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Comparing with traditional stochastic differential equations (SDEs) involving white noise, random differential equations (RDEs) with colored noise are claimed to have more practical meaning. This paper considers the event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities. Combining a tracking error-based dynamic gain with a relative threshold event triggered control mechanism (ETCM), the tracking control problem for the random systems is solved without Zeno behavior. The tracking error can be rendered small enough by tuning design parameters. First, a series of adaptive control laws are designed by using backstepping technique. Then, two special cases are considered and the main results are extended to MIMO systems. Finally, a simulation example confirms the validity of the results. To the best of the authors knowledge, this paper serves as the first attempt of event-based control for RDE systems.

The paper proposes a mathematical model aimed at calculating the effective elastic moduli of a micro-inhomogeneous two-component isotropic composite material, which components are connected randomly depending on the level of their relative volumetric contents. A stochastic equation is formulated for the connectivity parameter of the constituent components, according to which, with an increase in the volumetric content of the filler, individual inclusions build the structures of the matrix mixture in the form of interpenetrating frameworks, and then turn into a new binding matrix with individual inclusions from the material of the rest of the old matrix. The algorithm for the numerical solution of this stochastic differential equation is constructed in accordance with the Euler-Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random connectivity function of the constituent components of the composite material. A variant of the method aimed at calculating the mathematical expectation of a random connectivity function of the constituent components has been developed and the corresponding differential equation has been obtained for it. It is shown that the numerical solution of this equation and the average value of the production factor function calculated for all realizations of stochastic trajectories give close numerical values. New macroscopic constitutive relations are found for microinhomogeneous materials with a variable microstructure and their effective elastic moduli are calculated. It is noted that the formulas for these effective elastic moduli generalize the known results for isotropic composite materials. The values of the effective elastic moduli, constructed according to the expressions obtained in the paper, lie within the Khashin-Shtrikman range for the lower and upper bounds of the effective elastic moduli of the composite materials. The numerical analysis of the developed models showed a good agreement with the known experimental data.

This article studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semidefinite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward–backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSDEs. By virtue of solutions to two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems with the help of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed control laws.

The paper considers the linear quadratic regulation (LQR) and stabilization problems for Ito stochastic systems with two input channels of which one has input delay. The underlying problem actually falls into the field of asymmetric information control because of the nonidentical measurability induced by the input delay. In contrast with single-channel single-delay problems, the challenge of the problems under study lies in the interaction between the two channels which are measurable with respect to different filtrations. The key techniques conquering such difficulty are the stochastic maximum principle and the orthogonal decomposition and reorganization technique proposed in a companion paper. The authors provide a way to solve the delayed forward backward stochastic differential equation (D-FBSDE) arising from the maximum principle. The necessary and sufficient solvability condition and the optimal controller for the LQR problem are given in terms of a new Riccati differential equation established herein. Further, the necessary and sufficient stabilization condition in the mean square sense is provided and the optimal controller is given. The idea proposed in the paper can be extended to solve related control problems for stochastic systems with multiple input channels and multiple delays. This work was supported by Science and Technology Project of Qingdao West Coast New Area (2019–32, 2020–20, 2020-1-4), High-level Talent Team Project of Qingdao West Coast New Area (RCTD-JC-2019-05), Key Research and Development Program of Shandong Province (2020CXGC01208).

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.

This paper focuses on the passive fault-tolerant control for an uncertain switched nonlinear random system with multiple interval time-varying delays and intermittent faults in sensors and actuators. There are also nonlinear functions, exogenous disturbances and measurement noise in the system. There are few tries to realize noise-to-state exponentially mean-square stability for switched random nonlinear systems subject to multiple interval time-varying delays allowing various time delay cases. Random differential equations are more common than stochastic differential equations in practice. Thus, the suggested method is more practicable. First, a nonlinear dynamic output feedback fault-tolerant controller is constructed. Next, an augmented closed-loop system can be obtained in the framework of multiple interval time-varying delays. Then piecewise Lyapunov function is utilized to realize stability analysis for the closed-loop system. At last, this feasibility of this approach presented in this study is verified through two examples.

