Vector- and operator-valued backward stochastic equations with finite-variation drivers and a maximum principle for singular stochastic control in infinite dimensions
Abstract
We study a mixed regular--singular control problem for stochastic evolution equations in a Hilbert space with possibly unbounded random linear operators, a nonconvex regular-control domain, and a state-dependent singular coefficient. The singular control is an adapted nondecreasing càdlàg process whose terminal value need not be bounded. We prove well-posedness and weighted moment estimates for the forward and backward equations, and characterize the second-order adjoint by a conditionally expected operator-valued backward stochastic integral equation. An Itô-type formula for the quadratic form of this adjoint, together with spike and convex variations, yields a second-order Hamiltonian condition for the regular control, as well as nonnegativity and a contact condition for the optional singular Hamiltonian.
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BibTeX
@misc{hu2026vector,
title = {{Vector- and operator-valued backward stochastic equations with finite-variation drivers and a maximum principle for singular stochastic control in infinite dimensions}},
author = {Ying Hu and Guomin Liu and Shanjian Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01545},
eprint = {2610.01545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01545},
}
RIS
TY - UNPB
TI - Vector- and operator-valued backward stochastic equations with finite-variation drivers and a maximum principle for singular stochastic control in infinite dimensions
AU - Ying Hu
AU - Guomin Liu
AU - Shanjian Tang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.01545
UR - https://arxiv.org/abs/2610.01545
AB - We study a mixed regular--singular control problem for stochastic evolution equations in a Hilbert space with possibly unbounded random linear operators, a nonconvex regular-control domain, and a state-dependent singular coefficient. The singular control is an adapted nondecreasing càdlàg process whose terminal value need not be bounded. We prove well-posedness and weighted moment estimates for the forward and backward equations, and characterize the second-order adjoint by a conditionally expected operator-valued backward stochastic integral equation. An Itô-type formula for the quadratic form of this adjoint, together with spike and convex variations, yields a second-order Hamiltonian condition for the regular control, as well as nonnegativity and a contact condition for the optional singular Hamiltonian.
ER -