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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]

The deterministic and stochastic SIQS models with non-linear incidence are introduced. For deterministic model, the basic reproductive rate R0 is derived. Moreover, if R0≥1, the disease-free equilibrium is globally asymptotically stable and if R0>1, there exists a unique endemic equilibrium which is globally asymptotically stable. For stochastic model, sufficient condition for extinction of the disease that is regardless of the value of R0 is presented. In addition, if the intensities of the white noises are sufficiently small and R0>1, then there exists a unique stationary distribution to stochastic model. Numerical simulations are also carried out to confirm the analytical results. [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, we investigate two-species Lotka–Volterra delayed stochastic predator–prey systems, with and without pollution, denoted by and , respectively. We show that there exists a unique non-negative solution in each system that is permanent in time average under certain conditions. Moreover, the non-permanence of model is studied. Finally, computer simulations are carried out to verify our results. [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: In this paper, we focus on the stability problem for a class of stochastic delayed recurrent neural networks. Different from the traditional stability criteria, we introduce and study a new stability criterion: the mean-square exponential input-to-state stability. To the best of our knowledge, this new stability criterion has never been discussed in the field of stochastic recurrent neural networks. The main objective of the paper is to fill the gap. With the help of the Lyapunov–Krasovskii functional, stochastic analysis theory and Itô's formula, we prove that the addressed system is mean-square exponentially input-to-state stable. Moreover, two numerical examples and their simulations are presented to verify the theoretical results well. [Copyright &y& Elsevier]

Abstract: This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs). It can be considered as an improvement to the result given by Yuan C. et al. in [6]. [Copyright &y& Elsevier]

In this paper, we obtain the stochastic differentials of some of the most common continuous stochastic processes using Itô's formula. [ABSTRACT FROM AUTHOR]

The paper introduces a novel Itô’s formula for time- dependent tempered generalized functions. As an application, we study the heat equation when initial conditions are allowed to be a generalized tempered function. A new proof of the Üstunel-Itô’s formula for tempered distributions is also provided. [ABSTRACT FROM AUTHOR]

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]

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a stochastic multi-group Lotka–Volterra mutualistic system under regime switching. It is well known that the population is forced to expire when the perturbation is sufficiently large. The main aim here is to investigate its ergodic property and positive recurrence by stochastic Lyapunov functions under small perturbation, which can be used to explain some recurring phenomena in practice and thus provide a good description of permanence. The mean of the stationary distribution is estimated. Simulations are also carried out to confirm our analytical results. [Copyright &y& Elsevier]

Let SH be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [ f( SH), SH] (W) defined by $[f(S^H ),S^H ]_t^{(W)} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \tfrac{1} {{\varepsilon ^{2H} }}\int_0^t {\{ f(S_{s + \varepsilon }^H ) - f(S_s^H )\} (S_{s + \varepsilon }^H - S_s^H )ds^{2H} } , $ provided the limit exists in probability, where x ↦ f( x) is a measurable function. We construct a Banach space ℋ of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ ℋ. Moreover, the generalized Bouleau-Yor identity takes the form [Figure not available: see fulltext.] for all f ∈ ℋ, where ℒ H( x, t) is the weighted local time of SH. This allows us to write the generalized Itô’s formula for absolutely continuous functions with derivative belonging to ℋ. [ABSTRACT FROM AUTHOR]

We present in this paper an SIRS epidemic model with saturated incidence rate and disease-inflicted mortality. The Global stability of the endemic equilibrium state is proved by constructing a Lyapunov function. For the stochastic version, the global existence and positivity of the solution is showed, and the global stability in probability and pth moment of the system is proved under suitable conditions on the intensity of the white noise perturbation.

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]

Abstract: In this paper, we obtain sufficient criteria for the existence of periodic solutions to deterministic SIR and SEIR epidemic models with modified saturation incidence rates by means of using the continuation theorem based on coincidence degree theory, and we show that the solution is unique and globally stable. Second, we discuss their corresponding stochastic epidemic models with random perturbation have a unique global positive solution respectively, and we utilize stochastic Lyapunov functions to investigate the asymptotic behavior of the solution. [Copyright &y& Elsevier]

In this article we study a randomized three-dimensional Lotka-Volterra model with competitor-competitor-mutualist interaction. We show the existence, uniqueness, moment boundedness, stochastic boundedness and global asymptotic stability of positive global solutions for this stochastic model. Analytical results are validated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the problem of pricing and hedging of certain catastrophe (CAT) options. In particular, a self-financing hedging strategy that minimizes the risk in the loss period and replicates the option in the development period is proposed. [Copyright &y& Elsevier]

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka-Rosen-Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones. [ABSTRACT FROM AUTHOR]

Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Itô's formula in terms of the extended stochastic integrals is obtained. [ABSTRACT FROM AUTHOR]

