A calculus on Lévy exponents and selfdecomposability on Banach spaces

Infinite-dimensional calculus under weak spatial regularity of the processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations, when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces, having a product structure with the noise in a Hilbertian component, again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by pat…

Tools for Malliavin calculus in UMD Banach spaces

In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called UMD property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizatio…

A version of Hörmander's theorem in 2-smooth Banach spaces

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hörmander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result, we apply methods of the Malliavin calculus.

Smoothness of densities for path-dependent SDEs under Hörmander's condition

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander's bracket condition in $\mathbb R^n$ for non-anticipative SDE coefficients defi…

The covariation for Banach space valued processes and applications

This article focuses on a new concept of quadratic variation for processes taking values in a Banach space $B$ and a corresponding covariation. This is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace $χ$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of $\bar ν_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calc…

Wong--Zakai Approximations of Stochastic Allen-Cahn Equation

We establish a unconditional and optimal strong convergence rate of Wong--Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen--Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the fully use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation, in combination with the factorization method and stochastic calculus in martingale type 2 Banach spaces applied to deduce sharp error est…

Stochastic integration in Banach spaces - a survey

This paper presents a brief survey of the theory of stochastic integration in Banach spaces. Expositions of the stochastic integrals in martingale type 2 spaces and UMD spaces are presented, as well as some applications of the latter to vector-valued Malliavin calculus and the stochastic maximal regularity problem. A new proof of the stochastic maximal regularity theorem is included.

通过马尔可夫算子和正则化在斯科罗霍德空间上进行高斯近似

Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization

本文引入了Banach值随机元素的 carré du champ 运算符，用于控制满足轻微正则性假设的马尔可夫光滑随机元素与斯科罗霍德空间上高斯随机元素之间的有界Lipschitz距离。

Clark-Ocone type formula for non-semimartingales with finite quadratic variation

We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $τ>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-τ,0]$. If $X$ is a real finite quadratic variation process…

Integral functionals on nonseparable Banach spaces with applications

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results. Then, we provide an interchange formula between integration and infimum, which enables us to get explicit formulas for the conjugate and Clarke subdifferential of integral functionals. Applications to expected functionals from stochastic programming, optimality conditions for a calculus of variation problem and sweeping processes are given.

BSDE and generalized Dirichlet forms: the infinite dimensional case

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space $E$: $(\partial_t+L)u+f(\cdot,\cdot,u, A^{1/2}\nabla u)=0$ on $[0,T]\times E,\qquad u_T=ϕ$, where $L$ is a possibly degenerate second order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator $L$ to obtain a probabilistic representation of the solution $u$ by solving the corresponding backward stochastic differential equat…

Information Projection on Banach spaces with Applications to State Independent KL-Weighted Optimal Control

This paper studies constrained information projections on Banach spaces with respect to a Gaussian reference measure. Specifically our interest lies in characterizing projections of the reference measure, with respect to the KL-divergence, onto sets of measures corresponding to changes in the mean (or {\it shift measures}). As our main result, we give a portmanteau theorem that characterizes the relationship among several different formulations of this problem. In the general setting of Gaussian measures on a Banach space, we show that this information projection problem is equivalent to minim…