Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

AN AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS DRIVEN BY TIME-CHANGED LéVY NOISE

In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.

Fixed Points and Asymptotic Properties of Neutral Stochastic Delay Differential Equations

This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such an equation is pth moment asymptotically stable. These conditions do not require the boundedness of delays, nor derivation of delays. An example was also given for illustration.

Global Solutions and Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.

Suppressive Influence of Time- Space White Noise on the Explosion of Solutions of Stochastic Fokker- Planck Delay Differential Equations

It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.

Mean Square Stability of the Composite Milstein Method for Nonlinear Stochastic Differential Delay Equations

In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.

Issues in the Influence of Ito-type Noise on the Oscillation of Solutions of Delay Differential Pantograph Equations

In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of the new stochastic delay differential pantograph equation （SDDPE） is investigated. It is established that under certain conditions and with a highly positive probability, the new stochastic delay differential pantograph equation has an oscillatory solution influenced by the presence of the noise. This is not possible with the original deterministic system which has a non-oscillatory solution due to the absence of noise.

Asymptotic Analysis of a Stochastic Model of Mosquito-Borne Disease with the Use of Insecticides and Bet Nets

Ross’ epidemic model describing the transmission of malaria uses two classes of infection, one for humans and one for mosquitoes. This paper presents a stochastic extension of a deterministic vector-borne epidemic model based only on the class of human infectious. The consistency of the model is established by proving that the stochastic delay differential equation describing the model has a unique positive global solution. The extinction of the disease is studied through the analysis of the stability of the disease-free equilibrium state and the persistence of the model. Finally, we introduce some numerical simulations to illustrate the obtained results.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations with Markovian Switching

The aim of this paper is to the discussion of the exponential stability of a class of impulsive neutral stochastic functional differential equations with Markovian switching.Under the influence of impulsive disturbance,the solution for the system is discontinuous.By using the Razumikhin technique and stochastic analysis approaches,as well as combining the idea of mathematical induction and classification discussion,some sufficient conditions for the pth moment exponential stability and almost exponential stability of the systems are obtained.The stability conclusion is full time-delay.The results show that impulse,the point distance of impulse and Markovain switching affect the stability for the system.Finally,two examples are provided to illustrate the effectiveness of the results proposed.

Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lvy processes

In this paper, we study the stochastic maximum principle for optimal control prob- lem of anticipated forward-backward system with delay and Lovy processes as the random dis- turbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L^vy processes （AFBSDEDLs）, we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs＇ preliminary result with certain classical convex variational techniques, the corresponding maxi- mum principle is proved.

MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.

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 differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptoticstability in the pth mean and the pth moment exponential stability of such equations. Finally, an example isgiven to illustrate the effectiveness of our results.

The Existence and Uniqueness of the Solution for Neutral Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to establish the existence-and-uniqueness theorem for neutral stochastic functional differential equations with infinite delay at phase space BC（（-∞, 0]; R^n） An example is given for illustration.

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.

General Decay Pathwise Stability of Neutral Stochastic Differential Equations with Unbounded Delay

Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.

MEAN SQUARE STABILITY AND DISSIPATIVITY OF SPLIT- STEP THETA METHOD FOR NONLINEAR NEUTRAL STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH POISSON JUMPS

In this paper, a split-step 0 （SST） method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps （NSDDEwPJ）. The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with ∈ E （0, 2 -√2） is asymptotically mean square stable for all positive step sizes, and the SST method with ∈ E （2 -√2, 1） is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

NUMERICAL SOLUTIONS OF NONAUTONOMOUS STOCHASTIC DELAY DIFFERENTIAL EQUATIONS BY DISCONTINUOUS GALERKIN METHODS

This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential equations.We prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square sense.In addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations.Numerical tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.