Asset pricing model is the pillar of modern financial market price theory. It is of great practical and theoretical significance to solve the equilibrium price-dividend function of the asset pricing model. To solve the asset pricing model, this paper develops a novel neural network method called improved trigonometric neural network, which consists of three parts: the improved trigonometric function, the initial-condition extreme learning machine algorithm and the reduction algorithm. First, the equilibrium price-dividend function is described as a stochastic differential equation and this function is translated into a second-order ordinary differential equation equivalently. Second, the improved trigonometric neural network is used to solve the differential equation with initial conditions, where the improved trigonometric function is used to serve as the activation function. The reduction algorithm is proposed to obtain a simpler network structure and a faster computing speed. Third, numerical experiments of the improved trigonometric neural network show that the numerical solution of the price-dividend function will be obtained precisely, quickly and feasibly. Compared with several methods to solve asset pricing model, the improved trigonometric neural network has the highest accuracy and fastest speed.

The present article aims to study the effect of external parameters, such as the action of a constant force and the action of an external harmonic potential, on the trajectory of a charged particle in Brownian motion in a time-dependent magnetic field. The approach presented here builds on methods of statistical physics and produces an analytical and numerical general solution for the stochastic differential equations of motion.

In this paper, we model the spread of coronavirus (COVID -19) by introducing stochasticity into the deterministic differential equation susceptible -infected-recovered (SIR model). The stochastic SIR dynamics are expressed using Itô's formula. We then prove that this stochastic SIR has a unique global positive solution I(t).The main aim of this article is to study the spread of coronavirus COVID-19 in Iraq from 13/8/2020 to 13/9/2020. Our results provide a new insight into this issue, showing that the introduction of stochastic noise into the deterministic model for the spread of COVID-19 can cause the disease to die out, in scenarios where deterministic models predict disease persistence. These results were also clearly illustrated by Computer simulation.

The connections between stochastic differential equations in which continuous and discontinuous random processes serve as sources of randomness and deterministic equations for the probabilistic characteristics of solutions of these stochastic equations are studied. In the study, we use various approaches based on the stochastic change of variables formula (Ito’s formula), on the analysis of local infinitesimal characteristics of the process, and on the theory of semigroups of operators in combination with the generalized Fourier transform. This allows us to obtain direct and inverse integro-differential equations for various probabilistic characteristics.

In this paper, we study the smooth Wong-Zakai approximations given by a stochastic process via Wiener shift and mollifier of Brownian motions. We show that solutions of random differential equations driven by such processes generate random dynamical systems and converge in mean square to solutions of Stratonovich stochastic differential equations with sub-linear drift and bounded diffusion. With the help of this result, we obtain the existence of periodic solutions in distribution and stationary measures for time-inhomogeneous and time-homogeneous stochastic systems with dissipativity respectively. As an application, we verify Levinson's conjecture to second order stochastic Newtonian systems via Lyapunov's method and truncation method.

In this paper, we investigate mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence and uniqueness theorem are obta...

A stochastic competition system with harvesting and distributed delay is investigated, which is described by stochastic differential equations with distributed delay. The existence and uniqueness of a global positive solution are proved via Lyapunov functions, and an ergodic method is used to obtain that the system is asymptotically stable in distribution. By using the comparison theorem of stochastic differential equations and limit superior theory, sufficient conditions for persistence in mean and extinction of the stochastic competition system are established. We thereby obtain the optimal harvest strategy and maximum net economic revenue by the optimal harvesting theory of differential equations.

Reconstruction of gene regulatory networks (GRN) plays an important role in understanding the complexity, functionality and pathways of biological systems, which could support the design of new drugs for diseases. Because differential equation models are flexible androbust, these models have been utilized to identify biochemical reactions and gene regulatory networks. This paper investigates the differential equation models for reverse engineering gene regulatory networks. We introduce three kinds of differential equation models, including ordinary differential equation (ODE), time-delayed differential equation (TDDE) and stochastic differential equation (SDE). ODE models include linear ODE, nonlinear ODE and S-system model. We also discuss the evolutionary algorithms, which are utilized to search the optimal structures and parameters of differential equation models. This investigation could provide a comprehensive understanding of differential equation models, and lead to the discovery of novel differential equation models.

