Geometry of Stopping & Continuation Regions

Updated 11 January 2026

Geometry of stopping and continuation regions is the study of how decision boundaries separate optimal exercise from waiting, defined by analytic and geometric criteria.
The analysis employs free-boundary problems, scale functions, and variational inequalities to capture complex configurations such as double continuations, bubbles, and spikes.
Algorithmic methods including Monte Carlo simulations and neural approximations offer practical tools for resolving these regions in high-dimensional and time-inhomogeneous settings.

A stopping region is the set of states or phase-space points at which it is optimal to exercise, stop, or execute a control, while the continuation region is its complement, i.e., the locus where it is optimal to continue waiting or refrain from stopping. The geometry—topology, regularity, and boundary structure—of these regions is central to the analysis of optimal stopping problems in stochastic processes, mathematical finance, and control. In classical one-dimensional settings, the regions are separated by points (the free boundary), but in general setups they may comprise intervals, unions, or more intricate manifolds, with structure determined jointly by the process dynamics, payoff function, time horizon, and discounting. Of particular importance are scenarios leading to multiple or double continuation regions, bubbles, spikes, or bays, and the links between these geometric features and analytic properties such as smooth fit, variational inequalities, and scale function representations.

1. Analytical Characterization Through Scale Functions and Free Boundary Problems

In Lévy models, the value function for a perpetual American option with payoff $(K-\exp(X_t))^+$ or $(\exp(X_t)-K)^+$ can be explicitly characterized using the scale function $W^{(q)}$ and its integral $Z^{(q)}(x)$ , with $q<0$ (negative effective discount rate) (Donno et al., 2017). Here $X_t$ is a spectrally negative Lévy process, $\psi(\theta)$ its Lévy exponent, and $q$ the discount rate. The optimizer is of threshold-type: exercise upon first entry into a stopping interval $[x_1^*, x_2^*]$ , with continuation regions $(-\infty, x_1^*)$ and $(x_2^*, \infty)$ . Matching value and smooth fit at $x_1^*$ and $x_2^*$ leads to a linear system in the scale variable: $\begin{aligned} & A\,W^{(q)}(x_1^*)+B\,Z^{(q)}(x_1^*) = K-e^{x_1^*}, \ & A\,W^{(q)\prime}(x_1^*)+B\,Z^{(q)\prime}(x_1^*) = -e^{x_1^*}, \ & A\,W^{(q)}(x_2^*)+B\,Z^{(q)}(x_2^*) = K-e^{x_2^*}, \ & A\,W^{(q)\prime}(x_2^*)+B\,Z^{(q)\prime}(x_2^*) = -e^{x_2^*}. \end{aligned}$ The corresponding regions are:

Stopping region $S = [x_1^*, x_2^*]$
Continuation region $C = (-\infty, x_1^*) \cup (x_2^*, \infty)$

For relevant multi-stopping (Swing) problems, this structure is preserved recursively: each new exercise right produces a larger exercise interval, yielding nested continuation/stopping intervals (Donno et al., 2017).

2. Topologies: Double and Multi-Interval Continuation Regions

While classical American option problems exhibit either lower or upper semi-infinite continuation regions (a single free boundary), more complex models yield geometrically richer configurations:

Double continuation: For negative discounting (backward-running "time value") or under intermittent stopping constraints, as in Poisson-exercised options, the stopping region may be the middle interval, with continuation optimal both far in- and out-of-the-money (Donno et al., 2017, Palmowski et al., 2020).
Disconnected continuation ("bubble"): Discontinuities in volatility, drift, or payoff (e.g., oscillating Brownian motion with quadratic reward, or BM with broken drift) can induce a "bubble"—an interior continuation interval surrounded by stopping regions (Mordecki et al., 2019, Mordecki et al., 2018). Conditions for this bifurcation are precisely established, e.g., $\sigma_1^2 < \sigma_2^2 < 2\sigma_1^2$ and $2\sigma_1^2 < r < \sigma_2^2$ for OBM (Mordecki et al., 2019).
Multi-intervals: Very general Lévy and diffusive models, or complex payoffs, can support multi-interval continuation regions (Egami et al., 12 Jun 2025, Kosmala et al., 2023). Systematic resolution procedures (e.g., iterated Riesz decomposition or potential theory) decompose the region as a union of intervals determined by recursive free-boundary equations.

