Inequality For Stochastic Integrals Research Articles

HMM-based asynchronous synthesis for Interval type-2 fuzzy semi-Markovian jump systems with its applications: A bilateral looped functional methodology

This article mainly addresses the hidden-Markovian model-based asynchronous sampled-data control (SDC) synthesis problem for interval type-2 (IT2) fuzzy semi-Markovian jump systems (S-MJSs) with stochastic actuator failures (SAFs). Herein, the asynchronous factor resulted from twofold: (1) hidden Markovian model (HMM) and (2) imperfectly matched membership functions (IMMFs) between the original plant and the controller. Aiming at approximating the nonlinearity along with randomness with high accuracy and broadening practical application background, the IT2 fuzzy S-MJSs associated with IMMFs is established. At the beginning, a HMM is borrowed to portray the discrepancy between the corresponding plant and the desired controller. Additionally, in comparison with majority of some previous sampled-data control mechanisms, a intrinsic lag signal and gain fluctuations are purposefully introduced in the double asynchronous control strategy, which exploited initially to settle the control synthesis issue. By considering the information of sampling intervals x(tk)→x(t)→x(tk+1) along with x(tk−τ)→x(t−τ)→x(tk+1−τ) adequately, a fresh looped Lyapunov functional called bilateral looped Lyapunov functional is innovatively constructed. Whereafter, by suitably incorporating with the skills of stochastic analysis and relaxed integral inequality (including generalized free matrix-based integral inequality and canonical Bessel–Legendre integral inequality) together with symmetric slack matrices, numerically testable sufficient conditions are established in the form of a group of feasible multi-parameterized linear matrix inequalities (M-PLMIs). The theoretical developments are implies the hierarchy character, and the conservatism decreases with the increase of the degree of hierarchy. Conclusively, the design methodology brought forward herein is demonstrated by two practical examples, which show the potency and practicability.

New stochastic fractional integral and related inequalities of Jensen–Mercer and Hermite–Hadamard–Mercer type for convex stochastic processes

In this investigation, we unfold the Jensen–Mercer (mathtt{J-M}) inequality for convex stochastic processes via a new fractional integral operator. The incorporation of convex stochastic processes, the mathtt{J-M} inequality and a fractional integral operator having an exponential kernel brings a new direction to the theory of inequalities. With this in mind, estimations of Hermite–Hadamard–Mercer (mathtt{H-H-M})-type fractional inequalities involving convex stochastic processes are presented. In the context of the new fractional integral operator, we also investigate a novel identity for differentiable mappings. Then, a new related mathtt{H-H-M}-type inequality is presented using this identity as an auxiliary result. Applications to special means and matrices are also presented. These findings are particularly appealing from the perspective of optimization, as they provide a larger context to analyze optimization and mathematical programming problems.

Sampled-data based extended dissipative synchronization of stochastic complex dynamical networks

<p style='text-indent:20px;'>In this paper, extended dissipative (ED) synchronization is considered for stochastic complex dynamical networks (SCDNs) with variable coupling delay via sampled-data control (SDC). First, a suitable Lyapunov–Krasovskii functional (LKF) is constructed, then a new synchronization criterion is obtained through stochastic integral inequality (SII) and linear matrix inequality (LMI) techniques. Moreover, the ED synchronization criteria are established, which consolidates passivity, dissipativity, <inline-formula><tex-math id="M1">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M2">\begin{document}$ L_2-L_\infty $\end{document}</tex-math></inline-formula> performances in a unified structure. SDC gain matrices are also designed for each performance in ED criteria. Finally, the feasibility and usefulness of the derived theoretical results are shown through numerical simulations.</p>

A stochastic convolution integral inequality

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

Improved stochastic integral inequalities to stability analysis of stochastic genetic regulatory networks with mixed time‐varying delays

This study investigates the problem of mean-square asymptotic stability analysis of stochastic genetic regulatory networks with mixed time-varying delays. The innovation of this study lies in that several stochastic integral inequalities are improved based on auxiliary function integral inequalities. Thereby, less conservative mean-square asymptotic stability criteria are derived by constructing Lyapunov–Krasovskii functional with multiple integral terms and employing the proposed stochastic integral inequalities and the (reciprocal) convex combination technique. The obtained stability criteria are described in terms of linear matrix inequalities, which can be easily verified by using standard software tools. Finally, three numerical examples are provided to test and verify the effectiveness and less conservativeness of the proposed stability criteria.

L2-L∞ filtering for stochastic delayed systems with randomly occurring nonlinearities and sensor saturation

This work investigates L2-L∞ filtering problem for a class of stochastic systems with time-delay and randomly occurring phenomena. The purpose of the addressed problem is to design a full order filter to ensure the filtering error system is asymptotically mean-square stable with prescribed L2-L∞ performance. A novel stochastic time-delay system model is established, in which randomly occurring nonlinearities and sensor saturation phenomena are considered. Then, a novel functional containing negative definite terms is constructed to relax the constraints on the functional, and a new free-matrix-based stochastic integral inequality is also given. Meanwhile, a novel L2-L∞ performance analysis method making full use of delay information is proposed. As a result, less conservative conditions for the existence of filters are obtained, under which the L2-L∞ performance level can be achieved for the filtering error system. Numerical examples are employed to show the benefits of the proposed approach.

Extended dissipative state estimation of delayed stochastic neural networks

In this article, extended dissipative state estimation (EDSE) criteria is established for stochastic neural networks with variable delay states. The EDSE criteria includes passivity state estimation, dissipativity state estimation, H∞ state estimation and L2−L∞ state estimation performances. Sufficient conditions for EDSE criteria are established by choosing Lyapunov–Krasovskii functionals (LKF) and using linear matrix inequality (LMIs) technique. New stochastic double integral inequality is applied for the integral term in the derivative of the chosen LKF in main results, which helps to establish EDSE criterion for delayed stochastic neural networks. The feasibility and superiority of the proposed approach is shown by providing numerical examples.

Global dynamics in a stochastic three species food-chain model with harvesting and distributed delays

This paper proposes a stochastic three species food-chain model with harvesting and distributed delays. Some criteria for the global dynamics of all positive solutions, including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, are established by using the stochastic integral inequalities, Lyapunov function method, and the inequality estimation technique. Furthermore, the effects of harvesting are discussed, the optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY for short) are obtained. Finally, numerical examples are carried out to illustrate our main results.

A stochastic Gronwall lemma revisited

A stochastic integral inequality of the Gronwall type is established. The result complements a stochastic Gronwall lemma proved by Scheutzow3.

On Bernstein type inequalities for stochastic integrals of multivariate point processes

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).

Global attracting sets and stability of neutral stochastic functional differential equations driven by Rosenblatt process

We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.

Formula omitted] filtering for stochastic time-varying delay systems based on the Bessel–Legendre stochastic inequality

This paper deals with the L2−L∞ filtering problem for stochastic systems with time-varying delay. First, based on the Bessel–Legendre inequality, a new stochastic integral inequality is established, which is called Bessel–Legendre stochastic inequality. Then, a new Lyapunov–Krasovskii functional is constructed for a stochastic time-varying delay system by utilizing Legende polynomials. With the help of the Bessel–Legendre stochastic inequality and the new Lyapunov–krasovskii functional, an L2−L∞ filter is developed, which can guarantee the filtering error system to be asymptotically mean-square stable with a prescribed L2−L∞ performance level. Finally, numerical examples are given to illustrate the effectiveness of the proposed filtering approach. The example results show that the proposed approach is less conservative than existing ones in designing the L2−L∞ filter for the stochastic time-delay system.

Stochastic integrals and BDG’s inequalities in Orlicz-type spaces

In this paper we extend an inequality of Lenglart et al. (1980, Lemma 1.1) to general continuous adapted stochastic processes with values in topological spaces. Using this inequality we prove Burkholder–Davies–Gundy’s inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions. As an application, we show the well-posedness of stochastic heat equations in Orlicz spaces.

A powered Gronwall-type inequality and applications to stochastic differential equations

In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.

Refinements of mean-square stochastic integral inequalities on convex stochastic processes

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143–151, 2012; Aequat Math 86:91–98, 2013) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite–Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given.

L2–L∞ filtering for stochastic Markovian jump delay systems with nonlinear perturbations

This paper is concerned with the L2–L∞ filtering problem for Itô stochastic delayed Markovian jump systems subject to nonlinear parameter and sensor perturbations. The nonlinear perturbations in both state and measurement equations considered in the existing literature are generalizeed by including the cross information among the current state, the delayed state and the nonlinear perturbations. Based on a stochastic integral inequality and the convex analysis property, an L2–L∞ performance condition is presented to guarantee the mean-square exponential stability of the resulting filtering error system with prescribed L2–L∞ disturbance attenuation level. By utilizing the information of the time-varying delay, the delay is not estimated by the worst-case enlargement such that the conservatism is reduced. Then with the obtained performance analysis result, a stochastic L2–L∞ filter for the system under consideration is designed. Illustrative examples are given to demonstrate the usefulness of the developed approach.

Maximal weak-type inequality for stochastic integrals

Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates. (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$ (ii) If $X$ is arbitrary, then$$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.

Dissipativity analysis of stochastic neural networks with time delays

This paper is concerned with the dissipativity problem of stochastic neural networks with time delay. A new stochastic integral inequality is first proposed. By utilizing the delay partitioning technique combined with the stochastic integral inequalities, some sufficient conditions ensuring mean-square exponential stability and dissipativity are derived. Some special cases are also considered. All the given results in this paper are not only dependent upon the time delay, but also upon the number of delay partitions. Finally, some numerical examples are provided to illustrate the effectiveness and improvement of the proposed criteria.

Sharp and Strict $L^p$-Inequalities for Hilbert-Space-Valued Orthogonal Martingales

The paper contains the proofs of sharp moment estimates for Hilbert-space valued martingales under the assumptions of differential subordination and orthogonality. The results generalize those obtained by Banuelos and Wang. As an application, we sharpen an inequality for stochastic integrals with respect to Brownian motion.

Stochastic integral inequalities with applications

In this paper, we study some new stochastic inequalities involving the Itntegral and give some estimates for the solutions of controlled stochastic differential equations. As applications, we utilize the stochastic integral inequalities presented in this paper to show an existence theorem of the solution for a class of stochastic differential equations and to give necessary conditions that make the solution for a class of stochastic differential equations be a diffusion process.