Target-Generated Dirichlet Problems in State-Constrained Stochastic Control
Abstract
We study state-constrained stochastic control problems with a scalar surplus R ≥ 0. Controls for which the drift and volatility of R vanish at zero define a lower-dimensional Hamilton-Jacobi-Bellman equation. Starting from control-wise state-constraint viscosity inequalities, we prove that the upper and lower limits of the interior value are a subsolution and a supersolution of this boundary equation. Terminal compatibility and comparison identify their common limit. We require one-sided bounds on positive surplus drift and generator growth, together with domination by boundary generators. For compact or coercive controls, these follow from a local lower bound and continuity of the boundary control set. For a smooth stochastic target value w, the transformation R = Y − w(t,X) flattens the viable epigraph, and, under the stated hypotheses, the boundary control value supplies the Dirichlet datum. Applications include set-valued boundary controls, unbounded drift with quadratic costs, and redundant hedging instruments with a nonconstant target boundary.
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BibTeX
@misc{bayraktar2026targetgenerated,
title = {{Target-Generated Dirichlet Problems in State-Constrained Stochastic Control}},
author = {Erhan Bayraktar and Haichuan Ding and Ibrahim Ekren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05687},
eprint = {2610.05687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05687},
}
RIS
TY - UNPB
TI - Target-Generated Dirichlet Problems in State-Constrained Stochastic Control
AU - Erhan Bayraktar
AU - Haichuan Ding
AU - Ibrahim Ekren
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05687
UR - https://arxiv.org/abs/2610.05687
AB - We study state-constrained stochastic control problems with a scalar surplus R ≥ 0. Controls for which the drift and volatility of R vanish at zero define a lower-dimensional Hamilton-Jacobi-Bellman equation. Starting from control-wise state-constraint viscosity inequalities, we prove that the upper and lower limits of the interior value are a subsolution and a supersolution of this boundary equation. Terminal compatibility and comparison identify their common limit. We require one-sided bounds on positive surplus drift and generator growth, together with domination by boundary generators. For compact or coercive controls, these follow from a local lower bound and continuity of the boundary control set. For a smooth stochastic target value w, the transformation R = Y − w(t,X) flattens the viable epigraph, and, under the stated hypotheses, the boundary control value supplies the Dirichlet datum. Applications include set-valued boundary controls, unbounded drift with quadratic costs, and redundant hedging instruments with a nonconstant target boundary.
ER -