A Unified Theory of $\theta$-Expectations

TL;DR

This paper introduces a new class of non-linear expectations derived from chaotic dynamics, leading to a non-convex HJB equation that broadens the scope of stochastic control theories beyond traditional G-expectations.

Contribution

It develops a novel $ heta$-expectation framework from chaotic systems, revealing a non-convex HJB Hamiltonian with a unique structure not covered by existing sub-additive models.

Findings

Derived non-linear expectations from deterministic chaotic dynamics.

Proved convergence to a non-linear HJB equation in the viscosity sense.

Identified a non-convex Hamiltonian structure in the $ heta$-expectation framework.

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 $\theta$-expectation and constructively establishes a class of non-convex stochastic control problems fundamentally outside the sub-additive framework of G-expectations.