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A Unified Theory of $θ$-Expectations

Qian Qi

TL;DR

This work derives a new class of non-linear θ-expectations from deterministic chaotic dynamics by a rigorous homogenization to a fully non-linear Hamilton-Jacobi-Bellman equation. The authors reveal a rigid θ-Hamiltonian that is affine in the Hessian yet possibly non-convex in the gradient, establishing a novel non-convex stochastic-control framework outside Peng’s G-expectations. They develop a two-part architecture: a deterministic construction based on a uniformly hyperbolic billiard-type micro-dynamics with Nash–Moser regularity to obtain a smooth invariant splitting and spectral regularity, followed by a probabilistic representation via a non-linear martingale problem, nonlinear Feynman-Kac formulas, and a θ-BSDE. The Scale I–IV homogenization analysis provides a detailed macroscopic law with a Green-Kubo diffusivity and a gradient-driven potential, while Scale IV uncovers fluctuations described by a viscous Burgers-type equation, leading to a complete probabilistic hierarchy and universal behavior. The framework culminates in a robust bridge from deterministic chaos to non-convex stochastic control, along with a nonlinear, pathwise stochastic calculus and a nonlinear Feynman-Kac representation for the θ-expectation, with clear conditions for emergent non-convexity.

Abstract

We derive a new class of non-linear expectations from first-principles deterministic chaotic dynamics. The homogenization of the system's skew-adjoint microscopic generator is achieved using the spectral theory of transfer operators for uniformly hyperbolic flows. We prove convergence in the viscosity sense to a macroscopic evolution governed by a fully non-linear Hamilton-Jacobi-Bellman (HJB) equation. Our central result establishes that the HJB Hamiltonian possesses a rigid structure: affine in the Hessian but demonstrably non-convex in the gradient. This defines a new $θ$-expectation and constructively establishes a class of non-convex stochastic control problems fundamentally outside the sub-additive framework of G-expectations.

A Unified Theory of $θ$-Expectations

TL;DR

This work derives a new class of non-linear θ-expectations from deterministic chaotic dynamics by a rigorous homogenization to a fully non-linear Hamilton-Jacobi-Bellman equation. The authors reveal a rigid θ-Hamiltonian that is affine in the Hessian yet possibly non-convex in the gradient, establishing a novel non-convex stochastic-control framework outside Peng’s G-expectations. They develop a two-part architecture: a deterministic construction based on a uniformly hyperbolic billiard-type micro-dynamics with Nash–Moser regularity to obtain a smooth invariant splitting and spectral regularity, followed by a probabilistic representation via a non-linear martingale problem, nonlinear Feynman-Kac formulas, and a θ-BSDE. The Scale I–IV homogenization analysis provides a detailed macroscopic law with a Green-Kubo diffusivity and a gradient-driven potential, while Scale IV uncovers fluctuations described by a viscous Burgers-type equation, leading to a complete probabilistic hierarchy and universal behavior. The framework culminates in a robust bridge from deterministic chaos to non-convex stochastic control, along with a nonlinear, pathwise stochastic calculus and a nonlinear Feynman-Kac representation for the θ-expectation, with clear conditions for emergent non-convexity.

Abstract

We derive a new class of non-linear expectations from first-principles deterministic chaotic dynamics. The homogenization of the system's skew-adjoint microscopic generator is achieved using the spectral theory of transfer operators for uniformly hyperbolic flows. We prove convergence in the viscosity sense to a macroscopic evolution governed by a fully non-linear Hamilton-Jacobi-Bellman (HJB) equation. Our central result establishes that the HJB Hamiltonian possesses a rigid structure: affine in the Hessian but demonstrably non-convex in the gradient. This defines a new -expectation and constructively establishes a class of non-convex stochastic control problems fundamentally outside the sub-additive framework of G-expectations.

Paper Structure

This paper contains 106 sections, 78 theorems, 131 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Logical Architecture of the Theory
  3. \ref{['part:i']}: The Deterministic Construction
  4. \ref{['part:ii']}: The Probabilistic Representation
  5. The Nature of the theta-Expectation Framework
  6. Relationship to Non-Commutative Probability Theory.
  7. Relationship to the Theory of Regularity Structures.
  8. The Deterministic Construction
  9. Preliminaries: The Microscopic Framework
  10. The Geometric State Space
  11. The Deterministic Law of Motion
  12. The Interior Flow
  13. The Boundary Transformation
  14. Geometric Foundations for Well-Posed Dynamics
  15. Immediate Consequences of the Axioms
  16. ...and 91 more sections

Key Result

Theorem 1.1

The $\theta$-Hamiltonian derived from the homogenization of the deterministic system has the form: where $p = \nabla u$ and $X = \nabla^2 u$. The constituent terms have the following properties:

Theorems & Definitions (218)

  • Theorem 1.1: Structure of the $\theta$-Hamiltonian
  • Definition 2.1: The Macroscopic Base Manifold
  • Definition 2.2: The Microscopic Fiber Space
  • Definition 2.3: The Obstacle Defining Function
  • Definition 2.4: The Total and Physical State Spaces
  • Definition 2.5: The Flow Vector Field
  • Definition 2.6: Boundary Normal and Phase Space Partition
  • Definition 2.7: The Specular Reflection Map
  • Remark 2.9: The Foundational Role of the Geometric Axioms
  • Proposition 2.10: Smoothness of the Collision Boundary
  • ...and 208 more