Stochastic Delay Differential Equations Research Articles

Existence results for second‐order semilinear stochastic delay differential equation

This article is devoted towards some necessary conditions for the existence of the mild solution of ‘second‐order semilinear stochastic delay differential equations’ having a variable delay in the state with the help of Banach fixed point theorem and theory of cosine family which is strongly continuous. In the last section, we have discussed an example to understand the theory of provided results in a better manner.

Stability analysis of neutral stochastic delay differential equations via the vector Lyapunov function method

In this paper, we study the pth moment exponential stability (p-ES) and the almost sure exponential stability (a-ES) of neutral stochastic delay differential equations (NSDDEs). By using the vector Lyapunov function (VLF) method, we can prove that the global solution of NSDDEs exists when the linear growth condition is removed, and we also get some stability criteria for NSDDEs. In the presented stability conditions, the established L-operator differential inequality is based on the VLF and allows cross item to exist. An example is given to verify the correctness of the proposed results.

Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay

This paper investigates the asset-liability management problem for an ordinary robust insurance system between a reinsurer and an insurer under the Heston model with delay. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the Heston model. The insurer, as the follower of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. Under the consideration of the performance-related capital inflow/outflow, the wealth processes of the insurer and reinsurer are modeled by stochastic differential delay equations (SDDEs). This paper aims to find the equilibrium strategies for the reinsurer and insurer by maximizing the expected utility of the player’s terminal wealth with delay under the worst-case scenario of the alternative measures. By using the idea of backward induction and dynamic programming approach, the explicit expressions of the robust equilibrium strategies and value functions are derived. Finally, some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the robust optimal reinsurance and investment strategies are performed.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps.

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay

<p style='text-indent:20px;'>In present work, a step-by-step Legendre collocation method is employed to solve a class of nonlinear fractional stochastic delay differential equations (FSDDEs). The step-by-step method converts the nonlinear FSDDE into a non-delay nonlinear fractional stochastic differential equation (FSDE). Then, a Legendre collocation approach is considered to obtain the numerical solution in each step. By using a collocation scheme, the non-delay nonlinear FSDE is reduced to a nonlinear system. Moreover, the error analysis of this numerical approach is investigated and convergence rate is examined. The accuracy and reliability of this method is shown on three test examples and the effect of different noise measures is investigated. Finally, as an useful application, the proposed scheme is applied to obtain the numerical solution of a stochastic SIRS model.</p>

Optimal Reinsurance and Investment Strategy with Delay in Heston’s SV Model

In this paper, we consider an optimal investment and proportional reinsurance problem with delay, in which the insurer’s surplus process is described by a jump-diffusion model. The insurer can buy proportional reinsurance to transfer part of the insurance claims risk. In addition to reinsurance, she also can invests her surplus in a financial market, which is consisted of a risk-free asset and a risky asset described by Heston’s stochastic volatility (SV) model. Considering the performance-related capital flow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s target is to find the optimal investment and proportional reinsurance strategy to maximize the expected exponential utility of combined terminal wealth. We explicitly derive the optimal strategy and the value function. Finally, we provide some numerical examples to illustrate our results.

Optimal Time-Consistent Investment and Reinsurance Strategies with Default Risk and Delay under Heston’s SV Model

Considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate the mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence. We characterize the insurance market by two-dimensional dependent claims, the financial market by the Heston SV model, and default risk by reduced-form approach and then obtain the evolution equation of the insurer’s wealth. Based on the introduction of time delay, the insurer’s wealth dynamics characterized by a stochastic delay differential equation are obtained. Furthermore, applying stochastic control theory within the game-theoretic framework and stochastic control theory with delay, we derive optimal time-consistent investment and reinsurance strategies, as well as equilibrium value function and equilibrium efficient frontier. Finally, we use a numerical example to analyze the influence of parameters on the time-consistent equilibrium strategies and give an economic explanation.

About one method of stability investigation for nonlinear stochastic delay differential equations

AbstractStability of equilibria of a nonlinear delay differential equation is investigated under stochastic perturbations. The use of Stiltjes integrals allows to consider both discrete and distributed delays in the investigated equation. Obtained conditions of stability in probability are formulated in terms of linear matrix inequalities. The proposed method of investigation can be applied to a number of very popular in research mathematical models such as Mackey–Glass model, neoclassical growth model and e‐commerce model, Nicholson's blowflies equation, mosquito and glassy‐winged sharpshooter populations, SIR epidemic model, social epidemic models, and many similar other mathematical models which can be considered as particular cases of the investigated nonlinear equation.