3. Boundary Regularity: Smooth Fit, Monotonicity, and Bubbles/Spikes

The interface between stopping and continuation is explicitly characterized by value-matching and (where regularity permits) smooth-fit conditions. For one-dimensional Itô diffusions and processes with scale functions, these reduce to:

Value-matching: $V(c_i) = g(c_i)$
Smooth-fit: $V'(c_i\pm) = g'(c_i)$ if the process has unbounded variation; otherwise only continuity.

In finite-horizon problems, the stopping/continuation boundary can display bays (vertical continuation segments within stopping) or spikes (stopping spikes inside continuation), determined by the sign and atom structure of the signed measure $\mu = g''(dz) - 2\sigma^{-2}(z)r(z)g(z)dz$ (Angelis, 2020). At points $x_0$ with $\mu(\{x_0\}) > 0$ , the continuation region contains a vertical segment (bay), while for $\mu(\{x_0\}) < 0$ there is a stopping spike at $x_0$ .

In higher-dimensional and time-inhomogeneous jump-diffusions, boundaries are (locally) continuous surfaces in state-space under regularity and monotonicity conditions—formally, $x^*(t,y)$ is jointly continuous in $(t, y)$ under natural PDE and monotonicity assumptions (Cai et al., 2021).

4. Phase-Space and Variational Formulations

Many optimal stopping problems admit a variational inequality and free-boundary PDE: $\min \Big\{ V(x,t) - g(x,t), \, (\partial_t + \mathcal{L}) V(x,t) \Big\} = 0$ where $\mathcal{L}$ is the generator of the process. In phase-space, the optimal stopping time is often the first hitting time of a one-dimensional barrier: for Brownian motion and martingale transport under distribution constraints, the boundary is $t = b(x)$ , dividing $\mathbb{R} \times [0, \infty)$ into stopping $S = \{ (x,t) : t \geq b(x) \}$ and continuation $C = \{ (x,t) : t < b(x) \}$ (Beiglboeck et al., 2016).

In discrete time, stopping boundaries are lower-semicontinuous graphs in state coordinates, with randomized or "fuzzy" relaxations enabling convergence-theory for neural network approximations of boundaries (Soner et al., 2023).

5. Potential-Theoretic and Envelope Methods

The value function can often be constructed as the pointwise infimum (envelope) of a family of "potentials" (expectations over candidate stopping domains), subject to boundary constraints:

For spectrally negative Lévy processes, the Riesz decomposition gives $v(x) = \int G_q(x,y) \mu(dy) + h(x)$ , with the measure $\mu$ supported on the stopping region (Egami et al., 12 Jun 2025).
In both 1D/MD Brownian optimal stopping, the value is the infimum over Dirichlet/harmonic potentials majorizing the payoff (Moriarty, 14 May 2025, Kosmala et al., 2023). The free boundary is the locus of tangency between the envelope and the gain, mathematically determined by a system of double-tangency equations.

The maximum principle guarantees that continuation regions are open intervals or domains and that their boundaries are characterized by value/smooth-fit and, in exceptional cases, permit novel features (bays, spikes, bubbles).

6. Algorithmic and Numerical Implications

Constructing or approximating the stopping and continuation regions is fundamentally a geometric problem:

Sequential design and active learning use uncertainty reduction and local-loss criteria to sample more densely near the anticipated stopping boundary, especially in high dimensions or discrete time (Gramacy et al., 2013).
Monte Carlo neural methods approximate phase-space boundaries, with convergence and regularity robust under boundary perturbations (Soner et al., 2023).
Potential-theoretic and envelope algorithms solve for boundaries through recursive concave majorant or envelope-tangency systems, with explicit complexity bounds (often quadratic in grid size for 1D cases) (Kosmala et al., 2023, Moriarty, 14 May 2025).

7. Schematic Summary of Geometric Regimes

Scenario	Continuation region	Stopping region
Classical American (BS, single barrier)	$(-\infty, x^)$ or $(x^, +\infty)$	$[x^, +\infty)$ or $(-\infty, x^]$
Double continuation (negative discount)	$(-\infty, x_1^) \cup (x_2^, +\infty)$	$[x_1^, x_2^]$
Bubble ("bifurcation," OBM, broken drift)	$(-\infty, c_1) \cup (c_2, c_3)$	$[c_1, c_2] \cup [c_3, \infty)$
Multidimensional/harmonic envelope	Open set in $\mathbb{R}^d$ , boundary $\partial C$	Its complement; free boundary is C² hypersurface

This table encapsulates the generic geometries arising in optimal stopping, as governed by analytic, probabilistic, and path properties of the underlying process and the payoff structure.

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