Gundy Inequalities Research Articles

Comments on “Strong convergence rates for backward Euler on a class of nonlinear jump–diffusion problems” [J. Comput. Appl. Math. 205 (2007) 949–956

In this note, we correct an inaccuracy of Eq. (13) in the proof of Theorem 1 in the paper (Higham and Kloeden, 2007). We also give a complete proof of the theorem by applying the Burkholder–Davis–Gundy Inequality.

New martingale inequalities and applications to Fourier analysis

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on HφS(Ω) and HφM(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

Variational estimates for martingale paraproducts

We show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053).

Applications of pathwise Burkholder–Davis–Gundy inequalities

In this paper, after generalizing the pathwise Burkholder–Davis–Gundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities 1 to the Bessel process of order $\alpha\geq1$ 2 to the case of a random exponent $p$ 3 to martingales stopped at a time $\tau$ which belongs to a well studied class of random times

The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy (Acta Math. 124 (1970) 249–304) and Davis (Israel J. Math. 8 (1970) 187–190) for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}]\leq C_{p}\mathbb{E}[(B^{*}(\tau))^{p}]$ with $0<p<2$ we propose a connection of the optimal constant $C_{p}$ with an ordinary integro-differential equation which gives rise to a numerical method of finding this constant. Based on numerical evidence, we are able to calculate, for $p=1$, the explicit value of the optimal constant $C_{1}$, namely $C_{1}=1.27267\ldots$ . In the course of our analysis, we find a remarkable appearance of “non-smooth pasting” for a solution of a related ordinary integro-differential equation.

A maximal inequality for fractional Brownian motions

We extend the Burkholder–Davis–Gundy inequalities for fractional Brownian motions with the Hurst index less than 1/2, completing the proof of Novikov–Valkeila's conjecture that these inequalities hold for arbitrary Hurst index.

Inequalities for noncommutative differentially subordinate martingales

The classical differential subordination of martingales, introduced by Burkholder in the eighties, is generalized to the noncommutative setting. Working under this domination, we establish the strong-type inequalities with the constants of optimal order as p→1 and p→∞, and the corresponding endpoint weak-type (1,1) estimate. In contrast to the classical case, we need to introduce two different versions of noncommutative differential subordination, depending on the range of the exponents. For the Lp-estimate, 2≤p<∞, a certain weaker version is sufficient; on the other hand, the strong-type (p,p) inequality, 1<p<2, and the weak-type (1,1) estimate require a stronger version. As an application, we present a new proof of noncommutative Burkholder–Gundy inequalities. The main technical advance is a noncommutative version of the good λ-inequality and a certain summation argument. We expect that these techniques will be useful in other situation as well.

Quasi-martingale Inequalities in Noncommutative Symmetric Spaces

In this paper, we prove Doob’s inequality and Burkholder–Gundy inequalities for quasi-martingales in noncommutative symmetric spaces. We also consider the inclusion relations between $$\widetilde{\mathcal {H}}_E(\mathcal {M})$$ and $$\widetilde{h}_{E}(\mathcal {M})$$ , where $$\widetilde{\mathcal {H}}_E(\mathcal {M})$$ and $$\widetilde{h}_{E}(\mathcal {M})$$ are the Hardy spaces of quasi-martingales in noncommutative symmetric spaces.

Global Well-Posedness and Dynamical Behavior of Delayed Reaction-Diffusion BAM Neural Networks Driven by Wiener Processes

This paper studies the global existence and uniqueness as well as asymptotic behavior of the reaction-diffusion bidirectional associative memory neural networks with S-type distributed delays and infinite dimensional Wiener processes. The conspicuous characteristics of this system are neurons in one layer interacting with neurons in another layer and the noise which disturbed this system has both time and spatial structure. First, several inequalities are proposed and proved for the preparation of future study. Then, the system is coped in the framework of semigroup theory and functional space. Furthermore, the local existence and uniqueness of mild solution for this system is proven by using the contraction mapping principle coupled with many functional inequalities such as Young inequality, Burkholder–Davis–Gundy inequality, Poincare inequality. The global well-posedness are proven through a prior estimate from constructing appropriate Lyapunov–Krasovskii functional. Moreover, the existence of equilibrium is solved by using the topological degree theory and homotopy invariance. At last, the globally exponential stability of the equilibrium in the mean square sense is studied by constructing appropriate vector Lyapunov–Krasovskii functional and using an improved inequality proposed by us. The criteria of stability are given in the form of matrix form. It is easy to verify them in the computer and they will have a wider application. We give an example to examine the availability of our result, and the code is performed in Matlab. The approach used in this paper can also be extended to other systems.

