Principle For Stochastic Differential Equations Research Articles

Generalized outer synchronization and parameters identification of uncertain delayed hypernetworks with stochastic noises

AbstractSynchronization of hypernetworks has been extensively researched due to its potential applications in various contexts, such as secure communications and power grid engineering. However, the existing works thus far have primarily been devoted to inner synchronization or complete outer synchronization with identical node dynamics and topology structures. To investigate harmonious coexistence between two networks with nonidentical node dynamics and reduce control costs as well, this paper focuses on generalized outer synchronization (GOS) and parameters identification of uncertain delayed hypernetworks with stochastic noises. Regardless of whether the node parameters are known or not, based on Lyapunov stability and the LaSalle‐type invariance principle for stochastic differential equations with multiple delays, some sufficient conditions are exhibited to reach GOS almost surely via adaptive control schemes. Moreover, the unknown node parameter vectors are identified by the designed updating laws. In addition, some corollaries are deduced and indicate that our proposed methods contain single‐layer networks and complete synchronization methods as special cases. Furthermore, numerical simulations are provided to verify the effectiveness of our approaches. In particular, it is revealed that network parameters, including time delays, coupling strengths, and noise intensities, have an impact on the performance of the proposed algorithms.

Time-averaging principle for G-SDEs based on Lyapunov condition

In this paper, we tame the uncertainty about the volatility in time-averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) based on the Lyapunov condition. That means we treat the time-averaging principle for stochastic differential equations based on the Lyapunov condition in the presence of a family of probability measures, each corresponding to a different scenario for the volatility. The main tool for the mathematical analysis is the G-stochastic calculus, which is introduced in the book by Peng (Nonlinear Expectations and Stochastic Calculus Under Uncertainty. Springer, Berlin, 2019). We show that the solution of a standard equation converges to the solution of the corresponding averaging equation in the sense of sublinear expectation with the help of some properties of G-stochastic calculus. Numerical results obtained using PYTHON illustrate the efficiency of the averaging method.

On the averaging principle for stochastic differential equations driven by [formula omitted]-Lévy process

In this paper, we investigate the averaging principle for stochastic differential equation driven by G-Lévy process. By the BDG inequality for G-stochastic calculus with respect to G-Lévy process, we show that the solution of averaged stochastic differential equation driven by G-Lévy process converges to that of the standard one, under non-Lipschitz condition, in the mean square sense and also in capacity. An example is presented to illustrate the efficiency of the obtained results.

A moderate deviation principle for stochastic Hamiltonian systems

We prove a moderate deviation principle for stochastic differential equations (SDEs) with non-Lipschitz conditions. As an application of our result, we also study the stochastic Hamiltonian systems.

Deterministic and stochastic evolution of rumor propagation model with media coverage and class‐age‐dependent education

Considering the media coverage and age‐dependent education, a deterministic and a stochastic class‐age‐structured rumor propagation models are studied, respectively. First, the deterministic rumor propagation model is characterized by a coupled system of ordinary and partial differential equations. The positivity and boundness of solutions are proved, and the basic reproduction number is derived. Second, the stochastic rumor propagation model is formulated by stochastic differential equations. The existence of global positive solutions in model is discussed with the Itô's formula and stochastic Lyapunov function. Additionally, by utilizing comparison principle of stochastic differential equations and the strong law of large numbers, several sufficient conditions for extinction and persistence of the rumor are derived. Finally, numerical simulations are carried out for illustrating the results.

On the averaging principle for stochastic differential equations involving Caputo fractional derivative

In this paper, we investigate the averaging principle for Caputo-type fractional stochastic differential equations driven by Brownian motion. Different from the approach of integration by parts or decomposing integral interval to deal with the estimation of integral involving singular kernel in the existing literature, we show the desired averaging principle in the sense of mean square by using Hölder inequality via growth conditions on the nonlinear stochastic term. Finally, a simulation example is given to verify the theoretical results.

The averaging principle for stochastic differential equations driven by a Wiener process revisited

We consider a one-dimensional stochastic differential equation driven by a Wiener process, where the diffusion coefficient depends on an ergodic fast process. The averaging principle is satisfied: it is well-known that the slow component converges in distribution to the solution of an averaged equation, with generator determined by averaging the square of the diffusion coefficient.

On the Averaging Principle For Stochastic Differential Equations Driven by G-Levy Process

On the Averaging Principle For Stochastic Differential Equations Driven by G-Levy Process

Averaging principle for stochastic differential equations with monotone condition

In this paper, we consider the averaging principle for a class of stochastic differential equations (SDEs) with the nonlinear terms only satisfying the local Lipschitz and monotone conditions. Compared with the classical Lipschitz and linear growth conditions, conditions given in this paper are relatively weak, so the conclusions obtained in this paper are suitable for a wider class of SDEs.

Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs

In this paper, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to a fully nonlinear degenerate PDEs. Our approach is mainly based on G-stochastic analysis argument. As a byproduct, we also establish the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) with two time-scales. The results extend Khasminskii’s averaging principle to nonlinear case.

Averaging principle for stochastic differential equations in the random periodic regime

We present the validity of stochastic averaging principle for non-autonomous slow–fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. The ergodicity of time periodic measures of the fast motion on certain minimal Poincaré section is used to identify the averaging limit.

On the averaging principle for SDEs driven by G-Brownian motion with non-Lipschitz coefficients

In this paper, we aim to develop the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs for short) with non-Lipschitz coefficients. By the properties of G-Brownian motion and stochastic inequality, we prove that the solution of the averaged G-SDEs converges to that of the standard one in the mean-square sense and also in capacity. Finally, two examples are presented to illustrate our theory.

Averaging principle for stochastic differential equations under a weak condition.

The aim of this paper is to establish the averaging principle for stochastic differential equations under a general averaging condition, which is weaker than the traditional case. Under this condition, we establish an effective approximation for the solution of stochastic differential equations in mean square.

Large deviations for stochastic differential equations with general delayed generator

Abstract This paper is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].

Large deviation principle for SDEs with Dini continuous drifts

ABSTRACT In this paper, using Zvonkin-type transform, the large deviation principle is proved for stochastic differential equations (SDEs) with Dini continuous drifts, where the existed methods for large deviation principle are unavailable. The method and result are new in related fields. Moreover, the result is also extended to a class of degenerate SDEs with Dini continuous drifts.

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time t and ω. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.

The averaging principle for stochastic differential equations with Caputo fractional derivative

This paper presents an averaging principle for Caputo fractional stochastic differential equations (FSDEs) driven by Brown motion. Under some assumptions, the solutions to FSDEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. The analyses of solutions to systems before and after averaging, allow to extend the classical Khasminskii approach to Caputo fractional stochastic equations.

Weak order in averaging principle for stochastic differential equations with jumps

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence.

Synchronization of Complex Stochastic Coupled Systems Based on Graph Theory

Based on the LaSalle invariant principle of stochastic differential equations and the graph theory, several sufficient conditions are obtained by adaptive feedback control and impulsive control. The error dynamical system is globally almost surely asymptotically stable. The conclusion is new and feasible. Finally, an example is given to show the validity of the results.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.