Maximal type inequalities for linear stochastic Volterra equations

Moderate and $L^p$ maximal inequalities for diffusion processes and conformal martingales

The $L^p$ maximal inequalities for martingales are one of the classical results in the theory of stochastic processes. Here we establish the sharp moderate maximal inequalities for one-dimensional diffusion processes, which include the $L^p$ maximal inequalities as special cases. Moreover, we apply our theory to many specific examples, including the Ornstein-Uhlenbeck (OU) process, Brownian motion with drift, reflected Brownian motion with drift, Cox-Ingersoll-Ross process, radial OU process, and Bessel process. The results are further applied to establish the moderate maximal inequalities for…

On the maximal inequalities of Burkholder, Davis and Gundy

We give a proof of the maximal inequalities of Burkholder, Davis and Gundy for real as well as Hilbert-space-valued local martingales using almost only stochastic calculus. Some parts of the exposition, especially in the infinite dimensional case, appear to be original.

Moderate maximal inequalities for the Ornstein-Uhlenbeck process

The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck process, which include the $L^p$ maximum inequalities as special cases and generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local marti…

On maximal inequalities for purely discontinuous $L_q$-valued martingales

We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were first obtained by Dirksen (arXiv:1208:3885) using arguments based on inequalities for sums of independent Banach-space-valued random variables, geometric properties of Banach spaces, and decoupling inequalities. Our proofs are completely different and rely almost exclusively on classical stochastic analysis for real semimartingales.

On pathwise counterparts of Doob's maximal inequalities

In this paper, we present pathwise counterparts of Doob's maximal inequalities (on the probability of exceeding a level) for submartingales and supermartingales.

Maximal inequalities and convergence results on multidimensionally indexed demimartingales

We obtain some maximal probability and moment inequalities for multidimensionally indexed demimartingales. Although the class of single-indexed demimartingales has been studied extensively, no significant amount of work has been done for the corresponding multiindexed class of random variables. This work aims to fill in this gap in the literature by extending well-known inequalities and asymptotic results to this more general class of random variables.

Generalizations of Maximal Inequalities to Arbitrary Selection Rules

We present a generalization of the maximal inequalities that upper bound the expectation of the maximum of $n$ jointly distributed random variables. We control the expectation of a randomly selected random variable from $n$ jointly distributed random variables, and present bounds that are at least as tight as the classical maximal inequalities, and much tighter when the distribution of selection index is near deterministic. A new family of information theoretic measures were introduced in the process, which may be of independent interest.

Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-$p$ Hölder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-Hölder) regularity of the solution together with a maximal type bou…

Maximal Inequalities and Some Applications

A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. 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 als discuss some relation…

Noncommutative maximal inequalities associated with convex functions

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $L^p$ case to the case of…

On the Ritt property and weak type maximal inequalities for convolution powers on $\ell^1(\Z)$

In this paper we study the behaviour of convolution powers of probability measures $μ$ on $\Z$, such that $(μ(n))_{n\in \N}$ is completely monotone or such that $ν$ is centered with a second moment. In particular we exhibit many new examples of probability measures on $\Z$ having the so called Ritt property and whose convolution powers satisfy weak type maximal inequalities in $\ell^1(\Z)$.

谱正稳定过程局部时的随机Volterra方程

Stochastic Volterra Equations for the Local Times of Spectrally Positive Stable Processes

本文研究谱正稳定过程局部时的空间演化动力学，证明其正半直线局部时满足由Poisson随机测度驱动的随机Volterra方程，借此得到正则性、矩界等结果，并拓展对偶方法建立Laplace泛函的指数-仿射表示。

Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.

On negative dependence inequalities and maximal scores in round-robin tournaments

We extend Huber's (1963) inequality for the joint distribution function of negative dependent scores in round-robin tournaments. As a byproduct, this extension implies convergence in probability of the maximal score in round-robin tournaments in a more general setting.

Maximal inequalities for fractional Lévy and related processes

In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding `convoluted martingale' is $p$-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.

Bound on the maximal function associated to the law of the iterated logarithms for Bernoulli random fields

We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given.

Maximal inequalities for centered norms of sums of independent random vectors

Let $X_1,X_2,\ldots,X_n$ be independent random variables and $S_k=\sum_{i=1}^k X_i$. We show that for any constants $a_k$, \[ \Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.

