N. V. Krylov
We prove Itô's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
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The overarching goal of this paper is to establish a set-valued Itô's formula. As an application, we obtain the existence and uniqueness of solutions for the general set-valued backward stochastic differential equation which gives an answer to an open question proposed by Ararat et al. (C. Ararat, J. Ma and W.Q. Wu, Set-valued backward stochastic differential equation, arXiv:2007.15073).
We prove the Itô-Wentzell formula for processes with values in the space of generalized functions by using the stochastic Fubini theorem and the Itô-Wentzell formula for real-valued processes, appropriate versions of which are also proved.
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。
We consider the continuous parabolic Anderson model with the Gaussian fields under the measure-valued initial conditions, the covariances of which are nonhomogeneous in time and fractional rough in space. We mainly study the spatial behaviors for the Feynman-Kac formulas in Stratonovich's sense. Benefited from the application of Feynman-Kac formula based on Brownian bridge, the precise spatial asymptotics can be obtained in the broader conditions than before.
We construct superprocesses with dependent spatial motion (SDSMs) in Euclidean spaces $R^d$ with $d\ge1$ and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on $R^d$, their local times exist when $d\le3$. A Tanaka formula of the local time is also derived.
Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of $\left\|\boldsymbol{X}(t)\right\|_p$, the $L^p$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of $\left\|\boldsymbol{X}(t)\right\|^c_p,\ c>0, $ with trend $g(t)$ over $[0,T]$. Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and non-stationary are considered. Important examples include $\sum_{i=1}^d \left|X_i(t)\right|+g…
We generalise multivariate subordination of Lévy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued Lévy processes. The processes are explicitly characterised and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space as a subordination …
This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of integrands and random sets. In the convex case, duality correspondences between the projections and projections of epigraphs are given. These results are used to study projections of set-valued integrands. Consistently with the general theory of stochastic processes, projections are not constructed using reference measures on the optional and predictable sigma-a…
Functional calculus and martingale representation formula for integer-valued random measures
本文针对连续鞅与整值随机测度建立非预期微积分，构造泛函导数并推导鞅表示公式，展示其在整值随机测度相关问题中的多项应用。
We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius.
The classical representation of random variables as the Ito integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron-Martin space. The Ito integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense. However, the published paper contains a mistake which restricts the validity of the main result, a…
We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.
We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of the Ahlfors-Beurling operator is bounded below by p^*-1.
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
Time-Uniform Self-Normalized Concentration for Vector-Valued Processes
本文为满足 sub-$ψ$ 尾部条件的多变量过程构造了通用的自归一化集中不等式，推导出紧致的迭代对数上定律，并应用于在线线性回归和自回归建模等统计任务。
An operadic framework is developed to explain the inversion formula relating moments and cumulants in operator-valued free probability theory.
An Itô-type formula for some measure-valued processes and its application on controlled superprocesses
本文针对具有半鞅型分解的测度值过程建立Itô型公式，并将其用于推导二元分支受控超过程的HJB方程、验证定理及粘性解框架。
Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins
本文研究了二维向量值狄利克雷高斯自由场及其质量化晶格对应物，在避免子域内每个点的球体的条件下，证明了无质量场的范数表现出熵排斥，而其角分量在所有介观尺度上冻结。
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