Existence and uniqueness for a class of nonlinear population diffusion system

ABSTRACTIn this paper, we introduce a class of nonlinear population diffusion system. Existence and uniqueness of strong solution are proved for stochastic age-dependent population diffusion system in Hilbert space by use of Gronwall's lemma and Burkholder–Davis–Gundy's inequality.

On some maximal and integral inequalities for sub-fractional Brownian motion

ABSTRACTWe obtain a maximal inequality for sub-fractional Brownian motion with Hurst index analogous to the Burkholder–Davis–Gundy inequality for fractional Brownian motion derived by Novikov and Valkeila [Statist. Probab. Lett. 44(1999):47–54] and an integral inequality for Wiener integrals with respect to a sub-fractional Brownian motion with Hurst index .

Johnson–Schechtman inequalities for noncommutative martingales

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E(M) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E. We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E=L2(0,1). As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46]. We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p-convexity and q-concavity for 1≤p≤2, q=2 and p=2, 2<q<∞. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case.

A Note on pth Moment Estimates for Stochastic Functional Differential Equations in the Framework of G-Brownian Motion

This paper presents the study of path-wise and moment estimates for the solutions to stochastic functional differential equations in the framework of G-Brownian motion. Under the linear growth condition, the pth moment estimates for the solutions to GSFDEs are investigated. The properties of G-expectations, Holder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis–Gundy (BDG) inequalities are used to develop the theory of pth moment estimates. In addition, the continuity of pth moment and path-wise asymptotic estimates for the solutions to GSFDEs are shown.

On the pth moment estimates of solutions to stochastic functional differential equations in the G-framework.

The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis–Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.

A logarithmic bound for a stopped Brownian motion

We study an estimate that can be regarded as a logarithmic version of Burkholder–Davis–Gundy inequality. Namely, for any K > 0 we determine the best constant L(K) ∈ (0, ∞] for which the following holds. If B is a standard Brownian motion and τ is an adapted stopping time, thenwhere Ψ(x) = (x + 1)log (x + 1) − x. Using standard embedding theorems, we obtain a related logarithmic bound involving a continuous-path martingale and its square bracket.

The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

In this paper, we consider a class of stochastic pantograph differential equations with Lévy jumps (SPDEwLJs). By using the Burkholder–Davis–Gundy inequality and the Kunita’s inequality, we prove the existence and uniqueness of solutions to SPDEwLJs whose coefficients satisfying the Lipschitz conditions and the local Lipschitz conditions. Meantime, we establish the p-th exponential estimations and almost surely asymptotic estimations of solutions to SPDEwLJs.

Pathwise versions of the Burkholder–Davis–Gundy inequality

We present a new proof of the Burkholder-Davis-Gundy inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging.

Martingale inequalities in noncommutative symmetric spaces

We investigate the Burkholder–Gundy inequalities in a noncommutative symmetric space $${E(\mathcal{M})}$$ associated with a von Neumann algebra $${\mathcal{M}}$$ equipped with a faithful normal state. The results extend the Pisier–Xu noncommutative martingale inequalities, and generalize the classical inequalities in the commutative case.

On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.

A generalization of the Burkholder–Davis–Gundy inequalities

In this paper, by applying some improved inequalities, we extend the Burkholder–Davis–Gundy inequalities for α ∈ (0,1) to more general functions and submartingales. Moreover, a series of inequalities for a logarithmic function are also obtained correspondingly. Finally, we give an application to a stopped Brownian motion.