Stochastic Delay Equations Research Articles

Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations

Abstract Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

Statistical inference for discretely sampled stochastic functional differential equations with small noise

Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when a small dispersion coefficient ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document} and sample size n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document} simultaneously.

Finite time SF bifurcation and stability analysis for stochastic time-varying delay system

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective.

Robust optimal asset–liability management with delay and ambiguity aversion in a jump-diffusion market

In this paper, we consider a robust optimal asset–liability management problem with delay for an ambiguity-averse investor (AAI), who does not hold a firm belief towards not only the drift parameters but also the jump intensity in the wealth process. The objective of AAI is to maximise the robust value involving the expected utility of a weighted sum of terminal wealth and a penalisation of model uncertainty. In addition, the wealth process is modelled by a stochastic delay equation with jump via introducing the performance-related capital inflow/outflow. Applying the technique of robust optimal theory and Hamilton–Jacobi–Bellman equation, we derive the optimal results and the proof of verification theorem for the general utility functions and liability processes. In the case of the exponential utility function and a positive liability process, we prove the existence and uniqueness of the robust optimal investment strategy and find it naturally satisfies the constraint of no-short selling. Meanwhile, we adopt a special method to solve the corresponding Riccati equation and obtain the value function in explicit form. Furthermore, the results under the complete memory are discussed in detail. Finally, some numerical examples are provided to illustrate the influence of model parameters on the optimal strategy as well as the economic interpretation behind it.

Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

Model order reduction for (stochastic-) delay equations with error bounds

<p style='text-indent:20px;'>In this article, we analyze a structure-preserving model order reduction technique for deterministic and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring the framework of [<xref ref-type="bibr" rid="b6">6</xref>], we find error estimates for the difference between the dynamics of the full and reduced model. This analysis also yields new error bounds for bilinear systems and stochastic systems with multiplicative noise and non-zero initial states.</p>

Distribution-dependent stochastic differential delay equations in finite and infinite dimensions

We prove that distribution-dependent (also called McKean–Vlasov) stochastic delay equations of the form [Formula: see text] have unique (strong) solutions in finite as well as infinite-dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.

The convergence of a numerical scheme for additive fractional stochastic delay equations with [formula omitted

In this paper, we investigate the strong convergence of the exponential Euler method to stochastic delay differential equations with fractional Brownian motion (FSDDEs) of Hurst parameter H∈(12,1). We establish the strong convergence rate H of the method for FSDDEs to the exact solution. Also we justify our theoretical results with some numerical examples of these equations alongside insignificant step size.

Well-posedness, stability and sensitivities for stochastic delay equations: A generalized coupling approach

We develop a new generalized coupling approach to the study of stochastic delay equations with Hölder continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solutions, and establish weak ergodic rates for the corresponding segment processes. We also prove, under additional smoothness assumptions on the coefficients, stabilization rates for the sensitivities in the initial value of the corresponding semigroups.

Heavy Traffic Limits for Join-the-Shortest-Estimated-Queue Policy Using Delayed Information

We consider a load-balancing problem for a network of parallel queues in which information on the state of the queues is subject to a delay. In this setting, adopting a routing policy that performs well when applied to the current state of the queues can perform quite poorly when applied to the delayed state of the queues. Viewing this as a problem of control under partial observations, we propose using an estimate of the current queue lengths as the input to the join-the-shortest-queue policy. For a general class of estimation schemes, under heavy traffic conditions, we prove convergence of the diffusion-scaled process to a solution of a so-called diffusion model, in which an important step toward this goal establishes that the estimated queue lengths undergo state-space collapse. In some cases, our diffusion model is given by a novel stochastic delay equation with reflection, in which the Skorokhod boundary term appears with delay. We illustrate our results with examples of natural estimation schemes, discuss their implementability, and compare their relative performance using simulations.

A white noise approach to optimal insider control of systems with delay

We use a white noise approach to study the problem of optimal insider control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system. These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the logarithmic utility optimal insider portfolio in a generalized Black-Scholes type financial market. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable, i.e. the maximal utility is infinite. We consider two extensions to delay of this result and prove the following:•If the risky asset price is given by a stochastic delay equation, the resulting insider market is still not viable.•If, however, there is delay in the information flow to the insider, the market becomes viable.

Asymptotic behaviours of a stochastic delay equation driven by an fBm in Hilbert space

ABSTRACTIn this paper, we consider a class of stochastic delay equations in Hilbert spaces driven by fractional Brownian motion with Hurst parameter . We obtain a sufficient condition for controllability of the systems and prove that their mild solutions are exponentially stable in pth moment with by using adequately the characteristic of fractional Brownian motion and the fractional power of a linear operator A with . In particular, when the mild solutions are exponentially stable in mean square.

Stochastic systems with memory and jumps

Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our results concern existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus, such as the Itô formula. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. The study is presented in two different frameworks: we work with random variables in infinite dimensions, where the values are considered either in an appropriate Lp-type space or in the space of càdlàg paths. The choice of the value space is crucial from the modelling point of view, as the different settings allow for the treatment of different models of memory or delay. Our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation.

Stability of highly nonlinear hybrid stochastic integro-differential delay equations

For the past few decades, the stability criteria for the stochastic differential delay equations (SDDEs) have been studied intensively. Most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability criterion for highly nonlinear hybrid stochastic differential equations is investigated in Fei et al. (2017). In this paper, we investigate a class of highly nonlinear hybrid stochastic integro-differential delay equations (SIDDEs). First, we establish the stability and boundedness of hybrid stochastic integro-differential delay equations. Then the delay-dependent criteria of the stability and boundedness of solutions to SIDDEs are studied. Finally, an illustrative example is provided.

Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay

A system of stochastic delayed reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing is considered. We first prove the existence and uniqueness of a bi-spatial pullback attractor for these equations when the initial space is C−ρ,0,L2O and the terminate space is C−ρ,0,H01O. The asymptotic compactness of solutions in C−ρ,0,H01O is established by combining “positive and negative truncations” technique and some new estimates on solutions. Then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions. Finally, upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero.

Stochastic consensus of discrete-time second-order multi-agent systems with measurement noises and time delays

We study the consensus control of discrete-time second-order multi-agents systems with time delays and multiplicative noises, where the consensus protocol is designed by both the local relative position measurements and each agent’s absolute velocity. Due to the existence of time delays and multiplicative noises, the classical methods for deterministic models with time delays cannot work. In this paper, we apply stochastic stability theorem of discrete-time stochastic delay equations to find some explicit sufficient conditions for both mean square and almost sure consensus. It is proven that for any given noise intensities and time delays, the second-order multi-agent consensus can be achieved by choosing appropriate control gains in the relative position measurement and absolute velocity, respectively. Numerical simulation is given to demonstrate the effectiveness of the proposed protocols as well as the theoretical results.

Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

NUMERICAL SCHEME FOR BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS WITH TIME DELAYED COEFFICIENTS

In this paper, we present some assumptions to get the numerical scheme for backward doubly stochastic dierential delay equations (shortly-BDSDDEs), and we propose a scheme of BDSDDEs and discuss the numerical convergence and rate of convergence of our scheme.

Chatter detection in turning using persistent homology

This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes.

H2/H∞ control for stochastic systems with delay

This paper is concerned with the H2/H∞ control problem for stochastic linear systems with delay in state, control and external disturbance-dependent noise. A necessary and sufficient condition for the existence of a unique solution to the control problem is derived. The resulting solution is characterised by a kind of complex generalised forward–backward stochastic differential equations with stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the equivalent feedback solution via a new type of Riccati equations. To explain the theoretical results, we apply them to a population control problem.