Allowing for imprecision in the game-theoretic characterisation of the Poisson process

Abstract: In their 1993 paper 'Forecasting point and continuous processes: Prequential analysis' in Test, Vovk put forward a game-theoretic definition of the Poisson process. A key assumption therein is that the rate of the Poisson process is known or specified exactly. In contrast, I replace this assumption with the less stringent -- and arguably more realistic -- one that the available information about the process takes the form of bounds on the rate rather than a single, exact value. The resulting process has properties similar to the standard, 'precise' Poisson process, albeit with an imprecise flavour to them, thus justifying the moniker 'imprecise Poisson process'.

• The paper extends the classical Poisson process by incorporating imprecise rate bounds, allowing for realistic modeling when exact parameters are unavailable.
• It introduces game-theoretic capital processes and sublinear expectations to compute event increments and inter-arrival times under uncertainty.
• The work provides explicit computable expressions and generalizes renewal and Markov properties, paving the way for robust continuous-time probabilistic modeling.

Imprecision in the Game-Theoretic Characterization of the Poisson Process

Introduction and Motivation

The paper "Allowing for imprecision in the game-theoretic characterisation of the Poisson process" (2604.00598) extends the game-theoretic approach to stochastic processes by Vovk, Shafer, and others to account for imprecise information about the Poisson process rate parameter $\lambda$. Instead of fixing the rate exactly, as in the classical formulation, this work assumes only knowledge of bounds $\underline{\lambda} \leq \overline{\lambda}$ within which the true rate lies. The resulting generalization is referred to as the imprecise Poisson process.

This extension is motivated by practical scenarios where precise parameters are unavailable or unwarranted, and instead coherent bounds more realistically capture uncertainty. While local imprecision in discrete-time game-theoretic stochastic processes has been previously examined, this is, to knowledge, the first systematic extension for continuous-time Poisson processes, filling a notable gap.

Technical Framework

The foundation is Vovk's prequential approach, where a process is modeled as an adversarial game between Trader and Market, possibly with a Forecaster specifying local uncertainty. The observables are right-continuous, unit-jump, increasing, cadlag counting paths $(\mathcal{C})$, with variables and processes defined as functionals over paths.

The essence of the framework is the capital process: for allowed trading strategies $\mathcal{K}$, the conditional upper expectation $\overline{E}_K(f|\omega,\tau)$ at stopping time $\tau$ on counting path prefix $\omega$ is the infimum initial capital required for Trader to superhedge the target $f$ in all feasible futures extending from $\omega$. Coherency (no arbitrage) requires that no trading strategy allows unbounded profit with zero risk.

The two-sided capital process recovers the precise-rate Poisson case; the paper generalizes to one-sided strategies, crucial when only bounds on the rate are available. Here, Forecaster specifies $[\underline{\lambda},\overline{\lambda}]$ and Trader may enter into positive bets on increments bounded below by $\underline{\lambda} \leq \overline{\lambda}$0 and bounded above by $\underline{\lambda} \leq \overline{\lambda}$1 between stopping times $\underline{\lambda} \leq \overline{\lambda}$2. The structure of the allowed capital processes is shown to be closed under positive scalar combinations and addition and to satisfy coherence.

