Stochastic currents
type: Item
instance of: scholarly article
title: Stochastic currents
zbMATH Open document ID: 1087.60043
author: Mariano Giaquinta
author: Vincenzo Maria Tortorelli
author: Franco Flandoli
author: Massimiliano Gubinelli
published in: Stochastic Processes and their Applications
publication date: 2005-09-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0211458
review text: Let \(D^1\) be the space of \(1\)-forms with compact supports in the Euclidean space \(\mathbb{R}^d\), i.e., if \(\varphi\in D^1\), then \[ \widetilde\varphi= \sum^d_{i=1} \varphi_i dx_i,\tag{1} \] where for \(i= 1,\dots, n\), \(\varphi_i\in C^\infty_K(\mathbb{R}^d)\), the space of smooth functions with compact support. Given a probability space \((\Omega, A,P)\) the authors call ``stochastic current'' (S-current for short) a linear continuous mapping from the space \(D^1\) into \(L^0(\Omega)\) (the space of real-valued (r.v.) in \((\Omega,A,P)\) equipped with the topology of convergence in probability). Let us denote by \(D_1\) (resp. \(D\)) the space of S-currents, and so if \(\widetilde S\in D_1\), then \(\forall\varphi\in D^1\Rightarrow\widetilde S(\varphi)\in L^0(\Omega)\). Definition 4. The S-current \(\widetilde S\in D_1\) is said to have a ``pathwise regularization'' provided that there exists a measurable mapping \(S^r: \Omega\to D_1\), such that \[ [\widetilde S(\varphi)](\omega)= S^r[\omega](\varphi)\quad\text{a.s. \(P\)-\(\omega\)},\quad\forall\varphi\in D^1.\tag{2} \] The authors are interested in the following problem of pathwise regularization for S-currents. Let \[ F:= (\Omega, A,(A_t)_{t\in [0,T]}, P)\tag{3} \] be a filtration and denote by \(X= M+ V\) a given continuous semimartingale over \(F\), in which \(M= (M_t)\) (resp. \(V= (V_t)\)) is a continuous local martingale (resp. a continuous, adapted and of bounded variation process) and define: \[ \widetilde S(\varphi):= \int^T_0 \langle\varphi(X_t), d^s X_t\rangle,\tag{4} \] where \(d^s X_t\) denotes the Stratanovich-stochastic differential generated by \(X\); resp. \[ \widetilde I(\varphi):= \int^T_0 \langle\varphi(X_t), d^1 X_t\rangle,\tag{5} \] where \(d^1 X_t\) denotes the Itô-stochastic differential generated by \(X\). The main result is the following theorem, which gives sufficient conditions for the existence of a pathwide regularization of the S-current (4) (resp. (5)). Theorem 9. Under th…
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 2211396
zbMATH Keywords: geometric measure theory
zbMATH Keywords: pathwise stochastic integration
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
OpenAlex ID: W4212814737
arXiv ID: math/0211458
cites work: Q5565773
cites work: Processus linéaires, processus généralisés
cites work: Q4850166
cites work: On a probabilistic description of small scale structures in 3D fluids
cites work: The Gibbs ensemble of a vortex filament
cites work: Q3154974
cites work: Q4210546
cites work: Controlling rough paths
cites work: Differential equations driven by rough signals
cites work: On conditional diffusion processes
cites work: Decomposition of Dirichlet processes and its applications
cites work: Q3994358
cites work: Q4303982
cites work: A generalized class of Lyons-Zheng processes
cites work: Q3336268
DOI: 10.1016/J.SPA.2005.04.007
description: scientific article