提出新型非线性期望θ-期望，突破传统凸性限制。
A Unified Theory of $θ$-Expectations
- 从确定性混沌动力学推导出新期望类
- 证明其满足非凸的哈密顿-雅可比-贝尔曼方程
- 适用于非凸随机控制问题，适合研究复杂系统
我们从第一性原理的确定性混沌动力学出发，推导出一类新的非线性期望。通过均匀双曲流的转移算子谱理论，实现系统斜自伴微观生成元的均质化。证明在粘性意义下收敛到由完全非线性哈密顿-雅可比-贝尔曼（HJB）方程支配的宏观演化。核心结果表明，该HJB哈密顿量在海森矩阵上为仿射结构，但在梯度上明显非凸。这定义了一种新的θ-期望，并构造性地建立了一类根本上超出G-期望次可加框架的非凸随机控制问题。
原文摘要 · Abstract (English)
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.
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