Second-order Stochastic Differential Equation Research Articles

Exponential stability results for second-order impulsive neutral stochastic differential equations

This paper is aimed to discuss the existence and exponential stability of mild solutions for second-order impulsive neutral stochastic differential equations (INSDEs). Firstly, we derive the existence of mild solutions for INSDEs by using the Banach fixed point theorem. Then, the exponential stability in the pth moment of mild solution for the considered INSDEs is proved by means of the generalized impulsive-integral inequality, which can effectively improve some known existing ones. Finally, as an application, an example is given to illustrate the efficiency of the obtained theoretical results.

A Magnus-based integrator for Brownian parametric semi-linear oscillators

We introduce a numerical method for solving second-order stochastic differential equations of the form x¨=−ω2(t)x+f(t,x)+σ(t)ξ(t), describing a class of nonlinear oscillators with non-constant frequency, perturbed by white noise ξ(t). The proposed scheme takes advantages of the Magnus approach to construct an integrator for this stochastic oscillator. Its convergence properties are rigorously analyzed and selected numerical experiments on relevant stochastic oscillators are carried out, confirming the effectiveness and the competitive behavior of the proposed method, in comparison with standard integrators in the literature.

Representing a second-order Ito equation as an equation with a given force structure

The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting the Rayleigh function are obtained. The obtained results are illustrated by an example of motion of a symmetric satellite in a circular orbit, assuming a change in pitch under the action of gravitational and aerodynamic forces.

Well posedness of second-order non-instantaneous impulsive fractional neutral stochastic differential equations

In this manuscript, the authors established existence, uniqueness and stability results for the second-order non-instantaneous impulsive fractional neutral stochastic differential equations (NIIFNSDEs). At first, existence and uniqueness results are obtained by using Caputo fractional derivative (fractional calculus), stochastic technique and fixed point approach with appropriate hypotheses on non-linear continuous functions. Secondly, we discuss the Ulam-Hyers Rassias stability and henceforth, we study Ulam-Hyers stability for NIIFNSDEs. Finally, an example is provided to validate the theoretical findings.

Collocation method for stochastic delay differential equations

In this work, we present a collocation-based numerical approach for handling stochastic delay differential equations. We approximate the solution function, and after that, we carry out integrations between the predefined collocation points to achieve a mapping from the delayed state to the present state. We build the first and second moment mapping matrices based on the mapping, and we utilize the matrices to approximate the stationary first and second moments and their stability. Numerical studies of a first and second-order stochastic delay differential equation show the convergence and time complexity of the stochastic collocation method. The last section covers the issues and possible further improvements of the method.

Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay

This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.

Approximate controllability of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps

We consider the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps in a real separable Hilbert space. Under the sufficient conditions, we obtain approximate controllability results by virtue of the theory of a strongly continuous cosine family of bounded linear operators combined with stochastic inequality technique and the Sadovskii fixed point theorem. Finally, we illustrate the main results by an example.

Ulam–Hyers stability for second-order non-instantaneous impulsive fractional neutral stochastic differential equations

In this paper, sufficient conditions are established for the Ulam–Hyers stability of second-order non-instantaneous impulsive fractional neutral stochastic differential equations (NIIFNSDEs) with supremum norm in the pth means square sense. The existence of solution of NIIFNSDEs is derived by using the cosine family of linear operator, Itô’s formula, and Mönch fixed point theorem in infinite-dimensional space. Finally, an example is demonstrated to illustrate the obtained theoretical results.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter t on top of the temporal and spatial variables (s,y) and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point (t,s,y) but also at the diagonal line of the time domain (s,s,y). Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. We first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

Strong 1.5 order scheme for second-order stochastic differential equations without Levy area

In this work, a finite difference scheme combined with a truncated spectral expansion of white noise is presented for solving second-order stochastic differential equations (SDEs) with additive white noise. Consistency of the approximate equation, which is obtained by replacing the white noise with its spectral truncation, is considered in mean-square sense. Based on it, a full discrete scheme is constructed and the mean-square convergence order of numerical solution is investigated. Numerical examples show that the presented scheme is of 1.5-order convergence in mean-square sense, which verifies the theoretical findings and is a half order higher than converting the equation to a two-dimensional system with additive white noise and using the Euler-Maruyama scheme to solve the resulting problem. It is also illustrated that the piecewise version of the spectral approximation of white noise plus proper time discretization works well for long-time simulation.

A weak-intrusive stochastic finite element method for stochastic structural dynamics analysis

This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method.

Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

Controllability of a second-order non-autonomous stochastic semilinear system with several delays in control

This paper studies a second-order non-autonomous semilinear stochastic differential equation with several constant point delays in control. We prove the mild solution’s existence and uniqueness using the semigroup theory of bounded linear operators, evolution family, stochastic analysis techniques, and Banach contraction principle. Our goal is to discuss various types of controllability of the stochastic semilinear system with the associated linear system. In the end, an example is included as an application to demonstrate the result.

On the separation method in stochastic reconstruction problem

Abstract The inverse problem of reconstruction second-order Ito stochastic differential equations by the given properties of motion is solved when control is included in the drift coefficient. By using the separation method, the form of control parameters is determined that provides sufficient conditions for the existence of a given integral manifold for the nonlinear, linear, and scalar cases.

Стохастические дифференциальные уравнения второго порядка: Устойчивость, диссипативность и периодичность. IV. - Обзор

Настоящая статья является продолжением предыдущей статьи и представляет собой четвертую часть работы автора. В работе делается обзор результатов исследований качественных свойств решений стохастических дифференциальных уравнений и систем второго порядка. В первой части был дан краткий обзор результатов работ по стохастической устойчивости решений дифференциальных уравнений и систем второго порядка с использованием аппарата функций Ляпунова. Были приведены некоторые предварительные сведения из теории вероятностей и теории случайных процессов. Во второй части дана конструкция стохастических интегралов Ито и Стратоновича. В третьей части дано понятие стохастического дифференциала, приведена формула Ито дифференцирования сложной функции для стохастических дифференциалов, дано определение стохастического дифференциального уравнения в форме Ито и в форме Стратоновича, сформулирована теорема существования и единственности для решений стохастических дифференциальных уравнений. В настоящей, четвертой, части работы даются вкратце основные сведения из теории устойчивости стохастических дифференциальных уравнений Ито. Приводятся основные определения устойчивости в различных смыслах стохастических дифференциальных систем, формулируются основные общие теоремы об устойчивости в терминах существования функций Ляпунова, являющиеся стохастическими аналогами классических теорем Ляпунова об устойчивости. Дается понятие о стохастических диссипативных системах. Приводится теорема, дающая условия существования периодических и стационарных решений в терминах вспомогательных функций для дифференциальных уравнений со случайной периодической по времени правой частью, представляющей собой периодический или стационарный процесс. This paper is a continuation of the previous papers and presents the fourth part of the author’s work. The paper reviews results concerning qualitative properties of second-order stochastic differential equations and systems. In the first part we gave a short overview on stability of solutions of the second-order stochastic differential equations and systems by Lyapunov functions techniques and introduced some mathematical preliminaries from probability theory and stochastic processes. In the second part the construction of Ito’s and Stratonovich’s stochastic integrals is given. In the third part, analog of the chain rule for stochastic differentials (Ito’s formula) is presented. The stochastic differential equations in the sense of Ito and in the sense of Stratonovich are introduced. The existence and uniqueness theorem for solutions of stochastic differential equations is formulated. In the present fourth part of the work basic facts from the theory of stability of stochastic differential equations are briefly given. The basic definitions of stability in different senses of stochastic differential systems are presented, the basic general theorems on stability are formulated in terms of the existence of Lyapunov functions, which are stochastic analogs of the classical Lyapunov’s theorems on stability. The concept of stochastic dissipative systems is given. A theorem is formulated which gives conditions for existence of periodic and stationary solutions in terms of auxiliary functions for differential equations with a random periodic in time right-hand side, which is a periodic or stationary process.

Exponential stability of impulsive fractional neutral stochastic differential equations

In this paper, we investigate the existence of a mild solution and exponential stability for a class of second-order impulsive fractional neutral stochastic differential equations (IFNSDEs) driven by fractional Brownian motion in a real and separable Hilbert space together with the semigroup of a bounded linear operator and stochastic settings. Some meaningful sufficient conditions are derived, which generalize and enhance some existing results. Finally, to show the efficacy of our results, a numerical example is provided.

Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

On construction of a field of forces along given trajectories in the presence of random perturbations

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

ОБ ОБРАТНОЙ СТОХАСТИЧЕСКОЙ ЗАДАЧЕ ВОССТАНОВЛЕНИЯ

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

ОБ ОБРАТНОЙ СТОХАСТИЧЕСКОЙ ЗАДАЧЕ ВОССТАНОВЛЕНИЯ

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.