We consider the indefinite linear-quadratic optimal control problem for stochastic systems with random coefficients, where the corresponding cost parameters need not be definite matrices. Although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic Hamilton–Jacobi–Bellman (HJB) equation, our approach is simple and direct. Specifically, we develop a direct approach, also known as the completion of squares method, to characterize the explicit optimal solution and the optimal cost. The corresponding optimal solution is linear characterized in terms of the stochastic Riccati differential equation (SRDE) and the linear backward stochastic differential equation (BSDE). In our approach, the completion of squares method, together with the SRDE and the BSDE, allows us to construct an equivalent cost functional that is quadratic in the control $u$ , provided that an additional positive definiteness condition in terms of the SRDE holds. Then, the optimal control and the associated cost can be obtained by eliminating the quadratic term of $u$ in the equivalent cost functional. The additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term. We verify the optimal solution by solving the stochastic HJB equation.

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada–Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

In this paper, we build the Wong–Zakai approximation for Stratonovich-type stochastic differential equations driven by G-Brownian motion and obtain the quasi-sure convergence rate under Holder norm by a rough path argument. As a corollary, we obtain the quasi-continuity of solutions of random rough differential equations driven by lifted martingales under a sequence of singular measures.

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been a shadow existence to stochastic differential equations (SDEs) despite being able to model a wider and often physically more adequate range of disturbances. In this article, we study the safety verification problem over both finite time horizons and the infinite time horizon for RODEs incorporating Wiener processes. Concretely, we investigate the ${p}$ -safety problem, where we identify the set of initial states from which the probability to satisfy safety specifications is at least ${p}$ . Based on identifying a set of sample paths whose probability measure is larger than ${p}$ , we propose a method of reducing stochastic reachability to adversary reachability of ODEs for solving the ${p}$ -safety problem over finite time horizons. This method permits an efficient lifting of reach-set computation methods for perturbed ODEs to RODEs. In this method, the ${p}$ -safety problem over finite time horizons is reduced to the problem of inner-approximating robust backward reachable sets for ODEs with time-varying perturbation inputs. We then extend the method to the ${p}$ -safety problem over the infinite time horizon. Finally, we demonstrate our method on several examples.

The Caratheodory approximation for a type of Caputo fractional stochastic differential equations is considered. As is well known, under the Lipschitz and linear growth conditions, the existence and uniqueness of solutions for some type of differential equations can be established. However, this approach does not give an explicit expression for solutions; it is not applicable in practice sometimes. Therefore, it is important to seek the approximate solution. As an extending work for stochastic differential equations, in this paper, we consider Caratheodory’s approximate solution for a type of Caputo fractional stochastic differential equations.

We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases., 18 pages

Stochastic hybrid systems (SHSs) are a modelling framework for a cyber-physical system (CPS), used to simulate, validate, and verify safety critical controllers under uncertainty. Popular simulation tools can miss detecting discontinuities when simulating SHS, thereby producing incorrect outputs during simulation. We propose a novel adaptive step size simulation/integration technique for a subset of SHS—stochastic differential equations (SDEs) with discontinuous drift and diffusion coefficients. Each integration step, of the Euler-Maruyama numerical solution of the SDEs, is made dependent upon the values of the continuous variables inducing the discontinuity. This in turn guarantees convergence of the system trajectory towards the discontinuity without missing it. A thorough analysis and extensive benchmarking of the proposed integration technique shows the efficacy of the approach when simulating complex SHSs.

In order to better link the stochastic diffusion stochastic differential equation with securities investment, this paper proposes a securities portfolio optimization method of the stochastic diffusion stochastic differential equation. In this paper, the stock Shanghai composite index is taken as the research object, and a new differential equation is obtained using dynamic programming method, and the optimal feedback control and optimal index of the system are obtained. By automatically adjusting the values of various types of parameters, the optimal portfolio of securities can be obtained, thereby providing more reasonable research and prediction for stock prices.

