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We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution. [ABSTRACT FROM AUTHOR]

Motivated by P.L. Lions' seminal course on Mean Field Games (MFG) and the associated Master equation at Collège de France [19] , we investigate in the case of absence of control mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process. This BDSDE can be interpreted as randomly perturbed recursive cost functional which we associate with a forward mean-field SDE. Defining the value function V = V (t , x , μ) through this BDSDE, we prove that V = V (t , x , μ) is the unique classical solution of a backward stochastic PDE over [ 0 , T ] × R d × P 2 (R d). This backward SPDE extends in the non-control case the Master equation studied in earlier works by different authors. The associated MFG can be interpreted as a forward dynamics endowed with a randomly perturbed recursive cost functional, the solution of the MFG system (u (t , x) , m (t)) remains also in this extended case related with V = V (t , x , μ) by the relation u (t , x) = V (t , x , m (t)). Our main objective is to characterise the random field V as classical solution of the backward SPDE. For this we study the regularity of V over [ 0 , T ] × R d × P 2 (R d). However, unlike the pioneering paper on BDSDEs by Pardoux-Peng [25] where V does not depend on the measure, here we have only an L 2 -regularity of V with respect to its variables. Other difficulties to overcome and subtle technical proofs are related with the study of the L 2 -regularity and, hence, that of the solution of the underlying mean-field BDSDE. It adds that our BDSDE needs weaker assumptions on the coefficients than in [25]. Finally, let us point out that for our approach we extend the classical mean-field Itô formula to smooth functions of solutions of mean-field BDSDEs. This formula contains earlier (mean-field) Itô formulas as special cases. [ABSTRACT FROM AUTHOR]

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the n th exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô's formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear stochastic eco-epidemic model is presented to study the dynamics of harvested prey–predator model with group defence. One of the key roles of forming a group is to defend themselves from the attack of the predators. Since prey population spreads infection through direct contact; we consider two groups of prey population. Using some certain assumptions, a stochastic model is formulated by considering noisy intrinsic growth rate of susceptible pest population, death rate of infected pest and predator populations, respectively. Next, the existence and uniqueness of positive solution are performed using comparison theorem of stochastic differential equations (SDEs). The boundedness and uniform continuity of the system are analyzed. Then, the bionomic equilibrium and optimal harvesting policy are obtained by using stochastic maximum principle. We verify the conditions of extinction and persistence in mean with the numerical results. To verify our consideration with the reality, the effects of harvesting effort and catchability coefficient are observed. In addition, infected and uninfected regions are reported to specify the infection free system. [ABSTRACT FROM AUTHOR]

The estimates of N. V. Krylov for distributions of stochastic integrals by means of the Ld-norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W2d. We also show how to use these estimates for the investigation of L2-convergence of solutions of optional SDE's. [ABSTRACT FROM AUTHOR]

Guo et al. [8] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [8] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M‐matrix are used to study exponential stability and boundedness of the hybrid NPSDEs. [ABSTRACT FROM AUTHOR]

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution ( Y t , ξ , Z t , ξ , H t , ξ ) , ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) of the split equations are proved. The first and the second order derivatives of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to x , the derivative of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to the measure P ξ , and the derivative of the process ( ∂ μ Y t , x , P ξ ( y ) , ∂ μ Z t , x , P ξ ( y ) , ∂ μ H t , x , P ξ ( y ) ) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L 2 . The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V ( t , x , P ξ ) ≔ Y t t , x , P ξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type. [ABSTRACT FROM AUTHOR]

We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications. [ABSTRACT FROM AUTHOR]

A linear stochastic differential equation in an arbitrary separable Banach space is considered. To solve this equation, the corresponding linear stochastic differential equation for generalized random processes is constructed and its solution is produced as a generalized Itô process. The conditions under which the received generalized random process is the Itô process in a Banach space are found, and thus the solution of the considered linear stochastic differential equation is obtained. The heart of this approach is the conversion of the main equation in a Banach space to the equation for generalized random processes, to find the generalized solution, and then to learn the conditions under which the obtained generalized random process is the random process with values in a Banach space. [ABSTRACT FROM AUTHOR]

