Theory Of Stochastic Differential Equations Research Articles

Stability of the neoclassical growth model under perturbations of the type of Poisson’s jumps: Analytical and numerical analysis

The well known delay differential neoclassical growth model is considered under stochastic perturbations of the type of Poisson’s jumps. Stability conditions for positive and the zero equilibria of this model are obtained via the Gikhman and Skorokhod theory of stochastic differential equations with the Poisson measure and the general method of Lyapunov functionals construction. The obtained analytical results are illustrated by detail numerical simulations of solutions of the considered stochastic differential equation and construction of appropriate stability regions.

An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior

Using elements from the theory of ergodic backward stochastic differential equations (BSDEs), we study the behavior of forward entropic risk measures in stochastic factor models. We derive general representation results (via both BSDEs and convex duality) and examine their asymptotic behavior for risk positions of large maturities. We also compare them with their classical counterparts and provide a parity result.

Multivariate stochastic delay differential equations and CAR representations of CARMA processes

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a CAR(∞) representation. Furthermore, we show how the CAR(∞) representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps

This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps (NSFDEwPJs). The boundedness and almost sure exponential stability are also considered. In general, the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition. However, there are many equations that do not obey the linear growth condition. Therefore, our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role. Moreover, the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions. Finally, we present two examples to illustrate the effectiveness of our results.

Control design for stochastic one-sided Lipschitz differential inclusion system with time delay

ABSTRACTThis paper is concerned with the control design problem for stochastic one-sided Lipschitz differential inclusion system with time delay. Different from the current works on the differential inclusion system, the nonlinear function with time delay is assumed to be one-sided Lipschitz. First, the theory of stochastic differential equation is extended to stochastic differential inclusion and the stability criterion is given. Next, convex hull Lyapunov function is used to construct the nonlinear controllers, and the sufficient conditions for stability are derived by a set of bilinear matrix inequalities. Finally, three simulation examples are provided to verify the validity of the designed controllers.

Almost sure exponential synchronisation of networked harmonic oscillators via intermittent coupling subject to Markovian jumping

Synchronisation behaviours emerge from the interaction of oscillators. This study addresses the problem of almost sure exponential synchronisation for networked harmonic oscillators over switching communication networks. The coupling protocol relies on only the sampled velocity data and is periodical intermittent interactions over networks described by directed graphs subject to Markovian jumping. Next, the leader–following coupling protocol is proposed in order to approach a desired synchronisation trajectory. By using the stability theory of stochastic differential equations, some sufficient conditions are established to guarantee synchronisation in the almost sure sense. The almost sure exponential synchronisation can be achieved even when the coupling network and strength are switched between enabling and disabling the system to achieve complete synchronisation. Some numerical simulations are presented to illustrate the effectiveness of the theoretical results.

Numerical solutions of stochastic PDEs driven by arbitrary type of noise

So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Levy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.

Concentration of dynamic risk measures in a Brownian filtration

Motivated by liquidity risk in mathematical finance, Lacker (2015) introduced concentration inequalities for risk measures, i.e. upper bounds on the liquidity risk profile of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type inequalities in terms of the generator of the BSDE, which in the present case can grow arbitrarily fast.

On fuzzy type-1 and type-2 stochastic ordinary and partial differential equations and numerical solution

This paper develops the mathematical framework and the solution of a system of type-1 and type-2 fuzzy stochastic differential equations (T1FSDE and T2FSDE) and fuzzy stochastic partial differential equations (T1FSPDE and T2FSPDE). The theory of fuzzy stochastic differential equations is developed with fuzzy initial values, fuzzy boundary values and fuzzy parameters. Some natural phenomena which perturbs randomly in the influence of a white noise can be modelled as stochastic dynamic systems whose initial conditions and/or parameters may be imprecise in nature as well. The imprecision of initial values and/or parameters is generally modelled by fuzzy sets. In this paper, the concept fuzzy stochastic differential equations is developed with the introduction of fuzzy stochastic process, fuzzy stochastic random variable and fuzzy Brownian motion. The generalized $$\hbox {L}^{\mathrm{p}}$$ -integrability, based on the extension of the class of differentiable and integrable fuzzy functions, is applied and fuzzy stochastic differential equations are represented as fuzzy integral equations. A novel numerical scheme for simulations of fuzzy stochastic differential equations have also been developed. Some illustrative examples have been provided for different T1FSDE, T1FSPDE, T2FSDE and T2FSPDE models related to mathematical finance and problems in mathematical biology.

Unreliable communication in high-performance distributed multi-agent systems: A ingenious scheme in high computing

Designing of distributed consensus algorithms featuring accuracy, robustness, reliability, and speed of convergence is in high demand for various multi-agent applications. In this research, it has been investigated to device a novel design of distributed estimation algorithm which can tackle the problem of unreliable communication among multi-agents to achieve consensus on the average value of their initial values and must be capable of computing the total number of agents in the system under dynamically changing interaction topologies. A dynamically changing network topology is considered in this research with unreliable communication links, and four different scenarios are established to be analyzed for the proposed consensus-based distributed estimation algorithm. This study established a consensus for a dynamically changing interaction topology among agents, for addition of agents in the network with dynamically switching topology at any instant in communication, for removal of agents from the network with dynamically switching topology at any instant in communication, and for a fixed topology with link failure and a reconnection with the same agent after each iteration. The proposed algorithm paces up the rate of convergence by reducing the number of iteration, along with sure convergence of the designed algorithm using the concepts of stochastic differential equation theory, control system theory, algebraic graph theory, and algebraic matrix theory. Finally, in the end, simulation results are provided which are clear evidence to validate the effectiveness of theoretical results of the proposed algorithm in comparison to previously known consensus algorithms in terms of different performance parameters.

