Nonnegative Supermartingales: Theory & Applications

• Nonnegative supermartingales are stochastic processes that remain nonnegative and satisfy a mean-decreasing property, ensuring robust convergence and optimal stopping behavior.
• The Doob–Meyer decomposition and optional stopping theorem provide a framework to analyze these processes, extending classical martingale results to more general settings.
• Advanced inequalities, such as Ville’s inequality and its extensions, enable practical applications in sequential testing, financial modeling, and risk assessment.

A nonnegative supermartingale is a stochastic process that generalizes martingales by allowing for mean decrease, subject to an essential constraint of pathwise nonnegativity. Formally, let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0}, P)$ be a filtered probability space satisfying the usual conditions. A process $(M_t)_{t\ge 0}$ is called a nonnegative supermartingale if it is adapted, nonnegative ($M_t \ge 0$ a.s. for all $t$), and satisfies the supermartingale property $E[M_t \mid \mathcal{F}_s] \leq M_s$ for all $s \leq t$ (Howard et al., 2018). Nonnegative supermartingales arise in diverse domains, including stochastic analysis, mathematical finance, sequential analysis, and the theory of stochastic processes.

1. Foundational Theory and Key Properties

Nonnegative supermartingales inherit many of the critical structural features of classical supermartingales but impose additional robustness via their nonnegativity. Core properties include:

• Doob–Meyer decomposition: Any càdlàg, class D nonnegative supermartingale $X$ admits a unique decomposition $X = M - A$ where $M$ is a martingale and $A$ is a predictable, nondecreasing process with $A_0 = 0$ (Schachermayer, 2013).
• Optional Stopping: For any stopping time $\tau$, the supermartingale property ensures $E[X_\tau] \leq E[X_0]$.
• Almost sure convergence: Classical theorems guarantee that a nonnegative supermartingale converges a.s. to a finite limit under mild right-continuity and integrability assumptions. At stopping horizons (including random or foretellable times), nonnegative local supermartingales converge a.s. (Larsson et al., 2014).
• Maximal inequalities: Ville’s inequality provides uniform-in-time bounds for probabilities of exceeding thresholds: $P(\sup_t M_t \geq a) \leq E[M_0]/a$, with various refinements for nonconstant thresholds or lower floors (Koolen et al., 22 Feb 2025, Wang et al., 2023).

2. Decomposition, Extended Classes, and Measure-Theoretic Aspects

The Doob–Meyer decomposition $X = M - A$ formalizes the exact mechanism by which nonnegative supermartingales lose mass compared to martingales. In more advanced settings, the following generalizations and representations are critical:

• Optional strong supermartingales: General processes (possibly lacking right-continuity) that are optional, have integrable values at all stopping times, and satisfy $X_\sigma \geq E[X_\tau|\mathcal{F}_\sigma]$ for stopping times $0 \leq \sigma \leq \tau$ (Czichowsky et al., 2013).
• Föllmer measures: Nonnegative supermartingales can often play the role of Radon-Nikodym densities between measures, even when the supermartingale property drives them to zero asymptotically (Perkowski et al., 2013). Existence and uniqueness of such measure extensions depend on boundary mass-loss properties and the details of the Doob–Meyer decomposition.
• Extended nonnegative supermartingales: By allowing processes $M_n$ to take value $+\infty$ and interpreting conditional expectations in the sense of monotone limits, Wang and Ramdas define an "extended" class that yields maximal inequalities (via the extended Ville’s inequality) without assuming integrability (Wang et al., 2023). These tools are crucial for sequential methods involving improper mixture priors.

3. Maximal and Concentration Inequalities

The power of nonnegative supermartingales is epitomized in the strength of their maximal inequalities and resulting concentration bounds:

• Classical Ville’s inequality: For any nonnegative supermartingale $M_t$ and $a > 0$, $P(\sup_t M_t \geq a) \leq E[M_0]/a$. Extensions exist for floor-bounded and barrier problems, handling time-varying boundaries or lower curves through tight generalizations, establishing bounds such as

$$P\left(\exists n : M_n \geq u(n)\right) \leq 1 - \frac{u(0) - E[M_0]}{u(0) - \ell(0)} S(0)$$

where $\ell(n)$ and $u(n)$ are time-varying bounds, and $S(0)$ is an explicit product correction (Koolen et al., 22 Feb 2025).

• Time-uniform concentration: The construction of exponential supermartingales (e.g., $M_t = \exp\{\lambda S_t - \psi(\lambda)V_t\}$, with sub-$\psi$ cumulant bounds) allows for Chernoff/Hoeffding/Bennett/Bernstein/Freedman-type uniform-in-time tail inequalities (Howard et al., 2018). These results extend to matrix and Banach-space-valued learners via operator-valued supermartingales (Wang et al., 28 Jan 2024).
• Nonintegrable maximal bounds: The extended Ville’s inequality of Wang & Ramdas strengthens the classical version to

$$P(\exists n \geq m : M_n \geq C) \leq P(M_m \geq C) + \frac{1}{C} E[M_m 1_{M_m < C}]$$

yielding tighter control when integrability may fail or is not assumed (Wang et al., 2023).

