Martingale Inequalities Research Articles

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators

Let 0<q≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<q\\le \\infty $$\\end{document}, b be a slowly varying function and Φ:[0,∞)⟶[0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Phi : [0,\\infty ) \\longrightarrow [0,\\infty ) $$\\end{document} be an increasing function with Φ(0)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi (0)=0$$\\end{document} and limr→∞Φ(r)=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim \\limits _{r \\rightarrow \\infty }\\Phi (r)=\\infty $$\\end{document}. In this paper, we introduce a new class of function spaces LΦ,q,b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{\\Phi ,q,b}$$\\end{document} which unify and generalize the Lorentz-Karamata spaces with Φ(t)=tp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi (t)=t^p$$\\end{document} and the Orlicz-Lorentz spaces with b≡1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b\\equiv 1$$\\end{document}. Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition qΦ-1<1/q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_{\\Phi ^{-1}}<1/q$$\\end{document} in [24], respectively.

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

Stability and rate of decay for solutions to stochastic differential equations with Markov switching

In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html

:6-moment B-valued martingale inequalities on Lorentz spaces

:6-moment B-valued martingale inequalities on Lorentz spaces

The sharp weighted maximal inequalities for noncommutative martingales

The purpose of the paper is to establish weighted maximal Lp-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the Lp constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.

Almost Sure Exponential Stability of Uncertain Stochastic Hopfield Neural Networks Based on Subadditive Measures

For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield neural networks. Next, we investigate the almost sure exponential stability of uncertain stochastic Hopfield neural networks, using the Lyapunov method, Liu inequality, the Liu lemma, and exponential martingale inequality. In addition, we prove two sufficient conditions for almost sure exponential stability. Furthermore, we consider stabilization with linear uncertain stochastic perturbation and present some exceptional examples. Finally, our paper provides our conclusion.

Averaging Principle for ψ-Capuo Fractional Stochastic Delay Differential Equations with Poisson Jumps

In this paper, we study the averaging principle for ψ-Capuo fractional stochastic delay differential equations (FSDDEs) with Poisson jumps. Based on fractional calculus, Burkholder-Davis-Gundy’s inequality, Doob’s martingale inequality, and the Ho¨lder inequality, we prove that the solution of the averaged FSDDEs converges to that of the standard FSDDEs in the sense of Lp. Our result extends some known results in the literature. Finally, an example and simulation is performed to show the effectiveness of our result.

Martingale inequalities on Musielak–Orlicz Hardy spaces

AbstractGiven a probability space and a Musielak–Orlicz function , we investigate martingale inequalities in the framework of Musielak–Orlicz spaces by constructing atomic decompositions. Especially, the obtained results for Musielak–Orlicz functions with particular structure, including the variable Orlicz functions , , the variable double phase‐growth , and the perturbed variable exponent function are also new. Hence, we handle the martingale inequality for the above functional in a universal way.

Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

AbstractLet be a complete probability space and let be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces , which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces .

Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen’s theorem.We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.We introduce a notion of martingale triangle inequality. We show that Kantorovich–Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

The reverse Hölder inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems

In this paper, we study the connections between three concepts — the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.

PARAMETER ESTIMATION FOR A CLASS OF DIFFUSIONPROCESS FROM DISCRETE OBSERVATION

This paper is concerned with the parameter estimation problem for a class of diffusion process from discrete observations. The approximate likelihood function is given by using a Riemann sum and an Itˆo sum to approximate the integrals in the continuous-time likelihood function. The consistency of the maximum likelihood estimator and the asymptotic normality of the error of estimation are proved by applying the martingale moment inequality, Holder’s inequality, Chebyshev inequality, B-D-G inequality and uniform ergodic theorem. The results are applied to the hyperbolic process.

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space hp for all 0<p<2 by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of hp for all 0<p<1, and provide a constructive proof of the atomic decomposition for p=1 which resolves a main problem on the subject left open for the last twelve years. We also study (p,∞)c-atoms, and show that every (p,2)c-atom can be decomposed into a sum of (p,∞)c-atoms; consequently, for every 0<p≤1, the (p,q)c-atoms lead to the same atomic space for all 2≤q≤∞. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space hp (0<p<1) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.

Exponential Inequality of Marked Point Processes

This paper presents the uniform concentration inequality for the stochastic integral of marked point processes. We developed a new chaining method to obtain the results. Our main result is presented under an entropy condition for partitioning the index set of the integrands. Our result is an improvement of the work of van de Geer on exponential inequalities for martingales in 1995. As applications of the main result, we also obtained the uniform concentration inequality of functional indexed empirical processes and the Kakutani–Hellinger distance of the maximum likelihood estimator.

Multi-channel transmission scheduling with hopping scheme under uncertain channel states

This paper investigates the multi-channel transmission scheduling problem for remote state estimation based on a hopping scheme in cyber-physical systems. The smart sensor sends multiple subpackets over different orthogonal channels to the remote end simultaneously. Owing to the randomness and vulnerability of transmission environments, the uncertain multi-channel states are considered in this paper, which relaxes the assumption of existing deterministic models. The objective is to find an appropriate hopping scheme that minimizes the remote estimation error covariance. First, the multi-channel selection problem is modeled as a multi-arm bandits (MAB) matrix via taking the packet receiving probability as the gain. From the perspective of strategy and channel, two exponential-weight online learning algorithms are designed with the assistance of transmission energy switching policy. Then, based on Bernstein’s inequality for martingales and mini-batching loop, the upper bounds of algorithms’ regret values are analyzed under stochastic and adversarial channel states, respectively. Further, the estimator expression in iterative form and a sufficient condition for the error covariance to be bounded are derived. Finally, an example of unmanned vehicle moving demonstrates all the theoretical results.

Stochastic stabilization of hybrid neural networks by periodically intermittent control based on discrete-time state observations

This paper is concerned with stabilization of hybrid neural networks by intermittent control based on continuous or discrete-time state observations. By means of exponential martingale inequality and the ergodic property of the Markov chain, we establish a sufficient stability criterion on hybrid neural networks by intermittent control based on continuous-time state observations. Meantime, by M-matrix theory and comparison method, we show that hybrid neural networks can be stabilized by intermittent control based on discrete-time state observations. Finally, two examples are presented to illustrate our theory.

Mean-field type quadratic BSDEs

&lt;p style='text-indent:20px;'&gt;In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean. The general case can be treated by a fixed point argument.&lt;/p&gt;

PARTIAL PRACTICAL STABILITY AND ASYMPTOTIC STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY LÉVY NOISE WITH A GENERAL DECAY RATE

In this paper, we mainly study the almost sure partial practical stability of stochastic differential equations driven by Lévy noise with a general decay rate. By establishing a suitable Lyapunov function and using Exponential Martingale inequality and Borel-Cantelli theorem, giving some sufficient conditions that can guarantee the almost sure partial practical stability of equations. At the same time, we also study general conditions that guarantee the almost sure asymptotic stability of the equation. Finally, we also give two examples to illustrate our theoretical results.