Maximal Inequalities in Bilateral Grand Lebesque Spaces Over Unbounded Measure

In this paper non-asymptotic exact rearrangement invariant norm estimates are derived for the maximum distribution of the family elements of some rearrangement invariant (r.i.) space over unbounded measure in the entropy terms and in the terms of generic chaining. We consider some applications in the martingale theory and in the theory of Fourier series.

New Optional Stopping Theorems and Maximal Inequalities on Stochastic Processes

In this paper, we develop new optional stopping theorems for scenarios where the stopping rules are defined by bounded continuity regions. Moreover, we establish a wide variety of inequalities on the supremums and infimums of functions of stochastic processes over the whole range of time indexes.

Maximal inequalities and weighted BMO processes

For a general adapted integrable right-continuous with left limits (RCLL) process $(X_t)_{t\in[0,τ]}$ taking values in a metric space $(\mathcal E,d)$, we show (among other things) that for every $m\in(1,\infty)$ $$ \frac{m-1}{2m-1}\|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m\le \|\sup_{t\in[0,τ]}d(X_0,X_t)\|_m\le c\frac{m^2}{m-1} \|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m $$ with a universal constant $c$. This is a probabilistic version of Fefferman--Stein estimate for the sharp maximal functions. While the former inequality is derived easily from Doob's mart…

Weighted norm inequalities for fractional maximal operators--a Bellman function approach

We study classical weighted $L^p\to L^q$ inequalities for the fractional maximal operators on $\R^d$, proved originally by Muckenhoupt and Wheeden in the 70's. We establish a slightly stronger version of this inequality with the use of a novel extension of Bellman function method. More precisely, the estimate is deduced from the existence of a certain special function which enjoys appropriate majorization and concavity. From this result and an explicit version of the ``$A_{p-\varepsilon}$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Pérez and To…

Maximal inequalities and a law of the iterated logarithm for negatively associated random fields

The exponential inequality of the maximum partial sums is a key to establish the law of the iterated logarithm of negatively associated random variables. In the one-indexed random sequence case, such inequalities for negatively associated random variables are established by Shao (2000) by using his comparison theorem between negatively associated and independent random variables. In the multi-indexed random field case, the comparison theorem fails. The purpose of this paper is to establish the Kolmogorov exponential inequality as well a moment inequality of the maximum partial sums of a negati…

解耦极大不等式

Decoupling Maximal Inequalities

本文研究极大不等式在非独立随机变量下的估计问题，证明对非负随机变量而言，两两独立足以使极大不等式表现与独立版本相当，且条件可放宽至可量化违反的负相关情形。

非齐次Itô过程的时间依赖符号及相应的极大不等式

The Time-Dependent Symbol of a Non-Homogeneous Itô Process and corresponding Maximal Inequalities

本文为非齐次Itô过程建立了时间依赖符号的存在性，推导了极大不等式，并据此将Blumenthal-Getoor指数推广至非齐次情形，进而刻画路径的渐近行为、指数矩及p变差性质。

关于具有多项式增长的群上随机游走最大位移的集中不等式

Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

本文研究了具有多项式增长的群上随机游走最大位移的集中不等式，证明了在特定群类上随机游走的扩散性质。

在一般混合条件下经验过程的极大不等式

Maximal Inequalities for Empirical Processes under General Mixing Conditions

本文在一般混合过程中研究样本均值的上界，提出基于浓度速率和新复杂度度量的界，揭示混合速度影响浓度速率并导致相变，应用于获得新的Glivenko-Cantelli型结果。

针对独立和依赖负二项随机变量的 Kolmogorov 型极大不等式：精确界限、亚指数细化及对超分散计数数据的应用

Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data

本文发展了适用于独立和依赖负二项随机变量和的 Kolmogorov 型极大不等式，推导了精确界限和亚指数细化，展示了对超分散计数数据的应用价值。

随机Volterra方程的马尔可夫近似在Hölder范数下的收敛性

Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations

本文研究随机Volterra方程马尔可夫近似的收敛性，在Hölder范数下给出误差界，并证明基于sinc求积的近似误差以指数速率衰减，数值实验验证了理论结果。

图依赖经验过程的耦合与极大不等式

Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes

针对图依赖观测数据，提出一种结合图适应依赖系数与块分区着色的耦合构造，导出经验过程的极大不等式，并应用于网络自回归、非线性局部传播等模型。