Based on the stability theory of nonlinear differential equation, this paper analyzes the stability of the evolutionary game supply chain, and obtains the strategies that both sides should adopt in different situations. In this paper, aiming at the problem of various random interference factors in the attack and defense game system, the concept of Gauss white noise is used for reference, and it o random differential equation is used to establish the random attack and defense evolution game model. The influence of various random interference factors such as system environment and strategy change on the selection and evolution of attack and defense strategy is analyzed. According to the stability discrimination theorem of stochastic differential equation, the stability of attack and defense strategy is analyzed. In addition, a security strategy selection algorithm based on evolutionary game model of random attack and defense is designed. The results show that under different conditions, there is a unique evolutionary stable strategy in the attack and defense game system.

In this article, we study the existence of solutions and their stability of random impulsive stochastic functional differential equations (ISFDEs) driven by Poisson jumps with finite delays. Initially, we prove the existence of the mild solutions to the equations by using Banach fixed point theorem. Then, we study the stability of random ISFDEs through the continuous dependence of solutions on initial condition. Next, we investigate the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is presented to illustrate our results.

Uncertain differential equation is a type of differential equation driven by Liu process that is the counterpart of Wiener process in the framework of uncertainty theory. The stability theory is of particular interest among the properties of the solutions to uncertain differential equations. In this paper, we introduce the Lyapunov’s second method to study stability in measure and asymptotic stability of uncertain differential equation. Different from the existing results, we present two sufficient conditions in sense of Lyapunov stability, where the strong Lipschitz condition of the drift is no longer indispensable. Finally, illustrative examples are examined to certify the effectiveness of our theoretical findings.

Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at times $0, \tau, 2\tau, \ldots$ , so that the controlled hybrid SDE becomes asymptotically stable? It has been proved that this is possible if the drift and diffusion coefficients of the given hybrid SDE satisfy the linear growth condition. However, many hybrid SDEs in the real world do not satisfy this condition (namely, they are highly nonlinear) and there is no answer to the question, yet if the given SDE is highly nonlinear. The aim of this paper is to tackle the stabilization problem for a class of highly nonlinear hybrid SDEs. Under some reasonable conditions on the drift and diffusion coefficients, we show how to design the feedback control function and give an explicit bound on $\tau$ (the time duration between two consecutive state observations), whence the new theory established in this paper is implementable.

Noise is an inherent characteristic of electronic circuits. In this Letter, the authors propose a novel estimation technique for a noise influenced electronic circuit. Stochasticity is accounted for by utilising Ito's stochastic calculus. This novel estimation technique unifies the Carleman linearisation with the Fokker–Planck equation for the Ito stochastic differential equations. To show the effectiveness of the proposed method, numerical simulation results are presented for a benchmark tunnel diode oscillator circuit. This method is beneficial to other electronic circuits for estimating the state trajectories under the influence of randomness when observations are not available.

In this paper, we investigate the pricing problems of European spread options with the floating interest rate. In this model, uncertain differential equation and stochastic differential equation are used to describe the fluctuation of stock price and the floating interest rate, respectively. We derive the pricing formulas for spread options including the European spread call option and the European spread put option. Finally, numerical algorithms are provided to illustrate our results.

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton-Jacobi-Bellman equation in our non-Markovian framework., Comment: 36 pages

Recently, the deep learning method has been used for solving forward–backward stochastic differential equations (FBSDEs) and parabolic partial differential equations, as it has good accuracy and performance for high-dimensional problems. In this article, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms, and the numerical results show remarkable performance, especially for high-dimensional cases.

The paper studies the parametric stochastic roll motion in the random waves. The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed. Random sea surface is treated as a narrow-band stochastic process, and the stochastic parametric excitation is studied based on the effective wave theory. The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function. By using the stochastic averaging method, the differential equation of motion is transformed into Ito’s stochastic differential equation. The steady-state probability density function of roll motion is obtained, and the results are validated with the numerical simulation and model test.

