‐square stable for any stepsize Δt = τ/m when 1/2 ≤ θ ≤ 1, and they are exponentially mean‐square stable if the stepsize Δt ∈ (0, Δt0) when 0 ≤ θ < 1. Finally, some numerical experiments are given to illustrate the theoretical results.

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

There are few results on the numerical stability of nonlinear neutral stochastic delay differential equations (NSDDEs). The aim of this paper is to establish some new results on the numerical stability for nonlinear NSDDEs. It is proved that the semi-implicit Euler method is mean-square stable under suitable condition. The theoretical result is also confirmed by a numerical experiment.

Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average <x(t)|x(t-τ)>, where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.

Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

This paper deals with the almost sure exponential stability of the Euler-type methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/ q ( q > 1). This result is more general than what they deal with the jump of deterministic magnitude.

A STUDY OF A CLASS OF NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully.

EXISTENCE, UNIQUENESS AND ASYMPTOTIC PROPERTIES OF A CLASS OF NONLINEAR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING

The main aim of this paper is to give some new conditions under which a class of nonlinear stochastic differential delay equations with Markovian switching admit a unique solution and this solution has nice asymptotic properties, including the moment boundedness and the moment boundedness average in time of this solution. These new conditions show that the coefficients of the nonlinear stochastic differential delay equation with Markovian switching are polynomial or controlled by the polynomial functions or functionals. Two examples are also given for illustration.

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero. We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds. We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.

Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations

AbstractWe study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

The semi-implicit Euler method for stochastic differential delay equation with jumps

The class of jump-diffusion SDDEs that admits explicit solutions is rather limited. Consequently, there is a need for the systematic use of discrete time approximations in corresponding simulations. In this paper, we shall deal with convergence of the semi-implicit Euler method for Nonlinear stochastic differential delay equation driven by Wiener processes and Poisson processes. It is proved that the semi-implicit Euler method is convergent with strong order p = 1 2 .

On delay-dependent stability for a class of nonlinear stochastic delay-differential equations

Global asymptotic stability conditions for nonlinear stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.

Oscillation and Non-oscillation in Solutions of Nonlinear Stochastic Delay Differential Equations

This paper studies the oscillation and nonoscillation of solutions of a nonlinear stochastic delay differential equation, where the noise perturbation depends on the current state, and the drift depends on a delayed argument. When the restoring force towards equilibrium is relatively strong, all solutions oscillate, almost surely. However, if the restoring force is superlinear, positive solutions exist with positive probability, and for suitably chosen initial conditions, the probability of positive solutions can be made arbitrarily close to unity.

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differential delay equations with Markovian switching. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Finally, an example is given to illustrate the effectiveness of our results.

Nonlinear stochastic differential delay equations

The methods used by Adomian and his co-workers to solve linear and nonlinear stochastic differential equations will be demonstrated to be applicable to differential equations, deterministic or stochastic, involving delays (constant, time dependent, or random).

Stability of systems with nonlinear feedback through randomly time-varying delays

In a large class of multiloop control systems, many feedback loops are "closed" through a time-shared digital computer, using algorithms which require information from sources which are sampled at a rate which is not synchronized with the sampling of the individual "plants." This missynchronization, in conjunction with variations in the computer's task load caused by "interrupts," results in a randomly time-varying delay in the closing of the various feedback loops. Consequently, the dynamics of each controlled "plant" in such a system may be modeled by means of a stochastic delay-differential equation. This paper presents some new research results concerning the sample stability (as opposed to statistical, or ensemble stability) of nonlinear stochastic delay-differential equations.