Time and Killed Resolvents in Reflected Optimal Stopping with a Max Payoff

Technical Summary: Local Time and Killed Resolvents in Reflected Optimal Stopping with a Max Payoff

1. Problem Formulation
The paper investigates the infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion $X = (X_1, X_2)$ confined to the closed positive quadrant $\mathbb{R}^2_+$. The decision maker seeks a stopping time $\tau$ to maximize the expected discounted reward net of a running cost:
$$V(x) = \sup_{\tau} \mathbb{E}_x \left[ e^{-r\tau}G(X_\tau) - \int_0^\tau e^{-rs}c(X_s) ds \right]$$
where the payoff function is the non-smooth "max" type: $G(x_1, x_2) = x_1 \vee \alpha x_2$ with $\alpha > 0$. The diffusion is subject to normal reflection at the coordinate axes, and the payoff function possesses a convex Lipschitz kink along the diagonal set $\Delta = \{x_1 = \alpha x_2\}$.

The authors identify two primary challenges that break standard smooth free-boundary calculus:

1. Boundary Complexity: The state is confined to a quadrant with Neumann boundary conditions and a corner at the origin.
2. Payoff Singularity: The max-type reward is only piecewise affine. Its distributional second derivatives concentrate as a surface measure on the kink set $\Delta$, rendering the standard "stopping gain" object ill-defined as a simple function.

2. Methodological Framework
The analysis is structured into two distinct tiers to separate unconditional measure-theoretic results from conditional geometric assumptions:

• Tier I (Unconditional): Focuses on the measure-theoretic decomposition of the stopping gain and the potential-theoretic representation of the value function. These results rely solely on the standing assumptions of the reflected diffusion (local Lipschitz coefficients, uniform ellipticity, strong Markov property) and do not assume specific geometric properties of the stopping set.
• Tier II (Conditional): Addresses the geometry of the free boundary (e.g., epigraph structure, monotonicity, smooth fit). These results require explicit, model-specific hypotheses regarding the monotonicity of the stopping advantage $H = V - G$.

The methodology employs:

• Itô–Tanaka Formula: Applied to the process $Y_t = X_{1,t} - \alpha X_{2,t}$ to handle the non-smoothness of $|Y_t|$.
• Occupation Density and Co-area Identities: Used to convert local time terms into surface measures on the diagonal.
• Killed Resolvents: The value function is represented using a resolvent operator killed upon entry into the stopping set, rather than the unrestricted resolvent.
• Verification Theorems: Based on generalized Itô classes and measure-superharmonicity, avoiding reliance on a priori regularity of the value function.

3. Key Contributions and Results

• Measure-Valued Stopping Gain: The paper proves that the stopping gain object $\Gamma = c + rG - \mathcal{L}G$ is not a function but a signed Radon measure. It decomposes into an absolutely continuous part and a singular surface measure on the diagonal $\Delta$:
  $$\Gamma_\Delta(dx) = -\frac{q(x)}{2\sqrt{1+\alpha^2}} \sigma_\Delta(dx)$$
  where $q(x) = n^\top a(x) n$ with $n=(1, -\alpha)$, and $a(x)$ is the diffusion covariance matrix. Under local ellipticity, $q(x) > 0$, making the diagonal component strictly negative.

• Diagonal Avoidance Theorem: A direct consequence of the strict negativity of the diagonal measure is that no interior point of the kink set $\Delta$ can belong to the stopping set. The optimal strategy always involves continuing when on the diagonal. This is proven by showing that the local time gain (scaling as $\sqrt{h}$) from waiting an infinitesimal time $h$ at the kink dominates the running cost (scaling as $h$).

• Killed Resolvent Representation: The authors establish that the correct value representation is:
  $$V(x) = G(x) - R^C_r \Gamma(x)$$
  where $R^C_r$ is the resolvent killed at the first entry time $\tau_D$ into the stopping set. They rigorously demonstrate that the unrestricted reflected resolvent $R^R_r$ is generally incorrect because the reflected process may re-enter the continuation region after hitting the stopping set, accumulating "extra" occupation that the optimal stopping problem must not count. A closed-form counter-example using one-dimensional reflected Brownian motion quantifies this gap.

• Corner Term Analysis: The paper addresses the potential singularity at the origin $(0,0)$. Under specific assumptions (verified for non-degenerate reflected Brownian motion), the corner contribution to the Itô–Tanaka formula vanishes, ensuring the measure decomposition holds globally in the interior.

4. Verification and Numerical Experiments
To demonstrate the framework is non-empty and operational, the authors verify all Tier I results for a reflected Brownian motion benchmark with constant drift and covariance.

• Analytical Verification: They confirm the value function is globally Lipschitz, the diagonal measure is strictly negative, and the killed-resolvent representation holds.
• Numerical Experiments:
  1. Local Time Scaling: Monte Carlo simulations recover the theoretical $\sqrt{t}$ scaling of local time at the diagonal kink, validating the mechanism that forces continuation.
  2. Resolvent Gap: Simulations of a 1D reflected process reproduce the exact positive gap between the killed and unrestricted resolvents, isolating the re-entry mechanism.
  3. Diagonal Avoidance: Finite-difference solutions of the obstacle problem visualize the continuation advantage as a ridge along the diagonal, with the stopping set filling the domain but strictly avoiding the kink.

5. Significance and Claims
The paper claims to provide a rigorous correction to heuristic treatments of multidimensional optimal stopping with non-smooth payoffs. Its primary significance lies in:

1. Correcting the Stopping Gain: Establishing that $\Gamma$ is a signed measure with a strictly negative singular part on the kink, invalidating pointwise conditions like "$\Gamma \ge 0$ on the stopping set" without separating the singular component.
2. Correcting the Potential Representation: Proving that the value function must be represented via a killed resolvent, correcting the common misconception that the unrestricted reflected resolvent suffices.
3. Separating Geometry from Mechanism: By distinguishing between unconditional measure-theoretic truths (Tier I) and conditional geometric properties (Tier II), the paper avoids asserting automatic regularity for free boundaries in multidimensional reflected settings. It explicitly states that while the diagonal is never a stopping set, the global shape of the stopping set (e.g., whether it forms an epigraph) remains a model-specific problem requiring additional structural hypotheses.

The authors conclude that while the "no-stopping-on-the-kink" phenomenon is robust and unconditional, the full characterization of the free boundary geometry requires further structural assumptions on the stopping advantage $H = V - G$ that are not implied by the diffusion dynamics alone.