Impulsive mean square exponential synchronization of stochastic dynamical networks with hybrid time-varying delays

This paper investigates the mean square exponential synchronization problem for complex dynamical networks with stochastic disturbances and hybrid time-varying delays, both internal delay and coupling delay are considered in the model. At the same time, the coupled time-delay is also probabilistic in two time interval. Impulsive control method is applied to force all nodes synchronize to a chaotic orbit, and impulsive input delay is also taken into account. Based on the theory of stochastic differential equation, an impulsive differential inequality and some analysis techniques, several simple and useful criteria are derived to ensure mean square exponential synchronization of the stochastic dynamical networks. Furthermore, pinning impulsive strategy is studied. An effective method is introduced to select the controlled nodes at each impulsive constants. Numerical simulations are exploited to demonstrate the effectiveness of the theory results in this paper.

The General Data Assimilation Method, its Comparison with the Standard Scheme, and its Application to Dynamical Simulation in the Atlantic

A new data assimilation scheme developed earlier and based on the theory of diffusion stochastic processes and parabolic differential equations is presented and tested. This scheme is applied to the Hybrid Circulation Ocean Model (HYCOM) and altimetry data base Archiving, Validating and Interpolating Satellite Oceanography Data (AVISO) over the Atlantic. Several numerical experiments are conducted and their results are analyzed. It is shown that the method really assimilates data, makes the output oceanic fields closer to observations and, on the other hand, conserves the model integrals and balance. The tested method is also compared with the Ensemble Optimal Interpolation scheme (EnOI) as a counterpart of the standard Kalman filter method and it is shown that the proposed general method has several advantages, in particular, it provides a better forecast and requires less computational consumptions.

Forward backward SDEs in weak formulation

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.

Complete synchronization of the global coupled dynamical network induced by Poisson noises.

The different Poisson noise-induced complete synchronization of the global coupled dynamical network is investigated. Based on the stability theory of stochastic differential equations driven by Poisson process, we can prove that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1. Furthermore, numerical examples are provided to show the agreement between theoretical and numerical analysis.

Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach

We discuss an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero-coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. Finally, we derive the explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case.

Robust distributed cooperative control for DC mircogrids with time delays, noise disturbances, and switching topologies

This paper develops a robust consensus-based distributed cooperative control scheme for DC microgrids with asymmetric time delays and switching topologies in noisy environments. The proposed distributed cooperative control scheme, consisting of a leader-following consensus-based distributed voltage controller and a consensus-based distributed load current controller, is to respectively regulate the voltage of DC microgrids to the desired value and achieve the cost-based load current sharing through a sparse communication network (i.e., a distributed low bandwidth communication network) with asymmetric communication delays, switching topologies and noise disturbances. Moreover, the proposed controllers are implemented through sparse communication networks and thus meet the plug-and-play feature of microgrid power systems. Based on the Lyapunov–Krasovskii functional method and stochastic differential equation theory, the criteria for the stability analysis and delays boundedness to maintain the system stable are derived. The effectiveness of the proposed control methods are verified by the simulation of a DC microgrid system in MATLAB/SimPowerSystems.

Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems

• The dynamical networks with complex structure are investigated. • The condition of synchronization is acquired by theoretical analyses. • Theoretical result shows noise plays a positive role in attaining synchronization. • The key role of eigenvalues of coupling matrix in attaining synchronization is revealed. In this paper, the role of multiplicative noise in attaining complete synchronization on large complex networks of dynamical systems is investigated by theoretical analysis and numerical simulations. Based on the stability theory of stochastic differential equation, we prove that the multiplicative noise plays a positive role in attaining synchronization if the complex networks of dynamical systems are bounded. Moreover, the theoretical result shows that smaller second eigenvalue of coupling matrix is of benefit in attaining complete synchronization. To demonstrate the correctness of theoretical results, the coupled Lorenz systems, Hindmarsh–Rose neuronal systems and Rössler-like systems are performed as numerical examples.

Adaptive finite-time outer synchronization between two complex dynamical networks with noise perturbation

In this paper, the finite-time outer synchronization between two complex dynamical networks with noise perturbation is considered. Combing the adaptive and finite-time control technologies, a novel adaptive finite-time controller is designed. Based on the finite-time stability theory of stochastic differential equations, sufficient conditions for the finite-time stochastic outer synchronization between two complex networks with identical as well as nonidentical configurations are obtained, respectively. Finally, numerical examples are examined to illustrate the effectiveness and feasibility of the theoretical results.

Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations

In this paper, a generalized nonautonomous stochastic competitive system with impulsive perturbations is studied. By the theories of impulsive differential equations and stochastic differential equations, we have established some asymptotic properties of the system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on. In order to show the correctness and feasibility of the theoretical results, several numerical examples are presented. Finally, the effects of different white noise perturbations and different impulsive perturbations are discussed and illustrated.

Minimum energy paths for conformational changes of viral capsids.

In this work we study conformational changes of viral capsids using techniques of large deviations theory for stochastic differential equations. The viral capsid is a model of a complex system in which many units-the proteins forming the capsomers-interact by weak forces to form a structure with exceptional mechanical resistance. The destabilization of such a structure is interesting both, per se, since it is related either to infection or maturation processes and because it yields insights into the stability of complex structures in which the constitutive elements interact by weak attractive forces. We focus here on a simplified model of a dodecahedral viral capsid and assume that the capsomers are rigid plaquettes with one degree of freedom each. We compute the most probable transition path from the closed capsid to the final configuration using minimum energy paths and discuss the stability of intermediate states.