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 \begin{document}$ \tau $\end{document} is bounded by \begin{document}$ \tau^* $\end{document}. Finally, an example is presented to illustrate our theory.

In this paper, we are concerned with the combinations of the stochastic Itô-differential and the arbitrary (fractional) orders derivatives in a neutral differential equation with a stochastic, nonlinear, nonlocal integral condition. The existence of solutions will be proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the unique solution will be studied.

In this paper, a new variance assignment method has been proposed for stochastic non-linear systems which are described by stochastic differential equation. The variance governs the shape of the probability density function if the stochastic distribution is Gaussian. It means that the variance assignment is a special case for stochastic distribution control if the control law can be designed to force the probability distribution of investigated stochastic non-linear system as Gaussian. Motivated by Ornstein-Uhlenbeck process and Fokker-Planck-Kolmogorov equation, the backstepping design has been adopted for system stabilisation and the desired variance can be formulated using the free design parameter in the control law. Therefore, the variance assignment can be achieved and the closed-loop system is bounded in probability sense. In addition, the entropy assignment is also discussed as a generalised study of variance assignment. The simulation is given to validate the proposed control algorithm where the effectiveness has been demonstrated.

This paper presents a stochastic differential equation based filter design algorithm for system state estimation. Motivated by backstepping procedure and the variance assignment, the non-linear system filtering has been achieved. In particular, the system dynamics have been described by stochastic differential equation which results in the structure of the filter. The estimation error has been stabilised by backstepping design. Then, the non-linear process of the estimation error is converted to Ornstein-Uhlenbeck process using the filter compensative signal. In the ideal case, the estimation error is a Brownian motion, which leads to the variance assignment of Ornstein-Uhlenbeck process, thus the parameter of the proposed filter can be further optimised. The numerical simulation has been given to show the effectiveness of the proposed design algorithm.

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

One-dimensional homogeneous semi-Markov processes of diffusion type are considered. The transition function of such a process satisfies an ordinary second-order differential equation. It is supposed that the process does not break and has no intervals of constancy. Under these conditions, the Dirichlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1 and 0 at the boundaries of the interval. A criterion for the left-hand boundary of the interval to be unattainable is derived, and for the corresponding values 0 and 1, a criterion for the right-hand boundary of the interval to be unattainable is derived. This criterion applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

The article proposes mathematical models of the stochastic dynamics of the single-factor manufacturing enterprises development through internal and external investments. Balance equations for such enterprises are formulated, describing random processes of continuous increase in output and growth of production factors. The interaction of proportional, progressive and digressive depreciation with internal and external investments is investigated. Equations are obtained to determine the equilibrium state of the enterprise and the limiting values of the factors of production are calculated. The cases of the stable progressive development of the enterprise, the suspension of its work during the re-equipment of production and the temporary crisis of production shutdown during equipment replacement are considered. The algorithm for the numerical solution of stochastic differential equations of enterprise development is constructed in accordance with the Euler–Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random function of the production factor. A variant of the method for calculating the expectation of a random function of a factor of production is developed and the corresponding differential equation is obtained for it. It is shown that the numerical solution of this equation and the average value of the function of the production factor calculated from two hundred realizations of stochastic trajectories give almost identical results. Numerical analysis of the developed models showed good compliance with the known statistical data of the production enterprise.

In this paper we present a data-driven approach for uncertainty propagation. In particular, we consider stochastic differential equations with parametric uncertainty. Solution of the differential equation is approximated using maximum entropy (maxent) basis functions similar to polynomial chaos expansions. Maxent basis functions are derived from available data by maximization of information-theoretic entropy, therefore, there is no need to specify basis functions beforehand. We compare the proposed maxent based approach with existing methods.

