Sublinear Expectation Framework

• Sublinear expectation framework is a generalization of classical probability that defines a convex, non-additive expectation over multiple probability measures to capture model uncertainty.
• Its dual representation and sublinear Markov semigroups facilitate robust stochastic analysis, nonlinear PDEs, and applications in finance and risk management.
• The framework supports convergence theorems and law of large numbers, extending classical results to settings where ambiguity and non-additivity are intrinsic.

A sublinear expectation is a convex, positively homogeneous, monotone, and constant-preserving functional 𝔼: ℋ→ℝ defined on a linear space ℋ of real-valued random variables. Unlike classical linear expectation, which is characterized by a unique underlying probability measure, a sublinear expectation is typically represented as the upper envelope over a (possibly weakly compact) family of probability measures. The sublinear expectation framework generalizes classical probability in order to accommodate model uncertainty, non-additive beliefs, and ambiguity aversion, playing a key role in robust stochastic analysis, nonlinear PDEs, financial risk modeling, and statistics (Yang et al., 2024, Ren, 2011, Li et al., 2023).

1. Axiomatic Structure and Representation

A functional 𝔼: ℋ→ℝ is a sublinear expectation if:

• Monotonicity: X≥Y ⇒ 𝔼[X]≥𝔼[Y].
• Constant preservation: 𝔼[c]=c for c∈ℝ.
• Sub-additivity: 𝔼[X+Y]≤𝔼[X]+𝔼[Y].
• Positive homogeneity: 𝔼[λX]=λ𝔼[X], for all λ≥0.

In many settings, sublinear expectations admit a dual representation: $𝔼[X] = \sup_{P\in𝒫} E_P[X]$ where 𝒫 is a (weakly compact) family of probability measures on the underlying measurable space (Ren, 2011). This connects the framework to robust statistics and the theory of capacities.

2. Sublinear Markov Semigroups and Path Space Extensions

Given a countable state space X, a sublinear Markov semigroup (Tₜ) consists of contractions on L(X) that are sublinear expectations for each t≥0, satisfy the semigroup property, and often possess a uniform jump-rate bound. Starting with a consistent family of finite-dimensional upper expectations (induced by Tₜ and an initial upper expectation), one recursively builds a projective system. Under downward-continuity (for bounded finitary functions) and a modulus-of-continuity condition on jump indicators, one obtains a unique extension of the sublinear expectation to all measurable functions on the space of càdlàg paths (Erreygers, 2023).

Table: Key Properties for Extension

Property	Role	Sufficient Condition
Downward-continuity	Ensures limit	Holds for cylinder σ-alg.
Jump modulus-of-continuity bound	Path regularity	Uniformly bounded jump rate

These properties guarantee that the path-space extension is well defined and unique, and that the corresponding robust Daniell–Kolmogorov theorem holds.

3. Weak Compactness, Tightness, and Representation Theorems

A core structural property of sublinear expectations in path space is the existence of a weakly compact representing set of measures 𝒫. This is established through two complementary routes (Ren, 2011):

• Separation/Choquet–Daniell–Stone: The supremum over finitely additive linear forms extends to countably additive probabilities, whose image under the process yields a weakly compact family.
• Stochastic control representation: Expressing the process as controlled SDEs or stochastic integrals with coefficients varying over a compact set, one verifies tightness using Kolmogorov–Chentsov criteria.

The tightness of 𝒫 is what ensures that suprema over cylinder functionals have limits and that the associated stochastic processes possess càdlàg modifications with regularity inherited from the components.

4. Model Classes: G-Expectations, G-Lévy Processes, and Nonlinear Markov Models

A prototypical example is the G-expectation, parameterized by convex sets of volatility or Lévy triplets (Ren, 2011). In this context, the canonical process is a G-Lévy process with independent, stationary increments under 𝔼^G, and the generator takes the form: $$G(u,p,A) = \sup_{(ν,q,Q)∈𝕌} \left\{ \int [u(x+z) - u(x)] ν(dz) + ⟨p,q⟩ + (1/2) \text{tr}[A QQ^T] \right\}$$ Nonlinear integro-PDEs characterize distributions, and the sublinear expectation is the value function of the associated control problem. The framework subsumes robust random walks, nonlinear birth–death processes, and random G-expectations with path-dependent stochastic volatility (Erreygers, 2023, Ren, 2011).

5. Convergence Theorems and Law of Large Numbers

Sublinear expectations enable a hierarchy of convergence concepts:

• Lᵖ convergence → convergence in capacity → convergence in distribution The relevant capacity is defined as V(A) = 𝔼[1_A]; almost-sure convergence is expressed as quasi-surely outside sets of zero capacity.

With independence defined via the equation

$𝔼[\varphi(X,Y)] = 𝔼[𝔼[\varphi(x,Y)]_{x=X}]$

and i.i.d. sequences, strong laws of large numbers, maximal distribution limits, and nonlinear central limit theorems are obtained. Under mild regularity, empirical means fill out the entire uncertainty interval [\underlineμ, \overlineμ], and normalized sums converge in law to distributions governed by nonlinear PDEs or PIDEs, generalizing Lévy–Khintchine representation to the sublinear regime (Yang et al., 2024, Ren, 2011).

6. Applications and Significance

The sublinear expectation framework is essential in the study of

• Stochastic control under ambiguity: Value functions for nonlinear SDEs and PDEs, including G-Lévy and robust Markov processes.
• Finance and risk management: Dynamic risk measures, G-VaR estimations via maximal distribution MLEs (Li et al., 2023).
• Stochastic calculus and SPDEs under model uncertainty: Martingale representation, Girsanov transformations, and path regularity for processes with volatility/jump uncertainty (Ren, 2011).

The capacity-induced regularity allows the development of a nonlinear stochastic calculus (Itô formula, SDEs, backward SDEs) and justifies the extension of statistical theory (e.g., regression, variable selection) to robust, model-uncertain settings (Lin et al., 2013).

7. Extensions and Future Directions

The framework naturally extends to countable-state processes with robust transition operators (Erreygers, 2023), set-valued and multivariate expectations (Molchanov et al., 2019), and infinite-dimensional systems such as G-Gaussian random fields and robust field quantization (Hu, 2023). Ongoing research investigates nonlinear ergodic theory, mixing coefficients, and their implications for stochastic PDEs driven by G-noise (Huang et al., 2024).

The sublinear expectation formalism, by providing a rigorous mathematical underpinning for model ambiguity and robust reasoning beyond the classical linear paradigm, represents a fundamental shift in modern stochastic analysis, probability theory, and its applications.