Stochastic Delay Differential Equations Research Articles

Equilibrium pairs trading under delayed cointegration

Cointegration analysis is an econometric tool used to identify equilibrium among assets and construct a pairs trading portfolio. The discrete-time vector error correction model (VECM) for identifying cointegration includes lag difference terms as explanatory variables, thus permitting delayed adjustment of the deviations from equilibrium. The continuous-time limit of the VECM becomes a stochastic delay differential equation. We investigate the dynamic open-loop equilibrium mean–variance pairs trading strategy under such a delayed cointegration model. The existence and uniqueness results for the equilibrium are offered. We prove in general that the equilibrium strategy can lead to statistical arbitrage under certain conditions that are related to the roots of the corresponding characteristic equation. We obtain an explicit solution for a case with distributed delay. Our empirical study demonstrates the superiority of our strategy over its Markovian counterpart when the model selection result prefers a high-order VECM.

Stochastic differential reinsurance and investment games with delay under VaR constraints⋆

This article investigates stochastic differential reinsurance and investment games with delay under Value-at-Risk (VaR) constraints. In our model, three players and two games are considered, i.e., a non-zero-sum stochastic differential game between two insurers and a stochastic Stackelberg differential game between the reinsurer and the two insurers are investigated. The reinsurer can set the reinsurance premium price and invest her surplus in a financial market consisting of a risk-free asset and a risky asset. The two insurers can purchase proportional reinsurance and invest in the same financial market. Since the delay is considered in this article, the wealth processes of three players are described by the stochastic delay differential equations. In this article, we establish two optimization problems, one is to maximize the expected utility of the reinsurer’s terminal wealth with delay, and the other is to maximize the expected utility of the combination of each insurer’s terminal wealth and relative performance with delay. Furthermore, in order to control risk, we incorporate the VaR constraints into the optimizations and derive the corresponding value functions and Nash equilibrium strategies by using optimal control theory, dynamic programming principle, backward induction, Lagrange function and Karush–Kuhn–Tucker condition. Finally, several numerical examples are given.

Stability of highly nonlinear neutral stochastic delay systems with non-random switching signals

For the last few decades, many works on stability criteria have been established for neutral stochastic delay systems, which are modeled by stochastic differential delay equations. Some of these works do not involve switching signals, and others contain Markovian switching signals. This paper studies highly nonlinear neutral stochastic delay systems with non-random switching signals. Firstly, the rule of non-random switching signals is imposed. Then, by using the Lyapunov function method and the mode-dependent average dwell time approach, the existence and boundedness of unique global solution is discussed for highly nonlinear neutral stochastic delay systems with non-random switching signals. In addition, without the linear growth condition, asymptotic stability and exponential stability are obtained. In the last part of the article, an illustrative example is provided to show the effectiveness of the obtained results.

The truncated θ-Milstein method for nonautonomous and highly nonlinear stochastic differential delay equations

In this paper we propose and analyze a truncated θ-Milstein method for solving a class of non-autonomous stochastic differential delay equations with highly nonlinear coefficients. Under a generalized monotone condition of the coefficients, the convergence order of the truncated θ-Milstein method is obtained. We find that the convergence rate is dependent on the exponent of the Hölder conditions. To verify our theoretical findings, we provide a numerical example.

Robust optimal proportional reinsurance and investment problem for an insurer with delay and dependent risks

In this article, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and dependent risks, in which the insurer’s surplus process is assumed to follow a risk model with two dependent classes of insurance business. The insurer is allowed to purchase proportional reinsurance and invest his surplus in a financial market, which contains one risk-free asset and one risky asset whose price process satisfies a jump-diffusion model. Under the consideration of the performance-related capital inflow or outflow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s aim is to develop the robust optimal reinsurance and investment strategy for the insurer by maximizing the expected exponential utility of the combination of the average historical performance and terminal wealth under the worst-case scenario of the alternative measures. By adopting the stochastic dynamic programming technique, the expressions of the robust optimal strategy and value function are explicitly obtained. Finally, we present some numerical examples to illustrate the effects of some model parameters on the optimal strategy.

Delay-dependent Asymptotic Stability of Highly Nonlinear Stochastic Differential Delay Equations Driven by [formula omitted]-Brownian Motion

Based on the classical probability, the stability of stochastic differential delay equations (SDDEs) whose coefficients are growing at most linearly has been investigated intensively. Moreover, the delay-dependent stability of highly nonlinear hybrid stochastic differential equations (SDEs) has also been studied recently. In this paper, using the nonlinear expectation theory, we first explore the delay-dependent criteria on the asymptotic stability for a class of highly nonlinear SDDEs driven by G-Brownian motion (G-SDDEs). Then, the (weak) quasi-sure stability of solutions to G-SDDEs is developed. Finally, an example is analyzed by the φ-max-mean algorithm to illustrate our theoretical results.

Approximation to Stochastic Variance Reduced Gradient Langevin Dynamics by Stochastic Delay Differential Equations

We study in this paper weak approximations in Wasserstein-1 distance to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations, and obtain uniform error bounds. Our approach is via Malliavin calculus and a refined Lindeberg principle.

