Robust Daniell–Kolmogorov Theorem

• Robust Daniell–Kolmogorov Theorem is a generalization of the classical theorem that constructs stochastic processes using nonlinear, sublinear, or convex expectation functionals.
• It establishes measures or expectations directly on path spaces, ensuring path regularity through conditions like right-continuity in law and uniform tightness of jumps.
• Applications include nonlinear Markov processes, ambiguity models in decision theory, and G-expectation spaces for robust stochastic analysis.

The Robust Daniell–Kolmogorov Theorem generalizes the classical Daniell–Kolmogorov Extension Theorem to settings involving nonlinear expectations, sublinear or convex expectation functionals, or uncertainty models involving set-valued or multiple probability measures. This framework permits the construction and extension of stochastic processes with prescribed marginal distributions or expectations—including sublinear (robust) laws—directly on path spaces, often under weaker or different regularity and consistency assumptions than those found in the classical theory. Major variants include robust extensions for countable-state càdlàg path processes, extension theorems for nonlinear (convex) expectations, and Kolmogorov-type theorems for systems of weak*-closed sets of marginals. Applications include nonlinear Markov processes, ambiguity models in decision theory, and the construction of G-expectation spaces.

1. Classical Daniell–Kolmogorov Theorem and Motivation for Robustification

The classical Daniell–Kolmogorov Theorem asserts that given a consistent family of finite-dimensional distributions $\{\mu_u: u \in U_T\}$ indexed by tuples $u$ of time-points in $T$, there exists a stochastic process (measure on the infinite path space) with these marginals. However, in classical constructions, this measure is initially supported on the full product space (e.g., $S^T$), which is generally too large since the set of regular paths (such as continuous or càdlàg paths) is not all of $S^T$ and may not be measurable in the constructed extension. Nontrivial regularity (e.g., Kolmogorov–Chentsov criteria) and modification arguments are needed to recover sample-path regularity: a key subtlety in both the topology and measurability of the path space $D(T,S)$ or $C(T,S)$. This gap motivates robust versions that construct the correct measure or expectation space directly on the appropriate path-space, under conditions adapted to the desired regularity and set-indexed families of laws or expectations (Erreygers et al., 2023, Ivanenko et al., 2016, Denk et al., 2015).

2. Robust Daniell–Kolmogorov Theorem for Countable-State Càdlàg Processes

Let $S$ be a countable state space, $T \subseteq \mathbb{R}$ a time index set, $U_T$ the set of finite strictly increasing tuples from $u$0. The robust version, as in Erreygers & De Bock (Erreygers et al., 2023), addresses the measure-theoretic shortfall by constructing probability measures directly on the product $u$1-algebra over the path space $u$2 of all càdlàg maps $u$3. The main result is as follows:

Given a consistent family $u$4 of finite-dimensional distributions on $u$5 satisfying:

• R1 (Right-continuity in law): For every $u$6 that is a right-limit point,

$u$7

• R2 (Uniform tightness of jumps): For every $u$8 with $u$9,

$T$0

where $T$1 is the number of jumps of the piecewise-constant path $T$2 on the grid $T$3,

there exists a unique probability measure $T$4 on $T$5 (product $T$6-algebra over $T$7) with prescribed marginals $T$8 for all $T$9 and $S^T$0. Here, the process $S^T$1 is $S^T$2-almost surely càdlàg (Erreygers et al., 2023).

The proof establishes countable additivity of the finite premeasure on the algebra of cylinder sets in $S^T$3 using the measurable structure of the jump count and tightness (R2), so no subsequent modification or regularization of the process is required. The approach contrasts with the classical construction, in which the initial measure on $S^T$4 is later shown (under additional regularity) to concentrate on $S^T$5, usually invoking auxiliary limit arguments or inequalities.

3. Robust Extensions with Multiple Probabilities

Ivanenko and Pasichnichenko (Ivanenko et al., 2016) develop a robust Kolmogorov extension theorem for set-valued families of finite-dimensional distributions. Let $S^T$6 be an arbitrary index set, $S^T$7 a set of outcomes, $S^T$8 an algebra on $S^T$9, and $S^T$0 nonempty weak*-closed sets (in the sense of probability measures) for each finite tuple. The system satisfies:

• Permutation invariance (C1): For any permutation $S^T$1 of $S^T$2, coordinate shuffling maps admissible marginals in a compatible way.
• Marginal/extensional consistency (C2): Projections of joint marginals and their possible extensions must correspond to members in respective $S^T$3.

