{"ID":2889971,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.20353","arxiv_id":"2507.20353","title":"A Unified Theory of $θ$-Expectations","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.","short_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 macros...","url_abs":"https://arxiv.org/abs/2507.20353","url_pdf":"https://arxiv.org/pdf/2507.20353v2","authors":"[\"Qian Qi\"]","published":"2025-07-27T16:56:01Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"cs.AI\",\"cs.LG\",\"stat.ML\"]","methods":"[\"Large Language Model\"]","has_code":false}