Main Results

Properties of the Imprecise Poisson Process and Upper Expectations

• Monotonicity in Rate Bounds: If the rate interval is expanded, the conditional upper expectation cannot decrease. Formally, for nested rate bounds $\underline{\lambda} \leq \overline{\lambda}$3, $\underline{\lambda} \leq \overline{\lambda}$4.
• Expectation of Increments: For constant $\underline{\lambda} \leq \overline{\lambda}$5, the upper expectation of $\underline{\lambda} \leq \overline{\lambda}$6 is precisely $\underline{\lambda} \leq \overline{\lambda}$7 and the lower expectation is $\underline{\lambda} \leq \overline{\lambda}$8. This shows that the process "remembers" the uncertainty in rate at every time step, and upper/lower expectations propagate accordingly.
• Expectation of Inter-Arrival Times: The expectation (upper/lower) of the time $\underline{\lambda} \leq \overline{\lambda}$9 to the next jump after a stopping time is $(\mathcal{C})$0 for the upper and $(\mathcal{C})$1 for the lower, consistent with classical renewal theory but widened via the imprecise parameters.
• Strong Markov Property: The framework supports memoryless reasoning and strong Markovian shifts. For any stopping time, conditional upper expectations can be shifted and "restarted," preserving the coherence of the process.
• Law of Iterated Upper Expectations (Tower Property): For finitary, bounded functionals $(\mathcal{C})$2 depending on finitely many coordinates of $(\mathcal{C})$3, the tower law holds in the sublinear setting: the upper expectation at $(\mathcal{C})$4 of $(\mathcal{C})$5 is the upper expectation at $(\mathcal{C})$6 of the upper expectation at $(\mathcal{C})$7 given $(\mathcal{C})$8, paralleling classical probabilistic recursion.
• Connection to Sublinear Poisson Semigroup: Upper expectations of the form $(\mathcal{C})$9 can be represented as $\mathcal{K}$0, where $\mathcal{K}$1 is the Nisio (sublinear) semigroup generated by the family of Poisson semigroups parameterized by all $\mathcal{K}$2. This generalizes the Kolmogorov-Chapman equations and generator formalism to the sublinear/imprecise setting.

Explicit Computable Results

The paper provides several closed-form and algorithmic expressions for upper/lower expectations of functionals relevant to the Poisson process, e.g.:

• For increments: the bounds are linear in the interval length modulated by the extreme rates.
• For time-to-next-event: the expectation is the reciprocal of the respective rate bound.
• For finitary (stepwise) variables: a recursive computation via the sublinear semigroup is possible, enabling practical evaluation for path-dependent functionals.

Relation to Previous Work

This approach connects with and generalizes several earlier works on imprecise L\'evy and Markov processes:

• The framework of Hu & Peng (viscosity PDEs for sublinear expectations of L\'evy processes) and Neufeld & Nutz (upper envelopes of expectations over sets of characteristics).
• Sublinear Kolmogorov generators and semigroup methods, as advanced by Erreygers & De Bock and Nendel.
• Imprecise Markov chain theory, where the basic transition probabilities are replaced by credal sets, and lower or upper transition operators are studied.

A distinguishing aspect is that the game-theoretic foundation avoids technical measure-theoretic conditions, provides explicit operational meaning for conditional expectations, and enables results for non-finitary (e.g., stopping time-based) functionals, such as the inter-arrival time variable.

Implications and Future Directions

Practically, this work provides a rigorous, operational framework for modeling continuous-time event processes under parametric uncertainty, with applications in finance, insurance risk, reliability engineering, and any domain where count data is observed but homogeneity assumptions may be ill-supported.

Theoretically, the main innovation is the seamless integration of imprecision into the game-theoretic approach to continuous-time stochastic processes, extending the reach of robust and sublinear probability theory. The Markov, renewal, and semigroup properties carry over in a controlled, conservative manner, maintaining the essential structure but now over families of processes rather than a single stochastic specification.

Potential future directions include:

• Extension to more general renewal processes, possibly with imprecise interarrival time distributions.
• Development of computational methods and algorithms for evaluating the Nisio semigroup in high-dimensional or non-finitary cases.
• Study of strong laws, limit theorems, and asymptotics in the sublinear/game-theoretic context.
• Application to imprecise filtering, change point detection, and control.

Conclusion

This paper systematically develops the imprecise Poisson process in the game-theoretic probability framework, providing both foundational definitions and a suite of properties paralleling classical Poisson processes but now interpreted for robust, coherent uncertainty. The marriage of game-theoretic superhedging and sublinear semigroup methods enables both operational interpretations and practical computations, and positions the imprecise Poisson process as a core object in robust continuous-time probabilistic modeling.