Convergence rate of Euler\u2013Maruyama scheme for SDDEs of neutral type

In this paper, we are concerned with the convergence rate of Euler–Maruyama (EM) scheme for stochastic differential delay equations (SDDEs) of neutral type, where the neutral, drift, and diffusion terms are allowed to be of polynomial growth. More precisely, for SDDEs of neutral type driven by Brownian motions, we reveal that the convergence rate of the corresponding EM scheme is one-half; Whereas for SDDEs of neutral type driven by pure jump processes, we show that the best convergence rate of the associated EM scheme is slower than one-half. As a result, the convergence rate of general SDDEs of neutral type, which is dominated by pure jump process, is slower than one-half.

Mathematical analysis of some iterative methods for the reconstruction of memory kernels

We analyze three iterative methods that have been proposed in the computational physics community for the reconstruction of memory kernels in a stochastic delay differential equation known as the generalized Langevin equation. These methods use the autocorrelation function of the solution of this equation as input data. Although they have been demonstrated to be useful, a straightforward Laplace analysis does not support their conjectured convergence. We provide more detailed arguments to explain the good performance of these methods in practice. In the second part of this paper we investigate the solution of the generalized Langevin equation with a perturbed memory kernel. We establish sufficient conditions including error bounds such that the stochastic process corresponding to the perturbed problem converges to the unperturbed process in the mean square sense.

On nonnegative solutions of SDDEs with an application to CARMA processes

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

The Solution of Stochastic Time-Dependent First Order Delay Differential Equations using Block Simpson’s Methods

The Solution of Stochastic Time-Dependent First Order Delay Differential Equations using Block Simpson’s Methods

Stabilization of Nonlinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

Stabilization of Nonlinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

Asymptotic stability for certain stochastic delay differential equations of the third-order

Asymptotic stability for certain stochastic delay differential equations of the third-order

带Poisson跳的随机时滞微分方程的矩有界性

带Poisson跳的随机时滞微分方程的矩有界性

Mean-field stochastic H 2/H ∞ control with delay

The finite horizon control problem of mean-field type for delayed stochastic systems is considered in this paper. Firstly, we derive a sufficient mean-field generalised stochastic bounded real lemma (SBRL). Secondly, based on the generalized SBRL and the mean-field anticipated forward-backward stochastic delay differential equations, this paper establishes a sufficient condition for the existence of a solution to delayed stochastic control of mean-field type. Thirdly, optimal state feedback regulators are studied for the case with delays only in control variable and disturbance signal via the solvability of coupled matrix-valued equations. Finally, an example is employed to show the effectiveness of the results obtained.

Effects of noise and harmonic excitation on the growth of Bacillus subtilis biofilm

We study the dynamic growth behaviors of Bacillus subtilis biofilms by using a stochastic delay differential equation similar to the Mackey-Glass equation. We observe a wealth of dynamic behavior from a mathematical perspective. Firstly, the change of the time delay (i.e. the time that the growth state of the periphery to affect the stress of the interior cells needs and the time required for the stress signal coming from the interior to reach the peripheral cells) and the external excitation (i.e. the change in temperature) frequency will influence the size of Bacillus subtilis biofilm and the collective growth of the cell community. We also found the resonance phenomena in mathematics, which is of great help in understanding the behavior of stress levels in the biofilm periphery. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold reduction. Numerical simulations are carried out to ensure theoretical results. Finally, the effects of noise on stress levels in the biofilm periphery are analyzed and reveal more complex dynamics of the system.

STABILITY ANALYSIS BETWEEN THE HYBRID STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH JUMPS AND THE EULER-MARUYAMA METHOD

<abstract> The aim of this paper is to concern with the mean square exponential stability equivalence between the hybrid stochastic delay differential equations with jumps and the Euler-Maruyama method (EM-method). Precisely, under the global Lipschitz condition, it is shown that a stochastic delay differential equation with Markovian switching and jumps (SDDEwMJ) is mean square exponentially stable if and only if for some sufficiently small step size, its EM-method is mean square exponentially stable. Based on such a result, the mean square exponential stability of a SDDEwMJ can be investigated by the careful numerical simulations in practice without resorting to Lyapunov functions. Moreover, a numerical example is provided to confirm the obtained results. </abstract>

Dynamic asset-liability management problem in a continuous-time model with delay

ABSTRACT This paper investigates a dynamic continuous-time asset-liability management (ALM) problem with delay under the mean-variance criterion. The investor allocates her wealth in a financial market consisting of one risk-free asset and one risky asset, and she is subject to a random liability. The historical information of the wealth and liability affects the investor's wealth process, which is then governed by a stochastic differential delay equation. Firstly, a general ALM problem with delay is formulated and the extended Hamilton-Jacobi-Bellman system of equations is obtained. Secondly, we focus on a linear model and derive the closed-form expressions of the equilibrium investment strategy and the corresponding equilibrium value function. Meanwhile, we also derive the pre-commitment strategy for the mean-variance ALM problem with delay using the maximum principle. Finally, some numerical examples and sensitivity analysis are presented to illustrate the equilibrium investment strategies and the efficient frontiers under the equilibrium and pre-commitment frameworks.