AI 中文总结
提出仿射和乐群量子化（AHGQ）几何框架，统一仿射定价模型的坐标算子与Riccati变换表示，揭示其共同结构起源。
AI 中文摘要
仿射定价模型的解析可处理性通常通过两种互补的表述来体现：坐标空间定价算子，以及由广义Riccati方程控制的指数仿射变换表示。我们发展了“仿射和乐群量子化”（AHGQ）作为几何框架，其中这两种表述源于同一底层结构。该构造将仿射定价符号分为齐次二次部分和互补的仿射部分。前者生成有限维辛输运和中心扩张的李群，后者由薄路径群胚承载的乘法和乐表示。它们的组合确定了仿射Poincaré-Cartan形式。其特征动力学在动量变量下约化为广义Riccati系统及其标量振幅，而坐标表示则恢复标准仿射定价算子。代表性的高斯和平方根模型说明了该构造。其贡献是结构性的：AHGQ为连续路径、时间齐次仿射定价模型的坐标表示和变换表示提供了共同的几何起源。
英文摘要
The analytic tractability of affine pricing models is usually expressed through two complementary formulations: a coordinate-space pricing operator and an exponential-affine transform representation governed by generalized Riccati equations. We develop \emph{Affine Holonomy Group Quantization} (AHGQ) as a geometric framework in which these two formulations arise from the same underlying structure. The construction separates the affine pricing symbol into a homogeneous quadratic sector and a complementary affine sector. The first generates a finite-dimensional symplectic transport and a centrally extended Lie group, while the second is represented by a multiplicative holonomy carried by a thin-path groupoid. Their combination determines an affine Poincaré--Cartan form. Its characteristic dynamics reduce in momentum variables to the generalized Riccati system and its scalar amplitude, whereas the coordinate representation recovers the standard affine pricing operator. Representative Gaussian and square-root models illustrate the construction. The contribution is structural: AHGQ gives a common geometric origin to the coordinate and transform representations of continuous-path, time-homogeneous affine pricing models.
