Rodrigo Banuelos, Michal Brzozowski, Adam Osekowski
Let $X$ be a continuous-path martingale and let $Y$ be a stochastic integral, with respect to $X$, of some predictable process with values in $[-1,1]$. We provide an explicit formula for Burkholder's function associated with the weighted $L^2$ bound $$ \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}.$$
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We prove sharp maximal inequalities for $L^q$-valued stochastic integrals with respect to any Hilbert space-valued local martingale. Our proof relies on new Burkholder-Rosenthal type inequalities for martingales taking values in an $L^q$-space.
The aim of this note is to give some Burkholder-Davis-Gundy type inequalities which are valid for the Ito stochastic integral with respect to Banach valued Levy noise.
This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling-Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón-Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the …
A matrix Burkholder-Davis-Gundy inequality
本文提出一个矩阵值随机积分的谱范数不等式，扩展了非交换辛钦不等式，并利用弗里德曼不等式进行证明。
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 $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{$\star$} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E γ([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ γ([\![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 U…
The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal value of the constant $C_{p,d}$ in the inequality $$ Y_p\leq C_{p,d}X_p ?$$ We answer this question by considering a more general related problem for nonnegative submartingales. This enables us to study extension of the above inequality to the case when $d>1$ is not an integer, which has further interesting applications to stopped Bessel processes and to the beh…
In this paper we extend an inequality of Lenglart, Lépingle and Pratelli \cite[Lemma 1.1]{LLP} to general continuous adapted stochastic processes with values in topology spaces. By this inequality we show Burkholder-Davies-Gundy's inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions.
We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\R^n$ as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob's inequalities for martingales.
Let $\tilde{N}\_{t}$ be a standard compensated Poisson process on $[0,1]$. We prove a new characterization of anticipating integrals of the Skorohod type with respect to $\tilde{N}$, and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorohod integral processes with respect to $\tilde{N}$ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorohod integral proces…
We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.
We derive inequalities for time-discrete and time-continuous martingales that are similar to the well-known Burkholder inequalities. For the time-discrete case arbitrary martingales in $L^p(Ω)$ are treated, whereas in the time-continuous case martingales defined by Itô integrals w.r.t. a multi-dimensional Wiener process are considered. The estimates for the time-discrete martingales are related to the more general results by I. Pinelis (1994) and are proved to be sharp by a different and more elementary proof for this special setting. Further, for time-continuous martingales the presented ineq…
A BDG inequality for stochastic Volterra integrals
本文为随机Volterra积分建立了Burkholder-Davis-Gundy型不等式，给出有限与无限区间上的上界，并用于量化不同核的方程间路径距离及均值回归方程的均匀时间界。
In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected $\mathrm{L}^p$ estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals $\int_{0}^{t}H_s d(X_s Y_s)$ associated with certain continuous-time martingales…
We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.
Martingale-driven integrals and singular SPDEs
本文针对逼近柱Wiener过程的càdlàg鞅建立多重随机积分的混沌展开与矩界，并结合正则结构离散化框架证明粒子系统向奇异SPDE收敛。
Itô's Formula for Itô processes defined with respect to a cylindrical-martingale valued measure
基于作者近期发展的随机积分理论，证明了希尔伯特空间值Itô过程的Itô公式，并推导了Burkholder不等式及鞅值测度的Itô公式。
In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $γ$-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gund…
We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative continuous process which satisfies a linear integral inequality with an added continuous local martingale $M$ and a process $H$ on the right hand side, then for any $p \in (0,1)$ the $p$-th moment of the supremum of $Z$ is bounded by a constant $κ_p$ (which does not depend on $M$) times the $p$-th moment of the supremum of $H$. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical …
We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.
An $L^0$-approach to stochastic evolution equations
本文提出基于两个 Burkholder--Davis--Gundy 型不等式的 $L^0$ 框架，用于研究随机发展方程的逐路径时间正则性与数值逼近收敛速率，并在广义 Whittle--Matérn 型噪声的线性抛物方程上验证了有效性。
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 Levy area that can be described in terms of the asymmetry (non-reversibility) 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 Ito rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. …
In this paper, we introduce the concept of $S^{2}$-weighted pseudo almost automorphy for stochastic processes. We study the existence and uniqueness of square-mean weighted pseudo almost automorphic solutions for the semilinear stochastic integral equation $x(t)=\int_{-\infty}^{t}a(t-s)[Ax(s)+f(s,x(s))]ds+\int_{-\infty}^{t}a(t-s)φ(s,x(s))dw(s), \ t\in\mathbb{R}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family on a Hilbert space $H$, $w(t)$ is the two-sided $Q$-Wiener process, $f,φ: \mathbb{R}\times L^{2}(P,H)\rightarrow L^{2}(P,H)$ are two $S^{2}$-weig…
The quantum spectral curve equation associated to KP $τ$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer $G$-functions, or their asymptotic series. Using their Mellin integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.
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