This article discusses the exponential stability of nonlinear stochastic delay differential systems (SDDSs) whose coefficients obey the polynomial growth condition. Delay-dependent criteria on almost sure exponential stability and pth moment exponential stability of such SDDSs have been established. By applying some novel techniques, our criteria work for many SDDSs including some cases in which the ℒV operator has a complicated form, which seemingly prevents the existing results from being directly used. The range of order of moment exponential stability and the decay rate can be estimated through the coefficients of the system. [ABSTRACT FROM PUBLISHER]

This paper develops variational inequality approach to establish optimal stopping problems arising from the pricing of stock loan and capped stock loan models. The valuations of the stock loans and ranges of reasonable values of key parameters such as loan sizes, interest rates, fees for providing such a service and quantity of this automatic termination clause and relationships among these parameters as well as the optimal terminable stopping times are explicitly derived. The paper also gives two numerical samples to portray the results and the difference between stock loan model and capped stock loan model. [ABSTRACT FROM AUTHOR]

For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo-differential operators {Δ + bΔα/2; b ∈ [0, 1]} on ℝd that evolves continuously from Δ to Δ + Δα/2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to Δ + bΔα/2 (or, equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b1/α) in C1,1 open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for non-negative functions which are harmonic with respect to Δ + bΔα/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. [ABSTRACT FROM AUTHOR]

For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo-differential operators {Δ + bΔα/2; b ∈ [0, 1]} on ℝd that evolves continuously from Δ to Δ + Δα/2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to Δ + bΔα/2 (or, equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b1/α) in C1,1 open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for non-negative functions which are harmonic with respect to Δ + bΔα/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. [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: This paper addresses the existence and uniqueness of strong solutions to the stochastic variational inequality in a bounded domain with Dirichlet homogeneous conditions. Here for , , for . An application to the one-phase Stefan problem with a stochastic heat source is given. One studies also the corresponding stochastic parabolic equation with Signorini boundary conditions on . [Copyright &y& Elsevier]

In this paper, we discuss the asymptotic stability of nonlinear stochastic delay differential systems (SDDSs) whose coefficients obey the polynomial growth condition. By applying some novel techniques, we establish some easily verifiable conditions under which such SDDSs are almost surely asymptotically stable and pth moment asymptotically stable. A nontrivial example is provided to illustrate the effectiveness of our results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [ABSTRACT FROM AUTHOR]

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: In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis type functional response and logistic growth for predators. The deterministic model is already well-studied and we recall some important results here. We construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations. For the stochastic model, we show that the system admits unique positive global solution starting from the positive initial value. Then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes. Finally, we show that the system has a stationary distribution under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results. [Copyright &y& Elsevier]

The aim of this article is to study the Doob–Meyer decomposition theorem, ∇-stochastic integration and Ito's formula for stochastic processes defined on time scale. The obtained results can be considered as a first attempt on the stochastic calculus on time scale. [ABSTRACT FROM PUBLISHER]

Abstract: This paper analyzes the asymptotic behaviors of the stochastic population model with the Allee effect. We apply the Feller’s test to obtain the criteria of the asymptotic behaviors for this model and compare our results with those obtained by Lyapunov functions. At last, the numerical simulations conform our results. [Copyright &y& Elsevier]

Abstract: Under the framework of -expectation and -Brownian motion, we introduce Itô’s integral for stochastic processes without assuming quasi-continuity. Then we can obtain Itô’s integral on stopping time interval. This new formulation permits us to obtain Itô’s formula for a general -function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010)  as well as those of Gao (2009)  and Zhang et al. (2010) . [Copyright &y& Elsevier]

Abstract: In this paper we consider the stochastic differential equation (SDE) population model for interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004)  and Mao (2005) ) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the problem of non-existence of global solutions for a class of stochastic reaction–diffusion equations of Itô type. Under some sufficient conditions on the initial state, the nonlinear term and the multiplicative noise, it is proven that, in a bounded domain , there exist positive solutions whose mean -norm will blow up in finite time for , while, if , the previous result holds in any compact subset of . Two examples are given to illustrate some application of the theorems. [ABSTRACT FROM AUTHOR]

This paper discusses the issue of stability and stabilization for the non-linear stochastic systems with time-varying delay subject to passivity performance. Based on the fuzzy modelling approach, the non-linear systems can be represented by a stochastic T-S fuzzy model whose consequent part is described by linear differential equations with a multiplicative noise term. For analysing the stability of the stochastic T-S fuzzy model, the less conservative stability criterion derived by Itô's formula and the Lyapunov-Krasovskii function is proposed as a linear matrix inequality (LMI) problem. Furthermore, the passivity theory is employed to discuss the external disturbance effect on the considered system. Through applying the above techniques, the sufficiency conditions are derived to design fuzzy controllers such that the stochastic T-S fuzzy model with time delay is asymptotically stable and passive in the mean square. Finally, two numerical examples are proposed to show the application and effectiveness of the proposed fuzzy controller design method. [ABSTRACT FROM AUTHOR]

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]

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given. [ABSTRACT FROM AUTHOR]

Abstract: A stochastic ratio-dependent predator–prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itô’s formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [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]