Sublinear Markov Semigroups: Theory & Applications

• Sublinear Markov semigroups are nonlinear transition operators characterized by sublinearity, monotonicity, constant preservation, and the semigroup property.
• They employ path-space constructions and generator representations to extend classical Kolmogorov and Hille–Yosida theory to uncertain contexts.
• Their framework finds applications in robust finance, stochastic control, and risk management by capturing model uncertainty through nonlinear expectations.

A sublinear Markov semigroup is a family of nonlinear transition operators that extend the classical Markov semigroup formalism to the context of coherent risk measures and robust stochastic modeling, providing a nonlinear expectation-theoretic framework for continuous-time uncertain processes. Such semigroups, characterized by sublinearity, monotonicity, constant preservation, and the semigroup property, underpin path-space sublinear expectations and give rise to nonlinear analogs of Kolmogorov and Hille–Yosida theory, with applications in control, finance, and robust stochastic processes.

1. Definition and Structural Properties

Let $X$ be a countable state space. Consider $L = L(X)$, the Banach space of bounded real-valued functions $f: X \to \mathbb{R}$ with norm $\|f\| = \sup_{x \in X} |f(x)|$. A family of operators $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$, where $T_t: L \to L$, is a sublinear Markov semigroup if:

1. Upper-transition operators: For each $t \ge 0$ and $x \in X$, the map $f \mapsto (T_t f)(x)$ is a sublinear expectation on $L$; i.e., for all $L = L(X)$0, $L = L(X)$1 is isotone, constant-preserving, subadditive, and positively homogeneous.
2. Semigroup property: $L = L(X)$2 for all $L = L(X)$3, with $L = L(X)$4.
3. Downward continuity: For each $L = L(X)$5, every monotone decreasing net $L = L(X)$6 in $L = L(X)$7 satisfies $L = L(X)$8 pointwise on $L = L(X)$9.
4. Uniformly bounded rate (optional, often assumed):

$f: X \to \mathbb{R}$0

which implies uniform continuity $f: X \to \mathbb{R}$1 (Erreygers, 2023).

This structure extends directly to convex cones of functions in general state spaces and can be expressed in terms of upper expectations and transition operators (Criens et al., 2023, Erreygers, 2024). On general measurable state spaces, the operators act on bounded measurable or continuous functions, and the axioms generalize accordingly (Criens et al., 2022, Goldys et al., 2022).

2. Path-Space Construction and Daniell–Kolmogorov Extension

Given an initial sublinear expectation $f: X \to \mathbb{R}$2 on $f: X \to \mathbb{R}$3 and a sublinear Markov semigroup $f: X \to \mathbb{R}$4, a canonical path-space expectation is constructed on $f: X \to \mathbb{R}$5:

• Define finite-dimensional upper expectations $f: X \to \mathbb{R}$6 for each finite set $f: X \to \mathbb{R}$7 by backward recursion:

$f: X \to \mathbb{R}$8

where $f: X \to \mathbb{R}$9.

• The family $\|f\| = \sup_{x \in X} |f(x)|$0 is consistent, i.e., $\|f\| = \sup_{x \in X} |f(x)|$1 for $\|f\| = \sup_{x \in X} |f(x)|$2.
• By a robust Daniell–Kolmogorov theorem, $\|f\| = \sup_{x \in X} |f(x)|$3 extends uniquely to a sublinear expectation $\|f\| = \sup_{x \in X} |f(x)|$4 on the space $\|f\| = \sup_{x \in X} |f(x)|$5 of all $\|f\| = \sup_{x \in X} |f(x)|$6-measurable functions on path space bounded above or below, inheriting downward and upward continuity properties from $\|f\| = \sup_{x \in X} |f(x)|$7 (Erreygers, 2023).

This path-space extension is critical for defining nonlinear laws of processes and for studying time-consistent robust (upper) expectations.

3. Generator, Hille–Yosida Theory, and Exponential Formula

The (sublinear) infinitesimal generator $\|f\| = \sup_{x \in X} |f(x)|$8 of a uniformly continuous sublinear Markov semigroup is given by: $\|f\| = \sup_{x \in X} |f(x)|$9 The semigroup admits an exponential representation: $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$0 with convergence in operator norm, provided $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$1 is bounded (Erreygers, 2024). The generator satisfies a positive maximum principle and Hille–Yosida-type resolvent bounds $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$2 for all $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$3, ensuring that every bounded sublinear generator gives rise to a unique uniformly continuous semigroup and vice versa (Erreygers, 2023, Erreygers, 2024).

