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Representation Theorems for Convex Expectations and Semigroups on Path Space

David Criens, Michael Kupper

TL;DR

This paper develops a unified, path-space framework that identifies a one-to-one correspondence among convex expectations, penalty representations, and convex semigroups on two-sided path spaces. By proving a dynamic representation $\mathcal{E}_t(\varphi)(\omega)=\sup_{P}(E^P[\varphi]-\alpha(t,\omega,P))$ and establishing equivalences with homogeneous semigroups, it shows convex expectations are determined by finite-dimensional distributions and can be realized via relaxed control rules. The work further derives a Markovian comparison principle, connects to nonlinear Lévy processes, and establishes a Laplace principle for entropic risk measures under small noise, linking stochastic control, PDE/viscosity methods, and uncertainty quantification. Collectively, these results bridge control, analysis, and probability in a path-dependent, non-dominated setting, with implications for nonlinear Lévy dynamics, risk measures, and small-noise asymptotics.

Abstract

The objective of this paper is to investigate the connection between penalty functions from stochastic optimal control, convex semigroups from analysis and convex expectations from probability theory. Our main result provides a one-to-one relation between these objects. As an application, we use the representation via penality functions and duality arguments to show that convex expectations are determined by their finite dimensional distributions. To illustrate this structural result, we show that Hu and Peng's axiomatic description of $G$-Lévy processes in terms of finite dimensional distributions extends uniquely to the control approach introduced by Neufeld and Nutz. Finally, we show that convex expectations with a Markovian structure are fully determined by their one-dimensional distributions, which give rise to a classical semigroup on the state space. As an application of this result, we establish a Laplace principle for entropic risk measures associated to controlled diffusions.

Representation Theorems for Convex Expectations and Semigroups on Path Space

TL;DR

This paper develops a unified, path-space framework that identifies a one-to-one correspondence among convex expectations, penalty representations, and convex semigroups on two-sided path spaces. By proving a dynamic representation and establishing equivalences with homogeneous semigroups, it shows convex expectations are determined by finite-dimensional distributions and can be realized via relaxed control rules. The work further derives a Markovian comparison principle, connects to nonlinear Lévy processes, and establishes a Laplace principle for entropic risk measures under small noise, linking stochastic control, PDE/viscosity methods, and uncertainty quantification. Collectively, these results bridge control, analysis, and probability in a path-dependent, non-dominated setting, with implications for nonlinear Lévy dynamics, risk measures, and small-noise asymptotics.

Abstract

The objective of this paper is to investigate the connection between penalty functions from stochastic optimal control, convex semigroups from analysis and convex expectations from probability theory. Our main result provides a one-to-one relation between these objects. As an application, we use the representation via penality functions and duality arguments to show that convex expectations are determined by their finite dimensional distributions. To illustrate this structural result, we show that Hu and Peng's axiomatic description of -Lévy processes in terms of finite dimensional distributions extends uniquely to the control approach introduced by Neufeld and Nutz. Finally, we show that convex expectations with a Markovian structure are fully determined by their one-dimensional distributions, which give rise to a classical semigroup on the state space. As an application of this result, we establish a Laplace principle for entropic risk measures associated to controlled diffusions.

Paper Structure

This paper contains 14 sections, 22 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Convex expectations, representation penalty functions and semigroups
  3. The path space
  4. Convex expectations and representation penalty functions
  5. Uniqueness via finite-dimensional marginal distributions
  6. Relation between convex expectations and convex semigroups
  7. Convex expectations arising from relaxed control rules
  8. The sublinear case
  9. Nonlinear Lévy processes
  10. A comparison result for convex expectations with Markovian structure
  11. A Laplace principle for controlled diffusion processes
  12. Connection to Related Concepts
  13. On the tower property for convex expectations
  14. On the relation of the properties \ref{['PF7']} and \ref{['M3']}

Key Result

Lemma 2.3

Fix $t \in \mathbb{R}$ and a function $\psi \colon \Omega \to \mathbb{R}$. Then, $\psi \in \textit{USA} (\Omega, \mathcal{F}_t; \mathbb{R})$ if and only if $\psi \in \textit{USA} (\Omega; \mathbb{R})$ and $\psi = \psi (X_{\cdot \wedge t})$.

Theorems & Definitions (63)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 53 more