Sublinear Expectations Overview

• Sublinear expectations are nonlinear functionals characterized by monotonicity, constant preservation, subadditivity, and positive homogeneity, effectively modeling uncertainty.
• They are represented as a supremum over linear expectations, linking robust risk measures with generalized limit theorems through capacities and convergence theories.
• Applications include robust statistics, financial modeling under ambiguity, and nonlinear martingale theory, offering practical insights for risk analysis.

A sublinear expectation is a nonlinear functional on a linear space of random variables, formally satisfying monotonicity, constant preservation, subadditivity, and positive homogeneity. Sublinear expectations provide an axiomatic and robust framework to model model uncertainty (Knightian uncertainty), ambiguity aversion, and related risk attitudes in situations where classical probability (additive expectations) is too restrictive. Their analysis and limit theorems generalize classical probability theory, generating a wide range of new phenomena in stochastic processes, statistics, and mathematical finance.

1. Mathematical Foundation and Representation

A sublinear expectation on a space $\mathcal{H}$ of bounded random variables is a map

$\mathbb{E} : \mathcal{H} \to \mathbb{R}$

that for all $X,Y\in\mathcal{H}$, all $c\in\mathbb{R}$, and $\lambda\ge0$ satisfies:

• Monotonicity: $X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$
• Constant preservation: $\mathbb{E}[c]=c$
• Subadditivity: $\mathbb{E}[X+Y]\le\mathbb{E}[X]+\mathbb{E}[Y]$
• Positive homogeneity: $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X]$

The fundamental convex-analytic result is that every sublinear expectation can be represented as a supremum over a (possibly non-dominated, convex and weakly compact) family of linear expectations: $\mathbb{E}[X] = \sup_{P\in\mathcal{P}} E_P[X]$ where each $\mathbb{E} : \mathcal{H} \to \mathbb{R}$0 is the expectation with respect to some probability measure $\mathbb{E} : \mathcal{H} \to \mathbb{R}$1 (0803.2656, Erreygers, 2023, Cohen, 2011). This dual representation lies at the heart of the theory, connecting sublinear expectations to risk measures, robust statistics, and multiple-priors models. The associated upper capacity is $\mathbb{E} : \mathcal{H} \to \mathbb{R}$2, while the conjugate (lower) sublinear expectation is $\mathbb{E} : \mathcal{H} \to \mathbb{R}$3.

2. Key Concepts: Independence, Capacities, Convergence

2.1 Independence

Independence in sublinear expectation spaces departs from the Kolmogorov product measure construction and is defined via consistency with the upper expectation: $\mathbb{E} : \mathcal{H} \to \mathbb{R}$4 where $\mathbb{E} : \mathcal{H} \to \mathbb{R}$5 is said to be independent of $\mathbb{E} : \mathcal{H} \to \mathbb{R}$6 (0803.2656, Li, 2021). There is also a weaker notion called pseudo-independence defined via classical conditional expectation under each $\mathbb{E} : \mathcal{H} \to \mathbb{R}$7, which can be upgraded to full independence under an extended set of priors (Li, 2021).

2.2 Capacities and Their Role

Capacities replace the probability measure in limiting robust frameworks. The upper capacity $\mathbb{E} : \mathcal{H} \to \mathbb{R}$8 and the lower capacity $\mathbb{E} : \mathcal{H} \to \mathbb{R}$9 provide the basis for “almost sure” and convergence notions (quasi-surely, in capacity). Random variables $X,Y\in\mathcal{H}$0 converge in capacity to $X,Y\in\mathcal{H}$1 if $X,Y\in\mathcal{H}$2 for every $X,Y\in\mathcal{H}$3 (Hu et al., 2016).

The Choquet integral

$X,Y\in\mathcal{H}$4

is used to quantify non-additive expectations.

2.3 Convergence and Dominated Convergence

Modes of convergence in this setting include $X,Y\in\mathcal{H}$5 convergence, convergence in capacity, and convergence in distribution. Under the monotone continuity of $X,Y\in\mathcal{H}$6, $X,Y\in\mathcal{H}$7 convergence implies convergence in capacity, which in turn implies convergence in distribution. The sublinear dominated convergence theorem requires uniform integrability and convergence in capacity (Hu et al., 2016, Zhang, 2019).

3. Limit Theorems: Law of Large Numbers and Central Limit Theory

3.1 Law of Large Numbers (LLN)

Under i.i.d. or pseudo-independent sequences with sublinear expectation $X,Y\in\mathcal{H}$8, the law of large numbers has the form: $X,Y\in\mathcal{H}$9 for every continuous $c\in\mathbb{R}$0 with at most linear growth, where $c\in\mathbb{R}$1 is the interval of mean-uncertainty (the range of expectations under the prior family) (0803.2656, Hu, 2011, Li et al., 2015, Li, 2021). The convergence is characterized in terms of upper capacity, not with respect to a single probability measure (Zhang, 2023, Song, 2022).

Strong laws, weak laws, and Marcinkiewicz-type results (for partial sums appropriately normalized) hold in this framework, with quasi-sure or in capacity convergence replacing classical almost-sure convergence (Zhang et al., 2017, Li et al., 2015, Hu, 2011).

