Stopping Times of Boundaries: Relaxation and Continuity

TL;DR

This work analyzes stopping-time problems where optimal decisions are characterized by hitting a boundary, establishing continuity of the boundary-induced value $v(\tau_f)$ under both the supremum norm and a novel relaxed $L^{\infty}$ metric. It introduces a fuzzy region to relax hitting times, proving that the relaxed value $v(\mu_f^{\varepsilon})$ converges to the true value as $\varepsilon \to 0$, and proves convergence results for neural boundary methods via universal approximation (including when the boundary is semicontinuous through inf/sup convolutions). The results underpin the convergence of the Neural Optimal Stopping Boundary algorithm for Bermudan-style options and provide a rigorous framework for approximating semicontinuous free boundaries in high dimensions. The combination of relaxed stopping, continuity analysis, and neural approximation broadens the tractable scope of data-driven optimal stopping, with potential extensions to continuous-time American options.

Abstract

We study the properties of the free boundaries and the corresponding hitting times in the context of optimal stopping in discrete time. We first prove the continuity of the map from the boundaries to the expected value of the corresponding stopping policy both in the supremum norm and also in a weaker, novel topology induced by the relaxed $L^\infty$ metric that we introduce. The latter is particularly useful when the optimal stopping boundary is only proved to be semicontinuous. Secondly, we study the connection between the hitting times, and their relaxations as widely employed in recent numerical methods. All these results together with the universal approximation capability of neural networks and the notion of inf/sup convolution are then used to provide a convergence analysis for the algorithm in [Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2025] for the numerical resolution of the exercise regions arising in the analysis of Bermudan type option.

Figures (5)

• Figure 1: Stopping region of a max-call option on two symmetric assets. The upper connected component becomes an epigraph through $(\alpha,\Xi)$(right panel), where we identify $\Xi(x) = (\frac{x_1}{x_2},1) \in \mathop{\mathrm{\mathbb{R}}}\nolimits^2$ with its first component. Figures adapted from ReppenSonerTissotFB.
• Figure 2: Stopping region of a min-call option on two symmetric assets (one time section). We note that the stopping boundary is discontinuous at $\Xi(x) = (1,1)$.
• Figure 3: Left panel: fuzzy region (purple) separating the interior of the continuation region (blue) from the stopping region (red). Right panel: Illustration of $\chi^{\varepsilon} \circ d(\cdot; f)$ in the coordinates $(\Xi,\alpha)$, where $d(\cdot; f)$ is the distance function to $\mathop{\mathrm{\mathfrak{S}}}\nolimits_t(f)$ and $\chi^{\varepsilon}(\delta) = (1 - \delta/\varepsilon)^+ \wedge 1$.
• Figure 4: Illustration of the symmetric difference between the stopping regions $\mathop{\mathrm{\mathfrak{S}}}\nolimits_t(f)$ and $\mathop{\mathrm{\mathfrak{S}}}\nolimits_t(f')$ (purple region) contained in the interval $[f \wedge f',f \wedge f'+\iota)$ (orange region); see proof of Figure \ref{['thm:unifcontV']}. The stopping decisions with respect to $f$ and $f'$ coincide in the red (stop) and blue (continue) region.
• Figure 5: Inf convolution of $f(t,\cdot)$ for some $t\in \mathop{\mathrm{\mathcal{T}}}\nolimits$. The monotonicity of $\delta \to f_{\delta}$ implies that the hitting time $\tau_{f_{\delta}}$ increases to $\tau_{f}$ (pathwise) as $\delta \downarrow 0$.

Theorems & Definitions (40)

• Theorem
• Theorem
• Proposition 2.1
• Proposition 2.2
• proof
• Example 1
• Example 2
• Proposition 3.1
• proof
• Lemma 3.1