Quasi-shuffle algebras in non-commutative stochastic calculus

通过洗牌代数的条件单调累积量

Conditionally monotone cumulants via shuffle algebra

本文在非交换概率的洗牌代数框架下研究条件单调累积量，将其刻画为无穷小特征，证明加法卷积为结合运算，并给出联系c-自由与c-单调累积量的组合公式，同时引入t-单调累积量。

Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulan…

Flows and stochastic Taylor series in Ito calculus

For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary semimartingales. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Ito integrals of semimartingales. Whereas the Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Ito calculus is naturally indexed by surjections.…

A group-theoretical approach to conditionally free cumulants

In this work we extend the recently introduced group-theoretical approach to moment-cumulant relations in non-commutative probability theory to the notion of conditionally free cumulants. This approach is based on a particular combinatorial Hopf algebra which may be characterised as a non-cocommutative generalisation of the classical unshuffle Hopf algebra. Central to our work is the resulting non-commutative shuffle algebra structure on the graded dual. It implies an extension of the classical relation between the group of Hopf algebra characters and its Lie algebra of infinitesimal character…

非交换概率论中的代数群再探讨

Algebraic groups in non-commutative probability theory revisited

本文旨在统一非交换概率论中基于泛积和基于Hopf代数的两种代数群方法，利用代数群与shuffle群技术证明了自由、Boolean及单调概率理论中代数群概念的等价性，并给出了Hopf代数结构的显式公式。

自由积分学 I

Free Integral Calculus I

本文针对自由随机变量中任意非交换多项式预解式的条件期望，提出基于线性化和布尔累积量的闭式求解方法，并引入非交换微分学，附有实例与附录。

On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in [A. Deya and R. Schott, On the rough paths approach to non-commutative stochastic calculus, J. Funct. Anal., 2013], we construct a product L{é}vy area above the $q$-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailled comparison between the resulting "rough" integral and the stochastic "It{ô}" integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic integration with respect to $q$ Brownian motion, Probab. Theory Related Fields, 2003].

Homogeneous Lie Groups and Quantum Probability

Here we extend the algebro-geometric approach to free probability, started in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show that any universal convolution product of moments of independent (non)-commutative random variables defined on a graded connected dual semi-group is given by a pro-unipotent group scheme. We show that moment-cumulant formulae have a natural interpretation within the theory of homogeneous Lie groups, which we generalise for the present purpose, and are given by the log and exp map, respectively. Finally, we briefly discuss the universal role …

Skorohod and rough integration with respect to the non-commutative fractional Brownian motion

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the pr…

分支伊藤公式与自然伊藤-斯特拉托诺维奇同构

Branched Itô formula and natural Itô-Stratonovich isomorphism

本文提出分支伊藤公式与自然伊藤-斯特拉托诺维奇同构，扩展了半鞅的修正公式，通过分支粗糙路径理论建立代数结构特征化。