Given these consistency constraints, there exists a probability space $S^T$4, a weak*-closed set $S^T$5, and coordinate maps $S^T$6 such that for every finite tuple, the push-forward of $S^T$7 under $S^T$8 recovers the prescribed $S^T$9 (Ivanenko et al., 2016).

This construction generalizes the classical theorem to accommodate ambiguity (multi-prior, robust, or imprecise probability) settings, ensuring that all compatible finite-dimensional specifications are realized without requiring uniqueness of the global law.

4. Robust Kolmogorov Theorems for Nonlinear and Sublinear Expectations

Denk, Kupper, and Nendel (Denk et al., 2015) extend the classical Daniell–Stone and Kolmogorov extension theorems to the setting of nonlinear (convex or sublinear) expectations. For a projective system of convex pre-expectations $D(T,S)$0 defined on a family of finite-dimensional Riesz subspaces $D(T,S)$1 indexed by the finite subsets $D(T,S)$2 of a general index set $D(T,S)$3, subject to a consistency condition $D(T,S)$4 for $D(T,S)$5 and $D(T,S)$6, the following robust extension holds:

There exists a unique convex expectation $D(T,S)$7 on $D(T,S)$8 that (a) extends each $D(T,S)$9, (b) is continuous from below on all bounded measurable functions, and (c) is continuous from above on the monotone closure of the algebraic cylinder space. In the sublinear (coherent) case, if each $C(T,S)$0 for convex compact sets $C(T,S)$1, the extension ensures $C(T,S)$2 compact with the correct finite-dimensional projections. The dual representation

$C(T,S)$3

holds for $C(T,S)$4 (Denk et al., 2015).

This extension provides the foundation for nonlinear Markov processes, $C(T,S)$5-expectations, and similar robust or ambiguity-sensitive processes.

5. Key Definitions and Conditions

• Càdlàg path space $C(T,S)$6: Collection of $C(T,S)$7 such that for all $C(T,S)$8 which are left (resp. right-) limit points of $C(T,S)$9, the left (resp. right-) limits of $S$0 exist, and the path is right-continuous everywhere.
• Product $S$1-algebra $S$2: $S$3.
• Jump count $S$4: Counts the number of discontinuities in the path $S$5 along the finite grid $S$6.
• Regularity (R1, R2): Weak regularity assumptions on the finite-dimensional distributions, namely right-continuity in law and uniform tightness of the jump count, guarantee both the existence and the appropriateness of the path-space support (Erreygers et al., 2023).

6. Proof Outlines and Methodological Comparison

Table: Key Steps in Robust Extension Approaches

Approach	Step 1	Step 2	Step 3
Countable-state càdlàg (Erreygers et al., 2023)	Cylinder premeasure on $S$7	Prove countable additivity (R1/R2)	Carathéodory extension to probability
Multiple probabilities (Ivanenko et al., 2016)	Path space $S$8 construction	Compactness, intersection of closed sets	Extension yields set of global laws
Nonlinear/sublinear (Denk et al., 2015)	Fiber-wise nonlinear Daniell–Stone	Algebraic patching of expectations	Extension by convex duality

The robust extensions characteristically construct or patch measures or expectations directly on the chosen path or function space, under conditions tailored to ensure the target regularity or ambiguity structure. These methods avoid the classical necessity for path-modification arguments, as the measure/expectation is supported on the regular path space by construction.

7. Applications, Limitations, and Further Directions

Robust Daniell–Kolmogorov theorems underpin models involving ambiguity (multiple priors or credal sets in decision theory), nonlinear stochastic processes (including quasi-sure or sublinear expectation spaces), and the construction of processes with prescribed nonlinear kernels. In particular, constructions for robust Markov chains, controlled diffusions (e.g., $S$9-expectation processes), and ambiguous behavior models are permitted (Denk et al., 2015, Ivanenko et al., 2016).

Limitations include the challenge of accommodating uncountable state spaces $T \subseteq \mathbb{R}$0 (e.g., $T \subseteq \mathbb{R}$1), where discrete jump tightness must be replaced by modulus of continuity or Skorokhod-type tightness. Further generalizations may utilize other regularity classes (continuous/Hölder/BV-paths) or more abstract index sets, modifying regularity condition (R2) to fit the topology of the path space (Erreygers et al., 2023).

The core methodological insight is to build the measure, set of measures, or convex expectation directly on the appropriate regular path domain, imposing only the regularity/coherence needed for extension, thereby ensuring support and measurability properties by construction.