Randomised Stopping Times

• Randomised stopping times are rules that integrate external randomization into the decision process, broadening classical stopping time concepts in stochastic processes.
• They establish equivalence among various formulations like mixed and behavioral stopping times, ensuring consistent outcomes across different applications.
• Their applications span mathematical finance, sequential analysis, reinforcement learning, and Monte Carlo simulation, driving efficient computational methods and robust risk assessment.

A randomized (or randomised) stopping time is a generalization of the classical stopping time concept in probability theory and stochastic processes, where the decision to stop is itself randomised, often by introducing additional random variables or randomization procedures. This framework is foundational in optimal stopping theory, stochastic control, mathematical finance, sequential analysis, game theory, and Monte Carlo simulation, as well as in modeling complex dependencies across random events. Over the past two decades, the theory and applications of randomized stopping times have expanded greatly, yielding equivalence results, powerful new computational methods, and connections to martingale theory, control, and statistical inference.

1. Foundational Definitions and Representations

A randomised stopping time is a rule for terminating an observation of a stochastic process that incorporates external or internal randomization, typically extending classical stopping times. Several core definitions are found in the literature:

• Randomized Stopping Time (Discrete Time): An $\mathbb{F}$-adapted process %%%%1%%%% with $p_n(\omega) \geq 0$, $\mathbb{F}_n$-measurable, and $\sum_{n=0}^\infty p_n = 1$ a.s.; at state $\omega$ time $n$ is selected with probability $p_n(\omega)$. This can be realized as $$\tilde{N}_p(r,\omega) = \min \{ n \geq 0:\ r < \sum_{i=1}^n p_i(\omega)\}$$ where $r \sim \mathrm{Uniform}[0,1]$ (Solan et al., 2012).
• Behavior Stopping Time: Adapted process $B = (B_n)_{n=0}^\infty$ with $B_n \in [0,1]$. At each time $n$, if not yet stopped, stop with probability $B_n(\omega)$. The law of stopping at $n$ is $$p_n(\omega) = B_n(\omega) \prod_{j=1}^{n-1} (1-B_j(\omega))$$ (Solan et al., 2012).
• Mixed Stopping Time: A measurable function $\mu: [0,1] \times \Omega \to \mathbb{N}_0$ such that for every $r \in [0,1]$, $\mu(r, \cdot)$ is a classical stopping time; this corresponds to randomizing over a family of stopping times (Solan et al., 2012, Shmaya et al., 2014).
• Continuous Time: A mixed stopping time is a measurable map $p: \Omega \times [0,1] \to [0,T]$ so that for almost every $r$, $p(\cdot,r)$ is a stopping time; a randomized stopping time is a right-continuous, nondecreasing, $[0,1]$-valued adapted process $p_t$ with $p_0=0$, $p_T=1$ (Shmaya et al., 2014).

All these notions are shown to be equivalent in both discrete and continuous time (i.e., they induce the same law for the stopped process), modulo technical caveats on measurability (Solan et al., 2012, Shmaya et al., 2014). The equivalence is essential for robust modeling and for the interchangeability of computational representations.

2. Theoretical Properties and Equivalence

The main theoretical advances center on the distributional equivalence of different formulations:

• Detailed Distribution Equivalence: Two random stopping times (of any type) are equivalent if they induce the same measure on $(\Omega \times (\mathbb{N}_0 \cup \{\infty\}))$ (or $(\Omega \times [0,T])$ in continuous time), meaning all expected payoffs and the stochastic laws for the stopped process are identical for any stopping problem (Solan et al., 2012, Shmaya et al., 2014).
• Kuhn's Theorem Extension: In continuous time, this equivalence extends Kuhn's theorem, showing that randomization schemes (including mixed/behavior/randomized) have the same expressiveness for optimal stopping and game-theoretic applications (Shmaya et al., 2014).
• Canonical Representation for Non-Honest Times: For any random time $\rho$ (not necessarily a stopping time or honest), there exists a canonical pair $(K,L)$ with $K_t$ an $\mathbb{F}$-adapted, right-continuous nondecreasing process and $L$ a local martingale. The associated randomized stopping time is $\psi = \inf \{ t \geq 0 : K_t \geq U \}$ for $U \sim \mathrm{Uniform}[0,1]$ independent of $\mathbb{F}_\infty$. Distributionally, optional processes observed up to $\rho$ under $P$ have the same law as those observed up to $\psi$ under the measure $Q$ with density process $L$; i.e.,

$E_P[Y_{\rho \wedge t}] = E_Q[Y_{\psi \wedge t}].$

Thus, for distributional properties of processes up to a random time, it is always sufficient to use (possibly under a new measure) a randomised stopping time (Kardaras, 2010).

