# A Unified Theory of θ-Expectations

arXiv: https://arxiv.org/abs/2507.20353 · HTML: https://mlsanity.com/paper/2507.20353/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2025.

## Abstract (arXiv)

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.