具有最大型收益的反射最优停止：测度值停止收益与被杀预解式表示
Reflected Optimal Stopping with a Max-Type Payoff: Measure-Valued Stopping Gains and Killed Resolvent Representation
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中文总结 AI 辅助
研究二维正态反射扩散的无限期最优停止问题，其收益为\(G(x_1,x_2)=x_1\vee \alpha x_2\)。通过制定反射障碍问题等，证明验证定理，推导停止集结构，明确停止收益对象是测度，证明正确势表示为被杀预解式公式。
中文摘要 AI 辅助
我们研究了在象限中具有收益\(G(x_1,x_2)=x_1\vee \alpha x_2\)的二维正态反射扩散的无限期最优停止问题。该问题具有使常规自由边界分析复杂化的三个特征：坐标轴上的反射、真正的二维停止区域以及非光滑的最大型奖励。我们制定了相关的反射障碍问题，在明确的伊藤 - 克里洛夫 - 田中可容许性和测度超调和性假设下证明了一个验证定理，并推导了停止集的条件上图结构。主要技术要点是停止收益对象\(\Gamma=c+rG - \mathcal LG\)是一个带符号测度而非函数。其对角分量为\(\Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\sigma_\Delta(dx)\)，\(n=(1,-\alpha)\)。我们还证明了正确的势表示是被杀预解式公式\(V(x)=G(x)-R_r^{\mathcal C}\Gamma(x)\)，而非无限制的反射预解式。一个常系数反射布朗运动的例子明确说明了对角奇异项。
英文摘要
We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff \(G(x_1,x_2)=x_1\vee αx_2\). The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit Itô--Krylov--Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object \(Γ=c+rG-\mathcal LG\) is a signed measure rather than a function. Its diagonal component is $Γ^Δ(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+α^2}}σ_Δ(dx)$, $n=(1,-α)$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal C}Γ(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.