4. Applications across Probability, Statistics, and Finance

Nonnegative supermartingales serve as fundamental objects in multiple domains:

• Sequential testing and inference: All admissible constructions of anytime-valid tests, p-processes, e-processes, and confidence sequences for broad classes of null hypotheses must be based on nonnegative (super)martingales (Ramdas et al., 2020). For any level-$\alpha$ sequential test, the optimal decision rule is given by thresholding a nonnegative martingale via a first-passage process.
• Financial mathematics and asset pricing: The "supermartingale deflator" is the analytic tool underlying no-arbitrage and viability conditions in asset pricing models beyond the existence of a strictly positive numéraire portfolio (Bálint, 2020). Under transaction costs, the nonnegativity theorem for wealth processes involves the optional strong supermartingale property, with counterexamples showing sharp distinctions from the frictionless case (Schachermayer, 2013).
• Azéma supermartingales and random times: In advanced stochastic calculus, Azéma supermartingales associated to random times admit precise representations as ratios $U_t/U^*_t$ for nonnegative local martingales $U$ vanishing at infinity, underpinning results on the structure of optional projections and last-passage times (Song, 2016).
• Stochastic program verification: Lexicographic ranking supermartingales (LexRSMs) with single-component nonnegativity, fixability techniques, and their lazy variants are key to sound automated termination analysis of probabilistic programs under weaker nonnegativity conditions (Takisaka et al., 2023).

5. Extensions: Non-Probabilistic and Matrix-Valued Supermartingales

Recent advances extend the classical framework in fundamentally new directions.

• Non-probabilistic superhedging frameworks: By replacing expectation with sublinear superhedging operators and null-set functionals, supermartingales are defined solely in terms of pathwise inequalities, supporting decomposition and convergence theorems in the absence of underlying probability measures (Bender et al., 2023).
• Positive semidefinite matrix supermartingales: Matrix-valued extensions rely on the Loewner partial order. Analogues of Doob's and Ville's inequalities, together with spectral or trace-inequalities, yield a unified theory supporting time-uniform matrix concentration and self-normalized bounds for covariance estimation, random matrix processes, and multivariate sequential testing (Wang et al., 28 Jan 2024).
• Pathwise and trajectorial convergence: Tools such as Fatou limits, convex combination arguments, and Komlós-type lemmas, cement the centrality of nonnegative supermartingale limits in martingale theory and facilitate convergence in probability at stopping times (Czichowsky et al., 2013).

6. Modern Directions and Open Questions

Current research expands the boundaries of nonnegative supermartingale theory.

• Extended classes and nonintegrability: Generalizations to extended supermartingales facilitate maximal and optional stopping results for processes outside $L^1$, broadening the horizon for sequential analysis and nonparametric confidence sequences (Wang et al., 2023).
• Generalizations of Ville’s inequality: Tight, sharp inequalities for time-dependent lower and upper boundaries reflect refined control over boundary crossing probabilities, with direct applications to law-of-the-iterated-logarithm-type results and uniform-in-time inference (Koolen et al., 22 Feb 2025).
• Martingale property criteria with jumps: Necessary and sufficient Novikov–Kazamaki-type criteria for the uniform integrability of nonnegative local martingales with jumps have been characterized in terms of process characteristics and extended local integrability, refining classical sufficient conditions (Larsson et al., 2014).

Table: Canonical Results for Nonnegative Supermartingales

Property/Theorem	Statement/Formulation	arXiv Reference
Doob–Meyer Decomp.	$X = M - A$, $M$ martingale, $A$ increasing predictable	(Schachermayer, 2013)
Ville’s Inequality	$P(\sup_t M_t \geq a) \leq E[M_0]/a$	(Howard et al., 2018)
Extended Ville’s Inequality	$$P(\exists n\ge m: M_n\ge C)\le P(M_m\ge C)+E[M_m1_{M_m<C}]/C$$	(Wang et al., 2023)
Nonnegativity Theorem	$X_T\ge0$ a.s. $\Rightarrow X_t\ge0$ a.s. $\forall t$	(Schachermayer, 2013)
Fatou Limit/Convergence	Convex combinations of martingales converge to optional strong supermartingale	(Czichowsky et al., 2013)
Generalized Ville Bound	Cross monotone curves: explicit tight formula for hitting probability	(Koolen et al., 22 Feb 2025)

The theory of nonnegative supermartingales thus encapsulates foundational probabilistic structure with broad and deep ramifications for asymptotic analysis, optimal stopping, mathematical finance, sequential learning, and stochastic computation. Ongoing developments in extensions, nonintegrable cases, pathwise methodologies, and non-probabilistic formulations ensure continued relevance and expansion of their analytic power.