Gundy Inequalities Research Articles

Doob's inequality, Burkholder–Gundy inequality and martingale transforms on martingale local Morrey spaces

Doob's inequality, Burkholder–Gundy inequality and martingale transforms on martingale local Morrey spaces

Martingale Hardy Orlicz–Lorentz–Karamata spaces and applications in Fourier analysis

Abstract We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.

Φ-moment inequalities for noncommutative differentially subordinate martingales

We establish some Φ \Phi -moment inequalities for noncommutative differentially subordinate martingales. Let Φ \Phi be a p p -convex and q q -concave Orlicz function with 1 > p ≤ q > 2 1>p\leq q>2 . Suppose that x x and y y are two self-adjoint martingales such that y y is weakly differentially subordinate to x x . We show that, for N ≥ 0 N\geq 0 , τ [ Φ ( | y N | ) ] ≤ c p , q τ [ Φ ( | x N | ) ] , \begin{equation*} \tau \big [\Phi (|y_N|)\big ]\leq c_{p,q}\tau \big [\Phi (|x_N|)\big ], \end{equation*} where the constant c p , q c_{p,q} is of the best order when p = q p=q . The Φ \Phi -moment estimates for square functions of noncommutative differentially subordinate martingales are also obtained in this article. Our approach provides constructive proofs of noncommutative Φ \Phi -moment Burkholder–Gundy inequalities and Burkholder inequalities.

Well‐posedness of quantum stochastic differential equations driven by fermion Brownian motion in noncommutative Lp‐space

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space for . First, we investigate the existence and uniqueness of ‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of ‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions.

Second-order neutral impulsive stochastic evolution equations with infinite delay: existence, uniqueness and averaging principle

In this paper, a class of second-order neutral impulsive stochastic evolution equations with infinite delay driven by fractional Brownian motion and Poisson jumps is considered. We utilise the Carathéodory approximation technique, the Burkhölder–Davies–Gundy inequality and the Gronwall inequality to present the existence and uniqueness of mild solutions for the addressed system under Carathéodory conditions. Furthermore, we derive an averaging principle for the proposed system which is approximated by the simplified equation in the sense of mean square. Finally, an application is used to illustrate the obtained theoretical results.

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.

A stochastic Hepatitis B model with infinite Lévy jumps

In this paper, we present the following stochastic hepatitis B epidemic model driven by a standard Brownian motion and Lévy jump noise and we discuss the impact of infinite Lévy jumps by the definition of the new threshold. Large time estimates are established based on Kunita’s inequality rather than Burkholder–Davis–Gundy inequality for continuous diffusions. We prove the existence of a unique global positive solution. The impact of stochasticity and the effect of vaccination is taken into consideration in the formulation of sufficient conditions for the extinction and persistence of Hepatitis B. We illustrate our theoretical results by numerical simulations.

Stability analysis for a class of stochastic delay nonlinear systems driven by G-Lévy Process

The regularity and stability of the solution to a class of stochastic delay differential equation driven by G-Lévy processes are studied in this paper. Firstly, we introduce a new Burkholder–Davis–Gundy (BDG) inequality involving the jump measure. Secondly, we use the BDG inequality to establish the existence and uniqueness of the solution under non-Lipschitz condition. Thirdly, we establish the existence, uniqueness, quasi-sure exponential stability and pth moment exponential stability of the solution under local Lipschitz condition and one-sided polynomial growth condition.

Maximal inequalities and some applications

A maximal inequality is an inequality which involves the (absolute) supremum sups⩽t|Xs| or the running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process –, strong Markov processes and Gaussian processes. Using the Burkholder–Davis–Gundy inequalities we also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.

The existence and averaging principle for stochastic fractional differential equations with impulses

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch's fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–Bellman inequality, and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.

Stochastic fractional differential equations with generalized Caputo's derivative and impulsive effects

In this paper, impulsive stochastic fractional differential equations (ISFDEs) in Lp (p> 2) space are introduced. We present a general framework for finding solution for ISFDEs. Then, by using the Burkholder - Davis - Gundy inequality and Holder's inequality, we prove the existence and uniqueness of solution to ISFDE by fixed point theorem. We also investigate Lipschitz continuity of solutions with respect to initial values by using Gronwall inequality. Finally, we provide an application to illustrate the results we obtained.

