AI 中文总结
该研究以粗糙Heston模型为示例，证明局部随机粗糙波动率下条件密度SPDE的Itô-Wentzell约化仍成立，得到路径式Fokker-Planck公式，纯粗糙Heston情形下条件密度为显式对数正态形式，关联Rao-Blackwell化校准。
AI 中文摘要
本注记以粗糙Heston（rHeston）为主要明确示例，研究局部随机粗糙波动率模型中的条件密度方程及其路径变换。在给定的共同滤过、可测性、可预测性和空间正则性假设下，证明了条件密度SPDE的Itô-Wentzell随机PDE约化在局部随机粗糙波动率下仍成立。在固定共同环境实现及相关随机流后，变换后的方程成为具有路径依赖系数的确定性PDE，由此得到的路径式Fokker-Planck公式与Rao-Blackwell化校准自然关联。在纯粗糙Heston情形下，变换后的系数简化，且条件密度具有显式对数正态形式。
英文摘要
This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker--Planck formulation that connects naturally with Rao--Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form.
