有限时域退出时间问题中退化扩散控制的经典解
Classical Solutions for a Finite-Horizon Exit-Time Problem with Degenerate Diffusion Control
- Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究一维仿射扩散的有限时域退出时间控制问题，在无界控制同时作用于漂移和波动率导致HJB方程非一致抛物型的退化情形下，通过结构假设证明值函数为经典解，构造局部Lipschitz最优反馈并刻画退化集为曲线，同时证明不变性与不可达性并给出接近概率的精确渐近。\n
AI中文摘要：
我们研究一维仿射扩散的有限时域退出时间控制问题，其中无界控制同时作用于漂移和波动率。由于允许的控制可以抵消瞬时波动率，相应的HJB方程不是一致抛物型的。此外，退化位置不由模型系数预先规定，而是依赖于未知的值函数和最优反馈。我们不依赖粘性解公式，而是在适当的结构性假设下，证明值函数在整个状态域内部是$C^{1,2}$经典解，并且关于状态变量严格凸。进一步，我们构造了局部Lipschitz的最优反馈，明确地将退化集识别为一条曲线，并证明该曲线上的闭环漂移等于其状态坐标的时间导数。最后，我们证明了退出前的不变性和不可达性，并导出了接近概率的精确平方对数渐近行为。
英文摘要：
We study finite-horizon exit-time control of a one-dimensional affine diffusion, with an unbounded control acting on both drift and volatility. Since an admissible control can cancel the instantaneous volatility, the associated HJB equation is not uniformly parabolic. Moreover, the location of the degeneracy is not prescribed by the model coefficients but depends on the unknown value function and optimal feedback. Rather than relying on a viscosity-solution formulation, we work under suitable structural assumptions and prove that the value function is a $C^{1,2}$ classical solution throughout the interior of the state domain and is strictly convex in the state variable. Moreover, we construct a locally Lipschitz optimal feedback, explicitly identify the degenerate set as a curve, and show that the closed-loop drift on this curve equals the time derivative of its state coordinate. Finally, we prove invariance and inaccessibility before exit and derive exact squared-logarithmic asymptotics for approach probabilities.