Some results on finite-time stability of stochastic fractional-order delay differential equations

Finite-time stability of stochastic fractional-order delay differential equations is researched here. Firstly, we derive the equivalent form of the considered system by using the Laplace transformation and its inverse. Subsequently, by defining the maximum weighted norm in Banach space and using the principle of contraction mapping, we prove that the solution of researched system is unique. What's more, by virtue of Henry-Grönwall delay inequality and interval translation, we derive the criterion of finite-time stability for the system with and without impulses, respectively. Finally, as a verification, examples are provided to expound the correctness of the deduced results.

A Dynamical Theory for Singular Stochastic Delay Differential Equations I: Linear Equations and a Multiplicative Ergodic Theorem on Fields of Banach Spaces

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

An Averaging Principle for Stochastic Differential Delay Equations Driven by Time-Changed Lévy Noise

An Averaging Principle for Stochastic Differential Delay Equations Driven by Time-Changed Lévy Noise

Discrete feedback control for highly nonlinear neutral stochastic delay differential equations with Markovian switching

In this article, it is proved that feedback controllers can be designed to stabilize nonlinear neutral stochastic systems with Markovian switching (NSDDEwMS in short) only by using discrete observed state sequences. Due to the superlinear coefficients, the neutral term and the discrete observation data, many routine methods and techniques for the study of stochastic systems are not applicable. A new Lyapunov functional is constructed by using multiple M-matrices to prove that a given unstable NSDDEwMS can be stabilized if the control function can be designed to meet a couple of easy-to-be-verified rules. Finally, an example is given to illustrate the feasibility of the theoretical results.

Stability of stochastic nonlinear delay systems with delayed impulses

In this paper, we study the input-to-state stability of a class of stochastic delay differential equations with delay impulses. We consider the two cases where the continuous stochastic delay system is stable and unstable, and establish different types of stability conditions by using the average dwell time and the vector Lyapunov function method. Finally, two examples are used to verify the validity of our results.

Two-time-scale stochastic differential delay equations driven by multiplicative fractional Brownian noise: Averaging principle

The main goal of this article is to study an averaging principle for a class of two-time-scale stochastic differential delay equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter H∈(12,1) and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (12,1) can be defined as a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce a sequence of stopping times to control the size of multiplicative fractional Brownian noise. Then, inspired by the Khasminskii's approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.

Boundedness analysis of stochastic delay differential equations with Lévy noise

The present paper is concerned with the mean square asymptotic boundedness and twice power almost surely asymptotic boundedness of stochastic delay differential equations driven by Lévy noise. First, mean square asymptotic boundedness criteria of the solutions are established by the method of reduction and the generalized Itô formula. Then, based on the Chebyshev inequality and the Borel–Cantelli lemma, the twice power almost surely asymptotic boundedness criteria are also derived for the addressed equations. Finally, a example is provided to demonstrate the validity of the proposed results.

Least squares estimation for distribution-dependent stochastic differential delay equations

<p style='text-indent:20px;'>The parametric estimation of drift parameter for distribution - dependent stochastic differential delay equations with a small diffusion is presented. The principle technique of our investigation is to construct an appropriate contrast function and carry out a limiting type of argument to show the consistency and convergence rate of the least squares estimator of the drift parameter via interacting particle systems. In addition, two examples are constructed to demonstrate the effectiveness of our work.</p>

Robust equilibrium investment and reinsurance strategy with bounded memory and common shock dependence

In this paper, we consider the robust investment and reinsurance problem with bounded memory and risk co-shocks under a jump-diffusion risk model. The insurer is assumed to be ambiguity-averse and make the optimal decision under the mean-variance criterion. The insurance market is described by two-dimensional dependent claims while the risky asset is depicted by the jump-diffusion model. By introducing the performance in the past, we derive the wealth process depicted by a stochastic delay differential equation (SDDE). Applying the stochastic control theory under the game-theoretic framework, together with stochastic control theory with delay, the robust equilibrium investment-reinsurance strategy and the corresponding robust equilibrium value function are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment and reinsurance strategy.

Stochastic Delay Differential Equations: Analysis and Simulation Studies

Stochastic Delay Differential Equations: Analysis and Simulation Studies

Lyapunov functionals and practical stability for stochastic differential delay equations with general decay rate

This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper.

Stochastic Delay-Differential Equations As Bacterial Population Model with boundary feedback

In this work, we present a way of treating an infinite-dimensional stochastic bacterial population system with delay in state space. This model is a variant of the transport equation, in which each bacteria is described by its degree of maturity and its maturation velocity. Here we use the operator theory to reformulate the boundary deterministic bacterial population system as a Cauchy problem with unbounded perturbation of the generator, this allows us to use the natural feedback theory developed by Staffans and Weiss (Staffans (2005); Weiss (1994)) to prove the well-posedness. We show differently that this model is governed by a positive strongly continuous semigroup. Finally, we prove the well-posedness of the stochastic delay version of this model giving some sufficient conditions.

Stabilization by intermittent control for hybrid stochastic differential delay equations

<p style='text-indent:20px;'>This paper is concerned with stablization of hybrid differential equations by intermittent control based on delay observations. By M-matrix theory and intermittent control strategy, we establish a sufficient stability criterion on intermittent hybrid stochastic differential equations. Meantime, we show that hybrid differential equations can be stabilized by intermittent control based on delay observations if the delay time <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is bounded by <inline-formula><tex-math id="M2">\begin{document}$ \tau^* $\end{document}</tex-math></inline-formula>. Finally, an example is presented to illustrate our theory.</p>