3.2 Central Limit Theorem and Nonlinear Limit Laws

The central limit theorem under sublinear expectations establishes convergence of normalized sums to a $c\in\mathbb{R}$2-normal or more general $c\in\mathbb{R}$3-distribution law. Specifically, for i.i.d. $c\in\mathbb{R}$4 with zero mean under $c\in\mathbb{R}$5, the CLT states: $c\in\mathbb{R}$6 for all $c\in\mathbb{R}$7, where $c\in\mathbb{R}$8 jointly have a $c\in\mathbb{R}$9-distribution determined by nonlinear mean and covariance functionals (0803.2656).

Threshold moment conditions are necessary: the CLT fails if only finite variance holds but not uniform integrability (Li, 2021). The limit distribution need not be additive, and may be characterized as the unique viscosity solution to a fully nonlinear PDE (the $\lambda\ge0$0-heat equation) (0803.2656, Li et al., 20 Jun 2025).

Multi-dimensional versions and laws for sample means of random vectors, including exponential concentration (Azuma-Hoeffding and Bernstein inequalities), are available in the sublinear setting (Seyoum, 25 Feb 2026).

4. Martingale Theory, Nonlinear Dynamics, and Path Space

Dynamic risk, time consistency, and nonlinear martingale theory have been developed for sublinear expectations, both in discrete-time (Cohen et al., 2011) and on path space (Nutz et al., 2012). The optional sampling theorem, sublinear martingale convergence theorems, and backward stochastic difference equations (BSDEs) all admit robust generalizations.

On path space $\lambda\ge0$1, dynamic sublinear expectation families $\lambda\ge0$2 can be constructed to satisfy the tower property, even for highly non-Markovian or model-uncertain processes. The conditional expectation at stopping times and the optional sampling theorem follow without requiring exceptional null sets (Nutz et al., 2012).

Dual representations and aggregation properties hold, using Hahn decompositions and essential supremum over a family of priors (Cohen, 2011). When the family of priors is closed under pasting, time consistency for the sublinear expectation is equivalent to stability of the prior family (Cohen, 2011).

5. Applications: Robust Statistics, Risk Measures, and Model Uncertainty

Sublinear expectations underlie coherent and convex risk measures, robust statistics, and financial modeling under ambiguity. In regression, sublinear expectation generalizes least squares to mini-max criteria over priors, yielding robust point estimates and variable selection procedures that target worst-case mean squared error (Lin et al., 2013).

Limit theorems and moment inequalities (e.g., von Bahr-Esseen, Rosenthal-type) for partial sums and weighted sums under sublinear expectations provide the theoretical underpinning for simulation, tail risk quantification, and Monte Carlo methods under ambiguity (Wu et al., 2023, Zhang, 2014). The extension to set-valued random objects leads to the study of sublinear expectations for random closed sets and their dual representations in convex analysis and multivariate statistics (Molchanov et al., 2019).

Robust Markov chains, stochastic processes with imprecise transition rates, and model-uncertain jump processes are constructed using sublinear Markov semigroups, with a rigorous extension theory for countable state space and path space functionals (Erreygers, 2023).

6. Technical Advances and Open Problems

Recent developments include:

• Exponential concentration inequalities for the sample mean under sublinear expectations, achieving sub-Gaussian and Bernstein-type tail bounds as in the classical linear theory, but with net-covering and dimension-free variants (Seyoum, 25 Feb 2026).
• Limit theorems for triangular arrays and nonlinear expectations dominated by sublinear expectations, yielding fully nonlinear limit laws characterized by nonlinear PDEs, beyond the classical or sublinear settings (Li et al., 20 Jun 2025).
• Careful analysis shows that classical moment conditions (first moment for LLN, second moment for CLT) are necessary and sufficient in the sublinear context, with explicit counterexamples demonstrating sharpness (Li, 2021, Zhang, 2019).
• Convergence equivalence: Under monotone continuity or countable subadditivity of the capacity, the convergences in $\lambda\ge0$3, capacity, and distribution are largely equivalent, enabling a direct transfer of several classical limit results (Hu et al., 2016, Zhang, 2019).
• Functional law of large numbers, strong law with random limiting boundaries (random cluster points), and presence of path-dependent limit sets for averages in the strong law, demonstrating a qualitatively different asymptotic behavior under non-additivity (Zhang, 2023).

7. Table of Key Sublinear Expectation Properties

Property	Classical Probability ($\lambda\ge0$4)	Sublinear Expectation ($\lambda\ge0$5)
Additivity	$\lambda\ge0$6	$\lambda\ge0$7
Dual representation	Linear, single $\lambda\ge0$8	Supremum over family: $\lambda\ge0$9
Law of Large Numbers limit	Deterministic mean	Interval $X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$0
Central Limit Theorem	Normal law	$X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$1-normal / $X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$2-distribution
Independence	Product measure	Functional consistency (Peng; pseudo-)
Capacity (event probability)	$X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$3	$X\ge Y \implies \mathbb{E}[X]\ge\mathbb{E}[Y]$4

The sublinear expectation framework thus provides a mathematically rigorous, flexible extension of classical probability, essential for the analysis and modeling of systems under model uncertainty and ambiguity. Its deep connection with viscosity solutions of fully nonlinear PDEs, its impact on stochastic analysis and robust finance, and its nontrivial asymptotic phenomena make it a central object of modern probability theory and mathematical risk analysis.