By means of Itô's formula and a comparison theorem for integral equations, we study the blow up in finite time of semilinear stochastic differential equations of the form Here, is non-negative and non-decreasing by components, is a predictable and continuous process, is an -Brownian motion and is an -measurable random variable. The results of this paper can be seen as extensions of the Feller and Osgood criteria. [ABSTRACT FROM AUTHOR]

The main objective of this work is to include randomness into the model for dynamics of tuberculosis. Stochasticity to the model is introduced through perturbation of parameters which is a standard method in stochastic population modeling. The wellposedness analysis of SDEs includes the existence of non-negative solutions as required in the dynamics of population modeling. A detailed stability analysis of results, analytical properties and asymptotical behavior of solutions is also done. The mean reverting process is approximated for one of the variables and the mean and variance of the process are found. [ABSTRACT FROM AUTHOR]

In this paper we investigate the positivity and boundedness of the solution of a stochastic seasonal epidemic model for the respiratory syncytial virus (RSV ). The stochasticity in the model is due to fluctuating physical and social environments and is introduced by perturbing the transmission parameter of the seasonal disease. We show the existence and uniqueness of the positive solution of the stochastic seasonal epidemic model which is required in the modeling of populations since all populations must be positive from a biological point of view. In addition, the positivity and boundedness of solutions is important to other nonlinear models that arise in sciences and engineering. Numerical simulations of the stochastic model are performed using the Milstein numerical scheme and are included to support our analytic results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to discuss the Lp-estimate of solution of stochastic differential equations under nonlinear growth condition. Furthermore, almost surely asymptotic estimates for the solutions of these equations are given. More accurately, we shall estimate sample Lyapunov exponent almost surely. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided. [Copyright &y& Elsevier]

Abstract: In this paper, a new method is introduced to enhance feedback robustness against communication gain uncertainties. The method employs a fundamental property in stochastic differential equations to add a scaled stochastic dither under which tolerable gain uncertainties can be much enlarged, beyond the traditional deterministic optimal gain margin. Algorithms, stability, convergence, and robustness are presented for first-order systems. Extension to higher-dimensional systems is further discussed. Simulation results are used to illustrate the merits of this methodology. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to improve the results on stability in distribution of nonlinear stochastic differential equations in Yuan and Mao (2003). Both conditions for the stability in distribution given in that paper are weakened. [Copyright &y& Elsevier]

Abstract: We study the stability of receding horizon control for continuous-time non-linear stochastic differential equations. We illustrate the results with a simulation example in which we employ receding horizon control to design an investment strategy to repay a debt. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given. [Copyright &y& Elsevier]

The solution of a stochastic optimal control problem may be associated with that of the Hamilton–Jacobi–Bellman (HJB) equation, which is a second order partial differential equation subject to a terminal condition. When this equation is semilinear and satisfies certain other constraints, it can be solved via a nonlinear version of the Feynman–Kac formula. According to this approach, the solution to the HJB equation can be obtained by simulating an associated pair of partly coupled forward–backward stochastic differential equations. Although an elegant way to interpret and solve a partial differential equation, simulating the system of forward–backward equations can be computationally inefficient. In this work, the HJB equation pertaining to the optimal control problem is reformulated such that instead of the given terminal condition, it is now subject to an appropriate initial condition. In the process, while the total cost associated with the control problem remains unchanged, pathwise solutions may not. Associated with the new partial differential equation, we then derive a set of stochastic differential equations whose solutions move only forward in time. This approach has a significant computational advantage over the original formulation. Moreover, since the forward–backward approach generally requires simulating stochastic differential equations over the current to the terminal time at every step, the integration errors may accumulate and carry forward in estimating the control. This error is particularly high initially since the time of integration is the longest there. The proposed method, numerically implemented for the control of stochastically excited oscillators, is free of such errors and hence more robust. Unsurprisingly, it also exhibits lower sampling variance. [Display omitted] • A Monte Carlo scheme with forward time integration for stochastic optimal control. • Based on a modified HJB equation, now supplied with an initial condition. • Achieved through a smearing of the terminal cost by Ito's formula. • Vastly reduced computational cost and sampling variance. • Advantages illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions. [ABSTRACT FROM AUTHOR]

