Viability problems for SDEs in a general Gelfand triple setup using a variational approach
Abstract
We establish viability conditions for a broad general class of stochastic differential equations formulated within the variational approach,% dXt=A( Xt) dt+B( Xt) dWt. The underlying functional space setup is given by a general Gelfand triple V⊂ H⊂ V∗, where H is a separable Hilbert space in which the stochastic evolution equation (SEE) is considered. Our approach to deriving viability conditions is inspired by the techniques developed by Aubin and Da Prato for finite-dimensional forward stochastic differential equations. We establish necessary and sufficient conditions for the viability of closed random constraint sets in terms of adapted variational tangent (and contingent) sets. Furthermore, we illustrate the applicability of the proposed framework by discussing several important classes of stochastic differential equations that fit naturally within our variational setting.
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BibTeX
@misc{ciotir2026viability,
title = {{Viability problems for SDEs in a general Gelfand triple setup using a variational approach}},
author = {Ioana Ciotir and Aureliu Ionescu and Eduard Rotenstein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01582},
eprint = {2610.01582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01582},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Viability problems for SDEs in a general Gelfand triple setup using a variational approach
AU - Ioana Ciotir
AU - Aureliu Ionescu
AU - Eduard Rotenstein
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01582
UR - https://arxiv.org/abs/2610.01582
AB - We establish viability conditions for a broad general class of stochastic differential equations formulated within the variational approach,% dX_t=A( X_t) dt+B( X_t) dW_t. The underlying functional space setup is given by a general Gelfand triple V⊂ H⊂ V^∗, where H is a separable Hilbert space in which the stochastic evolution equation (SEE) is considered. Our approach to deriving viability conditions is inspired by the techniques developed by Aubin and Da Prato for finite-dimensional forward stochastic differential equations. We establish necessary and sufficient conditions for the viability of closed random constraint sets in terms of adapted variational tangent (and contingent) sets. Furthermore, we illustrate the applicability of the proposed framework by discussing several important classes of stochastic differential equations that fit naturally within our variational setting.
ER -