In infinite dimensions or weighted continuous spaces, the generator is defined via the strong derivative at zero in the mixed topology, and the nonlinear Euler–Trotter formula reconstructs $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$4 from its generator (Goldys et al., 2022).

4. Representation of the Generator and the Nonlinear HJB Structure

On general domains (including $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$5), sublinear Markov semigroups' pointwise generators have a sup-representation: $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$6 where the supremum is over a family of Lévy triplets $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$7, which encode local uncertainty in drift, diffusion, and jump (Kühn, 2019). This structure gives rise to nonlinear and nonlocal Hamilton–Jacobi–Bellman (HJB) equations.

For finite or countable spaces, the generator reduces to a bounded sublinear rate operator. For diffusion-type semigroups with set-valued volatility/drift, the generator takes the HJB form: $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$8 (Criens et al., 2022, Criens et al., 2023).

5. Nonlinear PDE Connections and Viscosity Solutions

For $\{T_t\}_{t \ge 0} \subseteq \mathcal{O}$9, the function $T_t: L \to L$0 solves the fully nonlinear PDE: $T_t: L \to L$1 where $T_t: L \to L$2 is the generator above. The solution is typically interpreted in the viscosity sense, and $T_t: L \to L$3 provides the unique (bounded) viscosity solution of the associated nonlinear HJB equation (Goldys et al., 2022, Criens et al., 2022, Criens et al., 2023, Kühn, 2019).

Moreover, a nonlinear Dynkin formula bounds finite-difference increments of $T_t: L \to L$4 in terms of the generator: $T_t: L \to L$5 (Kühn, 2019).

6. Nonlinear Markov Families and Stochastic Representations

Each sublinear Markov semigroup induces a family of nonlinear path-space expectations associated with a family of semimartingale laws with uncertain local characteristics: $T_t: L \to L$6 where $T_t: L \to L$7 is the set of laws for which the local characteristics (drift, volatility, jumps) vary over a prescribed set (model uncertainty) (Criens et al., 2023, Criens et al., 2022). These nonlinear Markov families satisfy dynamic programming and nonlinear Markov property, and establish a one-to-one correspondence with the nonlocal HJB generator. Classic examples include G-Brownian motion, sublinear Lévy processes, robust diffusions, and Nisio semigroups (Criens et al., 2023, Criens et al., 2022).

7. Applications and Examples

• Classical Markov Chains: If each $T_t: L \to L$8 is linear, we retrieve the classical Markov semigroup $T_t: L \to L$9.
• Imprecise CTMCs: For a bounded collection $t \ge 0$0 of rate matrices, the generator is the upper envelope $t \ge 0$1.
• Imprecise Poisson Processes: With jump rates in an interval $t \ge 0$2, the sublinear generator becomes $t \ge 0$3.
• Nonlinear Jump-Diffusions and Robust Control: The value function in stochastic control problems is a nonlinear semigroup with HJB generator, describing worst-case control performance.
• Ergodicity: Sublinear Markov semigroups admit an ergodic theory in terms of invariant sublinear expectations, spectral properties, and strong laws of large numbers, paralleling the classical setting but adapted to model uncertainty (Feng et al., 2017).
• Nonlinear Feller Theory: Sublinear semigroups may exhibit smoothing (Feller) properties and have measure-kernel representations with tightness, extending classical integral kernel theory to the nonlinear case (Criens et al., 2022, Goldys et al., 2022).
• Measure-Theoretic Representations: Even in the nonlinear context, $t \ge 0$4, where $t \ge 0$5 is a convex, compact subset of probability measures on $t \ge 0$6 (Goldys et al., 2022).

8. Outlook and Research Directions

Sublinear Markov semigroups are fundamental in model uncertainty, robust control, finance (risk measures), imprecise probability theory, and dynamic programming. Extensions to non-countable state spaces, infinite-dimensional systems, processes with unbounded generators, and path-dependent functionals are active areas of investigation (Erreygers, 2023, Criens et al., 2023, Goldys et al., 2022). The theory forms the analytic backbone for nonlinear expectations, nonlinear PDEs, and robust stochastic analysis, with tight analogs to linear semigroup theory but new phenomena—especially regarding regularity, uniqueness, and the propagation of nonsmoothness under sublinear transition dynamics.