3. Applications in Mathematical Finance, Control, and Sequential Analysis

Randomized stopping times have found broad application across several domains:

• Finance: They provide tractable representations for default times (which are generally non-stopping times with respect to filtration generated by asset prices), modelling systemic risk, optimal exercise problems for swing options, and reduced-form credit risk models (especially via Cox and Marshall–Olkin constructions for simultaneous default events) (Kardaras, 2010, Christensen et al., 2012, Protter et al., 2021).
• Optimal Stopping with Randomized Exercise Opportunities: In settings where opportunities to stop arrive stochastically (not at deterministic times), the stopping problem is reformulated as a discrete-time optimal stopping problem along the sequence of arrival times. Algorithms developed include random times least squares Monte Carlo (a variant of Longstaff–Schwartz), Markovian regression, and iterative policy improvement, all adapted to this random time horizon (Dekker et al., 2023).
• Sequential Analysis and Quality Control: The explicit link between distributions of stopping times and stopped sums under exponential tilting provides essential tools for boundary-crossing probabilities, analysis of random walks, and design of sequential sampling plans; e.g., in the context of $k$-run switching rules in manufacturing (Boutsikas et al., 2010).
• Reinforcement Learning and Exploratory Control: In entropy-regularized stopping, randomization is introduced by controlling a stochastic stopping intensity, leading to smooth HJB-type equations and facilitating reinforcement learning algorithms for high-dimensional American/Bermudan options (Dong, 2022).
• Statistical Inference: Inverse problems involving random observation at unknown stopping times can be addressed using nonparametric estimation techniques (e.g., via Mellin transforms) to reconstruct the law of the stopping time from sampled values, with provable minimax rates (Schulmann, 2019).
• Stochastic Thermodynamics: Martingale arguments at random stopping times yield integral fluctuation relations and universal bounds for entropy production, showing that even under feedback or adaptive stopping, standard fluctuation relations (and the second law at stopping times) are preserved (Neri et al., 2019).

4. Algorithmic Developments and Simulation

Efficient computational methods and advanced Monte Carlo schemes leverage randomized stopping times:

• Monte Carlo Pricing and Simulation: The convexification of the set of stopping rules via randomization allows for application of minimax and duality principles (e.g., Rogers-type dual for optimal stopping under risk measures such as AV@R) and performance of simulation-based pricing with provable upper and lower bounds (Belomestny et al., 2014, Dekker et al., 2023).
• Variance Reduction and Risk-sensitive AIS: For rare event simulation in problems with unbounded stopping times, naive zero-variance adaptive importance sampling may fail (producing degenerate or computationally infeasible estimators). Introducing risk-sensitive optimal control penalizes excessively long trajectories and ensures proposal measures remain absolutely continuous and generate feasible simulation paths (Hartmann et al., 13 Feb 2024).
• Randomized Refraction (Canadization) in Multiple Stopping: The replacement of deterministic refraction periods with randomly distributed (e.g., Erlang) waiting times (Canadization) allows analytic recursive solutions for multi-exercise problems driven by Lévy processes, speeding up computation and facilitating phase-type fitting in model calibration (Leung et al., 2015).

5. Advanced Structural and Game-Theoretic Insights

Randomized stopping times extend or resolve structural and equilibrium issues:

• Time-Inconsistency and Game-Theoretic Equilibria: For stopping problems with nonexponential (weighted) discounting, classical pure threshold (deterministic) strategies may not provide equilibrium (as smooth fit may fail). Introducing mixed stopping via a stopping intensity and local time pushes (i.e., singular intensities at boundaries) can restore equilibrium; existence of such equilibria is guaranteed under variational inequalities with smooth fit constraints (Bodnariu et al., 2022).
• Simultaneous Events and Dependency Modelling: The Marshall–Olkin construction and extensions (e.g., via bivariate Gumbel distributions) enable modeling of positively and negatively dependent stopping times, with explicit probability for simultaneous occurrences, central to credit risk, epidemiology (e.g., COVID-19 superspreading), and civil engineering (joint failure of components) (Protter et al., 2021).
• Honest Times and Representation as Randomized Stopping: Any honest time (in the sense of foretellable random times) that avoids all stopping times can be represented as the last maximum of a nonnegative local martingale with zero terminal value, and this characterization is robust even in the presence of jumps, provided paths attain their running supremum continuously. The conditional survival probability then has an explicit martingale representation (Kardaras, 2012).

