Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

Abstract: We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff (G(x_1,x_2)=x_1\veeαx_2). The main novelty is a measure-valued variational formulation: the stopping gain (Γ=c+rG-\mathcal LG) is shown to be a signed Radon measure whose singular component is supported on the kink diagonal ({x_1=αx_2}), and this component is computed explicitly. We prove that the value admits the killed-resolvent representation [ V=G-R_r^{\mathcal C}Γ, ] where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.

• The paper establishes that nonsmooth max-type payoffs in reflected stochastic control require a measure-valued obstacle approach to properly capture singularities.
• The authors leverage a killed-resolvent representation to ensure correct free-boundary determination, validated by numerical simulations and Monte Carlo experiments.
• The study demonstrates parameter sensitivity effects, showing that excluding the diagonal singular measure introduces significant boundary errors.

Measure-Valued Variational Formulation in Reflected Stochastic Optimal Stopping

Problem Context and Formulation

The paper "Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control" (2606.21575) presents a rigorous analysis of an infinite-horizon optimal stopping problem driven by a two-dimensional, normally reflected diffusion in the positive quadrant $\mathbb{R}_+^2$. The objective is to maximize expected discounted rewards given by a nonsmooth, convex max-type payoff $G(x_1, x_2) = x_1 \vee \alpha x_2$, subject to a running cost $c$ and discount rate $r$. The dynamic is governed by the reflected SDE:

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$

where $L_t$ enforces normal reflection at the boundary. The primary analytic peculiarity arises from the nonsmoothness of $G$, whose kink at the diagonal $\Delta = \{x_1 = \alpha x_2\}$ induces singular behavior in associated variational and potential-theoretic representations.

Measure-Valued Obstacle and Stopping Gain

The value function for the optimal stopping problem is characterized as the solution to a reflected variational inequality:

$\max\{\mathcal{L} V - r V - c, G - V\} = 0$

with normal reflection on $\{x_i = 0\}$ for $G(x_1, x_2) = x_1 \vee \alpha x_2$0, where $G(x_1, x_2) = x_1 \vee \alpha x_2$1 is the generator for the diffusion. The nonsmooth nature of $G(x_1, x_2) = x_1 \vee \alpha x_2$2 prohibits standard $G(x_1, x_2) = x_1 \vee \alpha x_2$3 analytic techniques; crucially, the obstacle's singularity on $G(x_1, x_2) = x_1 \vee \alpha x_2$4 forces the stopping gain $G(x_1, x_2) = x_1 \vee \alpha x_2$5 to be interpreted as a signed Radon measure. The singular component localized at $G(x_1, x_2) = x_1 \vee \alpha x_2$6 is explicitly computed via distributional calculus, yielding:

$G(x_1, x_2) = x_1 \vee \alpha x_2$7

where $G(x_1, x_2) = x_1 \vee \alpha x_2$8 and $G(x_1, x_2) = x_1 \vee \alpha x_2$9 denotes surface measure on $c$0. This term provides the measure-valued analogue of the local time accumulated by $c$1 at zero, essential for precise modeling near the kink.

Figure 1: Baseline model; left, continuation region $c$2 (shaded), stopping region $c$3, free boundary $c$4, and diagonal $c$5; right, stopping advantage $c$6.

Killed Resolvent Representation

A major conceptual and practical advance is the killed-resolvent representation:

$c$7

where $c$8 denotes the resolvent operator for the reflected diffusion, killed upon entry to the stopping set $c$9. This formula corrects the usual unrestricted-resolvent identity, which erroneously includes post-stopping occupation—an error that results in both analytic inconsistency and incorrect boundary determination. The representation employs the full measure-valued stopping gain, including both absolutely continuous and singular components, and is validated via both deterministic PDE solvers and path-space Monte Carlo estimators.

Figure 2: (a) Monte Carlo value $r$0 at the free boundary against FD value $r$1; (b) Diagonal local-time potential $r$2 via deterministic and MC local-time estimators.

Free-Boundary Characterization and Verification

Under explicit monotonicity assumptions—vertical monotonicity of the stopping advantage and nonemptiness of vertical stopping sections—the stopping set admits an epigraph representation:

$r$3

with $r$4 constituting the free boundary. The boundary is characterized not via classical smooth fit but by a killed-resolvent trace condition:

$r$5

The paper formalizes this as a verification theorem: if a candidate boundary $r$6 yields $r$7 satisfying domination, contact, strict continuation, normalized reflection, admissible growth, and measure-superharmonicity, then $r$8 coincides with the true value and the corresponding stopping time is optimal. This framework provides a rigorous certification protocol for free-boundary candidates.

Numerical Implementation and Diagnostic Experiments

The theoretical formulation is substantiated with detailed numerical analysis, combining deterministic solvers for the variational inequality and MC simulations for killed resolvent potentials. The key computational findings are:

• Effect of the diagonal singular measure: Omitting $r$9 incurs substantial boundary error ($dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$0 in baseline tests), falsifies domination ($dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$1 is violated), and collapses the continuation region near $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$2.

Figure 3: Effect of the diagonal singular measure; left, true boundary $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$3 vs $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$4; right, vertical sections $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$5 at $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$6.

• Killed vs unrestricted resolvent: The unrestricted resolvent $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$7 fails to satisfy the contact condition, with the candidate value exceeding $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$8 throughout the stopping region—demonstrating the necessity of the killed operator.
• Parameter sensitivity: The contribution of the diagonal singular measure varies with the crossing coefficient $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t + dL_t,$9 (controlled via covariance structure and payoff weight $L_t$0); as $L_t$1, $L_t$2 vanishes, directly verifying the theoretical link between correlation, singularity, and boundary location.

  Figure 4: Boundary discrepancy $L_t$3 as a function of crossing coefficient $L_t$4 (varied via correlation $L_t$5); diagonal effect vanishes as $L_t$6.

Practical and Theoretical Implications

The measure-valued obstacle formulation establishes that nonsmooth payoffs require precise treatment of singularities in reflected stochastic control. The killed-resolvent trace condition provides a robust approach for free-boundary determination and verification, avoiding pitfalls inherent in scalar, pointwise generator calculations or unrestricted resolvent approximations. Computationally, successful implementation demands explicit inclusion of pathwise local time and careful handling of singular measures.

The theoretical framework can readily support further generalizations, including oblique reflection in polyhedral domains, higher-dimensional max-type payoffs (yielding singular measures on multiple interfaces), and relaxation of monotonicity assumptions via advanced PDE techniques.

Conclusion

The paper rigorously demonstrates that reflected optimal stopping with nonsmooth max-type payoff is governed by a measure-valued obstacle problem. The interplay between reflection, nonsmooth obstacles, and singular stopping gains induces nonlocal, killed-resolvent free-boundary equations. Both analytic and computational evidence confirms that ignoring the measure-valued structure—whether via classical generator methods or unrestricted resolvent formulas—results in invalid value representations and boundaries. The verification theorem, coupled with explicit diagnostic experiments, establishes a principled pathway for optimal boundary certification in multidimensional reflected diffusion settings with nonsmooth rewards.