The main aim of this article is to deal with the almost-sure stability of stochastic differential delay equations. Our improved theorems give better results while conditions imposed on the Lyapunov function are much weaker, thus, it is easier to find a right Lyapunov function in application. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we investigate the problem of suppression explosive solutions by noise for nonlinear deterministic differential system. Given a deterministic differential system with coefficients satisfying a more general one-sided polynomial growth condition, we introduce Brownian noise feedback and therefore stochastically perturb this system into the nonlinear stochastic differential system . We show that appropriate guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic systems may explode in a finite time. Under some weaker conditions, we reveal that the single noise can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomially. [Copyright &y& Elsevier]

Abstract: The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. ). The numerical solutions under these conditions have also been discussed intensively (see e.g. ). Recently, Mao and Rassias established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap. [Copyright &y& Elsevier]

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on the robust passive stability and stabilization problems for uncertain nonlinear stochastic time-delay systems. Via the fuzzy modeling approach, the nonlinear stochastic system is described by Takagi–Sugeno (T–S) fuzzy model in which the consequent parts are presented by linear stochastic time-delay differential equation. With Lyapunov–Krasovskii function and improved Jensen''s inequality, the stability criteria are derived. In addition, the passivity theory is employed to discuss external disturbance effect on system for achieving attenuation performance. According to the proposed design method, the fuzzy controller is carried out by parallel distributed compensation (PDC) concept to guarantee the robust asymptotical stability and attenuation performance of system in the sense of mean square. Finally, a synchronous generator power system is presented to manifest the application and effectiveness of the proposed fuzzy control method. [Copyright &y& Elsevier]

Abstract: We trace Itô’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions. [Copyright &y& Elsevier]

We will concentrate on a 1-dimensional nonlinear filtering problem, which allows an explicit solution in terms of a stochastic differential equation for $$ \hat X_t $$. [ABSTRACT FROM AUTHOR]

Abstract: This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69–84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population. [Copyright &y& Elsevier]

In this article, we stochastically perturb the classical non-autonomous Lotka-Volterra model [image omitted] into the stochastic differential equation [image omitted] Different from most of existing articles, for example, [3, 20] the system parameters in this article are time-dependent. We will give a sufficient condition under which the stochastic differential equation will have a unique global positive solution. We will then establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction. [ABSTRACT FROM AUTHOR]

Abstract: We develop a notion of nonlinear expectation–-expectation–generated by a nonlinear heat equation with infinitesimal generator . We first study multi-dimensional -normal distributions. With this nonlinear distribution we can introduce our -expectation under which the canonical process is a multi-dimensional -Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our -Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our -expectation. [Copyright &y& Elsevier]

Abstract: In this paper we will develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We will show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to improve some results obtained by Mao [X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud. 7 (2000) 307–328]. Our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above. For example, we need only the local Lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions. To guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation (SFDE) under the weaker conditions imposed in this paper, we establish a generalised existence-and-uniqueness theorem which covers a wider class of nonlinear SFDEs as demonstrated by the examples discussed in this paper. Moreover, from our improved results follow some new criteria on the stochastic asymptotic stability for SFDEs. [Copyright &y& Elsevier]

The classical Khasminskii theorem (see [6]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied. [ABSTRACT FROM AUTHOR]

In financial markets traders often protect their position from a significant decline by using a trailing stop. Assume the trader is long the market (owns the security). A trailing stop is an order to sell the security at the market, if the price of the security drops to the stop price. The stop price is always less than the market price when the stop is entered. As the price fluctuates, the stop is raised to remain a fixed distance from the maximum price at which the security trades. In this paper we consider two models for the price process: a discrete time random walk and continuous time Brownian motion, both with positive drift. For these price processes we compute the distribution, mean, and variance of the gain to the trader as well as the duration of the trade when a trailing stop strategy is used. Also discussed is the question of optimizing the distance from the current price to the stop. [ABSTRACT FROM AUTHOR]