6. Mathematical Formulations and Explicit Results

Key mathematical structures and representative formulas encountered are:

• Canonical Representation via Uniform Randomization:

$$\psi = \inf \{ t \geq 0 : K_t \geq U \}, \qquad Q[\psi > t | \mathcal{F}_t] = 1 - K_t$$

with $U \sim \mathrm{Uniform}[0,1]$ and $K_t$ adapted, nondecreasing (Kardaras, 2010).

• Distributional Equivalence: For any optional process $Y$,

$E_P[Y_{\rho \wedge t}] = E_Q[Y_{\psi \wedge t}]$

and so, up to the law of the process stopped at the random time, it suffices to work with a randomized stopping time (Kardaras, 2010).

• Game-Equivalent and Detailed Distribution: Randomized, mixed, and behavioral stopping times are equivalent (have identical outcome distributions), and for any process $X$,

$E\left[ \sum_n p_n(\omega) X_n \right]$

is the expected payoff for all equivalent representations (Solan et al., 2012).

• Continuous-Time Stopping Times:
  • Mixed: $p: \Omega \times [0,1] \to [0,T]$, $p(\cdot, r)$ stopping for almost every $r$.
  • Randomized: $p_t$ adapted, right-continuous, nondecreasing, $p_T = 1$.
  • Equivalence via induced measure: $$d_p(A \times [0, t]) = \int_A p_t(\omega) dP(\omega)$$ (Shmaya et al., 2014).
• Optimal Stopping under Model Uncertainty (OCE):

$$\sup_{\tau \in \mathcal{T}} \inf_{x \in \mathbb{R}} E[\Phi^*(x + Y_\tau) - x] = \inf_{x \in \mathbb{R}} \sup_{\tau \in \mathcal{T}} E[\Phi^*(x + Y_\tau) - x]$$

with randomized stopping times convexifying the decision set (Belomestny et al., 2014).

• Recursive Structure in Multiple Stopping with Randomized Delays:

$$Z_n(\sigma) = \operatorname{ess\,sup}_t E \left[ Y(t) + E(Z_{n-1}(T + \delta_1) | \mathcal{F}_t) \mid \mathcal{F}_\sigma \right]$$

and the need for filtration extension at each randomized waiting period (Christensen et al., 2012).

7. Open Problems and Directions for Further Research

Several research directions and open challenges arise:

• Extension to High-Dimensional and Non-Markovian Settings: While algorithms for randomized optimal stopping are increasingly scalable, efficient handling of high-dimensional, path-dependent, or infinite-dimensional processes remains a major problem, particularly for practical RL or rare event simulation (Dong, 2022, Hartmann et al., 13 Feb 2024).
• Time-Inconsistent Stopping and Equilibrium Analysis: Deeper connections between mixed strategies, local time randomization, and dynamic consistency in sequential decision problems require further paper, especially for real options under nonstandard discounting (Bodnariu et al., 2022).
• Joint Laws and Multivariate Dependence: The full characterization of joint distributions for multiple dependent stopping times, both with positive and negative correlation structures, remains an active topic, particularly relevant to systemic risk in finance, epidemic modeling, and reliability (Protter et al., 2021).
• Algorithmic Equivalence and Efficiency: Theoretical and practical implications of switching between randomized, mixed, or behaviorally defined stopping times for algorithmic performance; the potential for adaptive algorithmic selection to enhance simulation or learning-based strategies is largely unexplored.
• Measure-theoretic Foundations and Robustness: Subtleties in measurability, especially in the context of stopping games or incomplete/inadequate filtrations, call for further foundational development (Solan et al., 2012).

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Randomized stopping times unify, generalize, and enhance the theory and practice of optimal stopping, providing an essential toolkit for both stochastic modeling and computational methods. Their mathematical structure—combining probabilistic, analytic, and combinatorial tools—yields robust equivalences, flexible representations, and computationally tractable algorithms indispensable for a wide range of applications in probability, finance, statistics, and physics.