C^{2,α} solution to the proportional transaction cost problem with two risky assets

Xinfu Chen; Shuaijie Qian

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arXiv · 2609.23028

C^{2,α} solution to the proportional transaction cost problem with two risky assets

Xinfu Chen·Shuaijie Qian

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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}. This paper concerns a two-dimensional portfolio selection setup, but our approach can potentially be extended to more general multi-dimensional singular control problems when the value function is concave.

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BibTeX RIS

Xinfu Chen, Shuaijie Qian. 2026-09-19. C^{2,α} solution to the proportional transaction cost problem with two risky assets. https://arxiv.org/abs/2609.23028

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