Stochastic Delay Differential Equations Research Articles

Stability in terms of two measures of solutions to stochastic partial differential delay equations with switching

In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed‐index sequence monotonicity condition and under the average dwell‐time switching are considered.

Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler–Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Time-delayed feedback control of diffusion in random walkers.

Time delay in general leads to instability in some systems, while specific feedback with delay can control fluctuated motion in nonlinear deterministic systems to a stable state. In this paper, we consider a stochastic process, i.e., a random walk, and observe its diffusion phenomenon with time-delayed feedback. As a result, the diffusion coefficient decreases with increasing delay time. We analytically illustrate this suppression of diffusion by using stochastic delay differential equations and justify the feasibility of this suppression by applying time-delayed feedback to a molecular dynamics model.

Optimal Control Problem for Risk‐Sensitive Mean‐Field Stochastic Delay Differential Equation with Partial Information

AbstractThis paper deals with the risk‐sensitive control problem for mean‐field stochastic delay differential equations (MF‐SDDEs) with partial information. Firstly, under the assumptions that the control domain is not convex and the value function is non‐smooth, we establish a stochastic maximum principle (SMP). Then, by means of Itô's formula and some continuous dependence, we prove the existence and uniqueness results for another type of MF‐SDDEs. Meanwhile, the verification theorem for the MF‐SDDEs is obtained by using a clever construction of the Hamiltonian function. Finally, based on our verification theorem, a linear‐quadratic system is investigated and the optimal control is also derived by the stochastic filtering technique.

Multivalued stochastic delay differential equations and related stochastic control problems

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):Specify that in this case the coefficients at time t depends also on previous values of X(t) through Y (t) and Z(t). Also X is constrained with the help of a bounded variation feedback law K to stay in the convex set .Afterwards we consider optimal control problems where the state X is a solution of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.

Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion

In this paper, we prove an existence and uniqueness result of mild solution for a stochastic delay differential equation in a Hilbert space driven by a fractional Brownian motion with the Hurst parameter H>1∕2 and with a non-deterministic diffusion coefficient. We also prove under a sufficient condition that the law of the norm of the solution admits a density with respect to Lebesgue measure on R.

Tamed EM scheme of neutral stochastic differential delay equations

In this paper, we investigate the convergence of the tamed Euler-Maruyama (EM) scheme for a class of neutral stochastic differential delay equations. The strong convergence results of the tamed EM scheme are presented under global and local non-Lipschitz conditions, respectively. Moreover, under a global Lipschitz condition, we provide the rate of the convergence of tamed EM, which is smaller than the rate convergence of classical EM scheme one half.

Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations with Poisson Jumps

Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations with Poisson Jumps

Strong convergence of the split-step theta method for neutral stochastic delay differential equations

Neutral stochastic delay differential equations often appear in various fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step theta (SST) method for the neutral stochastic delay differential equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the SST method with θ∈[0,1] strongly converges to the exact solution with the order 12. Some numerical results are presented to confirm the obtained results.

Delay dependent stability of highly nonlinear hybrid stochastic systems

There are lots of papers on the delay dependent stability criteria for differential delay equations (DDEs), stochastic differential delay equations (SDDEs) and hybrid SDDEs. A common feature of these existing criteria is that they can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In other words, there is so far no delay-dependent stability criterion on nonlinear equations without the linear growth condition (we will refer to such equations as highly nonlinear ones). This paper is the first to establish delay dependent criteria for highly nonlinear hybrid SDDEs. It is therefore a breakthrough in the stability study of highly nonlinear hybrid SDDEs.

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

This paper investigates an optimal control of an infinite horizon system governed by mean-field backward stochastic differential equation with delay and partial information. Firstly, we establish the existence and uniqueness results for a mean-field backward stochastic differential equation (BSDE) with average delay. Then a class of mean-field time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint equations via duality relation. Meanwhile, necessary and sufficient conditions for optimal control under partial information on infinite horizon are derived. Finally, we apply the theoretical results to study linear-quadratic control problem on infinite horizon to obtain the optimal control, which is explicitly expressed by the solution of a mean-field forward-backward stochastic differential filtering equation.

Caratheodory's approximate solution to stochastic differential delay equation

Caratheodory's approximate solution to stochastic differential delay equation

Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations

ABSTRACTIn this paper, we are interested in numerical methods with variable step-size for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. The problem of computer memory hold, when the numerical methods with constant step-size are applied to the SPDEs. In this work, we construct split-step theta (SSθ) methods with variable step-size for SPDEs. The boundedness and strong convergence of the numerical methods are investigated under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. It is proved that, the SSθ methods with variable step-size for converge strongly to the exact solution. In addition, the strong order 0.5 is given under mild assumptions. The mean-square stability (MS-Stability) of the numerical methods with is given. Finally, some illustrative numerical examples are presented to show the efficiency of the methods, and how MS-Stability of SSθ methods depends on the parameter theta for both linear and nonlinear models.

Instability in time-delayed switched systems induced by fast and random switching

In this paper, we consider a switched system comprising finitely or infinitely many subsystems described by linear time-delayed differential equations and a rule that orchestrates the system switching randomly among these subsystems, where the switching times are also randomly chosen. We first construct a counterintuitive example where even though all the time-delayed subsystems are exponentially stable, the behaviors of the randomly switched system change from stable dynamics to unstable dynamics with a decrease of the dwell time. Then by using the theories of stochastic processes and delay differential equations, we present a general result on when this fast and random switching induced instability should occur and we extend this to the case of nonlinear time-delayed switched systems as well.

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an \(L^2\)-valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman–Kac representation theorem under mild assumptions of differentiability.

Moderate deviations for neutral stochastic differential delay equations with jumps

A moderate deviation principle for neutral stochastic differential delay equations driven by Poisson random measure is established. The weak convergence method introduced by Budhiraja et al. (2016) plays a key role.

Governing equations for Probability densities of stochastic differential equations with discrete time delays

The time evolution of probability densities for solutions to stochastic differential equations (SDEs) without delay is usually described by Fokker-Planck equations, which require the adjoint of the infinitesimal generator for the solutions. However, Fokker-Planck equations do not exist for stochastic delay differential equations (SDDEs) since the solutions to SDDEs are not Markov processes and have no corresponding infinitesimal generators. In this paper, we address the open question of finding governing equations for probability densities of SDDEs with discrete time delays. In the governing equation, densities for SDDEs with discrete time delays are expressed in terms of those for SDEs without delay. The latter have been well studied and can be obtained by solving the corresponding Fokker-Planck equations. The governing equation is given in a simple form that facilitates theoretical analysis and numerical computation. Some example are presented to illustrate the proposed governing equations.

Portfolio Optimization Problem with Delay under Cox-Ingersoll-Ross Model

This paper considers a portfolio optimization problem with delay. The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model. In addition, considering the history information related to investment performance, the dynamic of wealth is modeled by stochastic delay differential equation. The investor’s objective is to maximize her expected utility for a linear combination of the terminal wealth and the average performance. By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closed-form expressions of optimal strategy and optimal value function for CRRA utility are derived. Finally, a numerical example is provided to show our results.

Error Analysis for D-Leaping Scheme of Chemical Reaction System with Delay

We perform an error analysis in both strong and weak senses for D-leaping scheme of chemical reactions with delays within the framework of stochastic delay differential equations (SDDEs). In order ...

Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps

In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.