This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $χ$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_πB_{2})^{\ast}$), we define the so-called $χ$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $χ$-covariation is called $χ$-quadratic variation. The notion of $χ$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $χ$ is the whole space $(B_{1}\hat{\otimes}_πB_{1})^{\ast}$ then the $χ$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $χ$-covariation of various processes for several examples of $χ$ with a particular attention to the case $B_{1}=B_{2}=C([-τ,0])$ for some $τ>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-τ,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-τ,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert-Schmidt norm. In addition, we show that the conditions on the vola…
In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued…
We present some regularity properties (convexity, weak/weak* compactness and preservation of weak/weak* upper hemicontinuity) for Bochner/Gelfand conditional expectation of Banach valued correspondences under the nowhere equivalence condition. These regularity properties for Bochner/Gelfand integral of Banach valued correspondences are obtained as corollaries. Similar properties for regular conditional distributions are also covered by the corresponding results for Gelfand conditional expectation of correspondences. We prove the necessity of the nowhere equivalence condition for any of these p…
In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double …
本文证明了在“Flows driven by rough-routes”一文中提出的C^1-近似流机制，并应用于有限维粗糙路径驱动的巴拿赫空间-值粗糙微分方程的大部分结果的改进和推广，可以用来处理无限维巴拿赫空间-值弱几何Holder p-粗糙路径驱动的粗糙微分方程，对于任意p>2，用一种简单的方法完全地给出了Lyons的理论。
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1…
This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $ϕ$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented…
