AI 中文总结
本文利用值函数的凹性，将比例交易成本下投资组合选择的梯度约束变分不等式转化为障碍问题，证明了其解具有 C^{2,α} 正则性，并可能推广到多维情形。
AI 中文摘要
本文研究由特定奇异控制问题（比例交易成本下的投资组合选择）产生的变分不等式。该变分不等式是一个梯度约束的偏微分方程（PDE）。通常，文献仅保证此类PDE解的 W^{2,∞} 正则性，并且存在更高正则性的反例。本文通过利用值函数的凹性，将该梯度约束问题与障碍问题联系起来，最终证明解是 C^{2,α} 的。本文考虑二维投资组合选择设置，但我们的方法在值函数为凹时，有可能推广到更一般的多维奇异控制问题。
英文摘要
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.