In this technical note, we consider linear exponential quadratic (LEQ) control for mean field stochastic differential equations (MFSDEs). The MFSDE includes the expectation value of state and control, and the objective functional is exponential of a quadratic functional in state, control, and their expectations. We obtain the explicit optimal solution as well as the optimal cost. The corresponding optimal solution is linear in state and its expectation, which is characterized by the Riccati differential equations (RDEs). The results are obtained by showing that after applying the completion of squares method, the remaining exponentiated stochastic integral and additional RDE terms can be eliminated together by taking expectation since they constitute the associated Radon–Nikodym derivative. As an extension of the problem, the LEQ zero-sum differential game is considered, for which we obtain the explicit optimal solution (saddle-point equilibrium) as well as the optimal cost.

The phenomenology of population extinction is one of the central themes in population biology which it is an inherently stochastic event. In the present investigation, we study this problem for three different stochastic models built from a single Lotka–Volterra deterministic model. More concretely, we study their mean-extinction time which satisfies the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients; hence, we can only compute numerical approximations. We suggest a finite element method using FreeFem++ . Our analysis and numerical results allow us to conclude that there are important differences between the three models. These differences enable us to choose the most “natural way” to turn a the deterministic model into a stochastic model.

The paper deals with a mathematical model for two interacting populations. Under the assumption of fast stochastic contacting of populations, we derive stochastic Poisson-type differential equations with a small parameter and propose an approximative algorithm for quantitative analysis of population dynamics that consists of two steps. First, we derive an ordinary differential equation for mean value of each population growth and analyse the average asymptotic population behaviour. Then, applying diffusion approximation procedure, we derive a stochastic Ito differential equation for small random deviations on the average motion in a form of a linear non-homogeneous Ito stochastic differential equation and analyse the probabilistic characteristics of the Gaussian process given by this equation.

We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of a G-Wishart process defined according to a G-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the G-Brownian motion matrix, we assume that, in our model, their quadratic covariations are equal to zero. An intermediate result, which states that the eigenvalues never collide, is also obtained. This extends Bru’s results obtained for the classical Wishart process in 1989.

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious t...

In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix.

An optimal control problem for backward doubly stochastic system is considered, where the control domain is not necessarily convex. By the method of classical spike variation and duality technique, one necessary condition and one sufficient condition are established for this kind of optimal control problem. The related adjoint process is characterised by the solution of a forward doubly stochastic differential equation, which formulates a forward–backward doubly stochastic differential equation coupled with the state equation. As an illustration, the authors' theoretical results are applied to study an optimal harvesting problem and a linear-quadratic optimal control problem. Moreover, the corresponding maximum principle with an initial state constraint is obtained by Ekeland's variational principle.

Forecasting non-stationary stochastic time series represents a rather complex problem. The reason is that such temporal series are not only self-similar but also exhibit a Long-Range Dependence (LRD). As it is known, the Fractional Brown Motion (FBM) can generate a non-stationary stochastic time series with self-similarity and LRD. In this study we investigate the properties of the LRD for identification of self-similarity and the LRD of non-stationary stochastic series by Hurst exponent. Parameter estimation is proposed for Stochastic differential Equation (SDE) of FBM based on Maximum Likelihood Estimation (MLE), and proves the convergence of MLE. The SDE is discretized.The difference equation constructed is the prediction model of the iterative format based on FBM. Monte Carlo simulation is applied to check the validity and accuracy of parameter estimation. We also give a practical example to demonstrate the appropriateness of the predictive model.

Recently, the study of opinion dynamics on complex networks has attracted a great deal of attention. In this brief, we take an interest in how an opinion forms in a social network with stochastic noises. We investigate a stochastic opinion dynamics model, in which individuals obtain information from their neighbors in a social network. We model the social network as a strongly connected graph. Each individual has an initial opinion and iteratively exchanges her opinion based on a stubborn opinion, her neighbors’ opinions and small stochastic noises continually. We first construct a general stochastic opinion evolution problem with multiplicative noises as a stochastic differential equation and then investigate the stochastic stability of this equation. We consider whether an opinion may form as a result of stochastic disturbance and local interactions. We also establish two sufficient conditions which ensure that the average opinion and the error vector system for all individuals are stochastic stable, respectively. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.