PreprintarXiv licencearXiv
Regularity of the Value Function in Multi-Asset Portfolio Selection with Proportional Transaction Costs
Abstract
This paper concerns the variational inequality arising from a specific singular control problem: portfolio selection under proportional transaction costs. This variational inequality is a gradient-constrained partial differential equation (PDE). Generally, the literature only guarantees the W^{2, \infty} regularity of the solution to such a PDE, and counterexamples exist for higher regularity. In this paper, by exploiting the concavity of the value function, we connect this gradient-constrained problem to an obstacle problem, and finally show that the solution is C^{2,α} except at the corner points where multiple gradient constraints bind. Especially, if there are two assets, then a global C^{2,α} solution can be guaranteed.
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BibTeX
@misc{chen2026regularity,
title = {{Regularity of the Value Function in Multi-Asset Portfolio Selection with Proportional Transaction Costs}},
author = {Xinfu Chen and Shuaijie Qian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23028},
eprint = {2609.23028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23028},
}
RIS
TY - UNPB
TI - Regularity of the Value Function in Multi-Asset Portfolio Selection with Proportional Transaction Costs
AU - Xinfu Chen
AU - Shuaijie Qian
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.23028
UR - https://arxiv.org/abs/2609.23028
AB - This paper concerns the variational inequality arising from a specific singular control problem: portfolio selection under proportional transaction costs. This variational inequality is a gradient-constrained partial differential equation (PDE). Generally, the literature only guarantees the W^{2, \infty} regularity of the solution to such a PDE, and counterexamples exist for higher regularity. In this paper, by exploiting the concavity of the value function, we connect this gradient-constrained problem to an obstacle problem, and finally show that the solution is C^{2,α} except at the corner points where multiple gradient constraints bind. Especially, if there are two assets, then a global C^{2,α} solution can be guaranteed.
ER -