One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations with mild nonlinearity are given.
Accepté le :
DOI : 10.1051/cocv/2018065
Mots-clés : Wiener process, convex function, subdifferential, stochastic total variation flow
@article{COCV_2019__25__A71_0,
author = {Barbu, Viorel},
title = {A variational approach to nonlinear stochastic differential equations with linear multiplicative noise},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018065},
mrnumber = {4036658},
zbl = {1437.60035},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018065/}
}
TY - JOUR AU - Barbu, Viorel TI - A variational approach to nonlinear stochastic differential equations with linear multiplicative noise JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018065/ DO - 10.1051/cocv/2018065 LA - en ID - COCV_2019__25__A71_0 ER -
%0 Journal Article %A Barbu, Viorel %T A variational approach to nonlinear stochastic differential equations with linear multiplicative noise %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018065/ %R 10.1051/cocv/2018065 %G en %F COCV_2019__25__A71_0
Barbu, Viorel. A variational approach to nonlinear stochastic differential equations with linear multiplicative noise. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 71. doi : 10.1051/cocv/2018065. http://www.numdam.org/articles/10.1051/cocv/2018065/
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