A two-dimensional stochastic fractional non-local diffusion lattice model with delays

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

The exponential behavior and stability of the stochastic three-dimensional primitive equations driven by Lévy noise

This paper establishes the exponential behavior and stability of the stochastic three-dimensional primitive equations driven by Lévy noise via Burkholder–Davis–Gundy inequality and Itô formula. In particular, we prove that under some conditions on the forcing terms, the weak solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

Additive functionals as rough paths

We consider additive functionals of stationary Markov processes and show that under Kipnis–Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First, we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible Ornstein–Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step, we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder–Davis–Gundy inequality for local martingale rough paths of (In Séminaire de Probabilités XLI (2008) 421–438 Springer; In Probability and Analysis in Interacting Physical Systems (2019) 17–48 Springer; J. Differential Equations 264 (2018) 6226–6301) to the case where only the integrator is a local martingale.

Stochastic SIR Lévy Jump Model with Heavy-Tailed Increments

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy inequality for continuous processes, and allows for the treatment of infinite Lévy measures by the definition of new threshold \(\mathcal {\bar{R}}_0\). An SIR model driven by a tempered stable process is presented as an example of application with the ability to model sudden disease outbreak, illustrated by numerical simulations. The results show that persistence and extinction are dependent not only on the variance of the processes increments, but also on the shapes of their distributions.

Burkholder\u2013Davis\u2013Gundy Inequalities in UMD Banach Spaces

In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M_0=0, we show that the following two-sided inequality holds for all 1le p<infty : Here gamma ([![M]!]_t) is the L^2-norm of the unique Gaussian measure on X having [![M]!]_t(x^*,y^*):= [langle M,x^*rangle , langle M,y^*rangle ]_t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of (star ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (star ) holds for all 0<p<infty , and that for purely discontinuous martingales the right-hand side of (star ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (star ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (star ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

Tree Martingales in Noncommutative Probability Spaces

We introduce the notion of a tree martingale in a noncommutative probability space and prove the Burkholder–Gundy inequalities for tree martingales in symmetric operator spaces. In particular, we establish some inequalities in this setting via an approach based on the concept of generalized singular valued functions of noncommutative random valuables.

Stability of Traveling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations

In this paper we establish the meta-stability of traveling waves for a class of reaction-diffusion equations forced by a multiplicative noise term. In particular, we show that the phase-tracking technique developed in [C. H. S. Hamster and H. J. Hupkes, SIAM J. Appl. Dyn. Syst., 18, pp. 205--278; Phys. D, 401, 132233] can be maintained over timescales that are exponentially long with respect to the noise intensity. This is achieved by combining the generic chaining principle with a mild version of the Burkholder--Davis--Gundy inequality to establish logarithmic supremum bounds for stochastic convolutions in the critical regularity regime.

Dynamic stability analysis of stochastic fractional-order memristor fuzzy BAM neural networks with delay and leakage terms

This paper deals with the uniform stability in mean square of stochastic fractional-order memristor fuzzy BAM neural networks with delay and leakage terms. By employing the ideas of Cauchy–Schwartz inequality, Burkholder–Davis–Gundy inequality, analysis technique, some sufficient conditions are derived to ensure the uniform stability in mean square of such networks. The existence, uniqueness and stability of its equilibrium point are also showed. Two different fractional-order derivatives between the U-layer and V-layer are taken into consideration synchronously with fractional order 1/2 ≤ α < 1. Two examples are also provided to illustrate the effectiveness of our results.

Estimation for incomplete information stochastic systems from discrete observations

This paper is concerned with the estimation problem for incomplete information stochastic systems from discrete observations. The suboptimal estimation of the state is obtained by constructing the extended Kalman filtering equation. The approximate likelihood function is given by using a Riemann sum and an Itô 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, Hölder’s inequality, the Chebyshev inequality, the Burkholder–Davis–Gundy inequality and the uniform ergodic theorem. An example is provided to verify the effectiveness of the estimation methods.