Related papers: Calculus via regularizations in Banach spaces and …
First, we revisit functional It\^o/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fourni\'e, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by…
Functional It\^o calculus was introduced in order to expand a functional $F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$ consists in…
This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be…
In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$…
Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on…
This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure It\^o and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite…
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 M\'etivier and Pellaumail. Those notions are…
The purpose of this work is to extend the formalism of stochastic calculus to the case of spaces with local anisotropy (modeled as vector bundles with compatible nonlinear and distinguished connections and metric structures and containing…
We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded…
Recently, functional It\=o calculus has been introduced and developed in finite dimension for functionals of continuous semimartingales. With different techniques, we develop a functional It\=o calculus for functionals of Hilbert…
In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation…
Learning methods in Banach spaces are often formulated as regularization problems which minimize the sum of a data fidelity term in a Banach norm and a regularization term in another Banach norm. Due to the infinite dimensional nature of…
This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and…
The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric…
This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between…
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the…
We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…
This survey paper is a structured concise summary of four of our recent papers on the stochastic regularity of diffusions that are associated to regular strongly local (but not necessarily symmetric) Dirichlet forms. Here by stochastic…
In this paper we derive higher order convergence rates in terms of the Bregman distance for Tikhonov like convex regularisation for linear operator equations on Banach spaces. The approach is based on the idea of variational inequalities,…
This paper addresses Tikhonov like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving…