发表机构
University of Freiburg; Brandenburg University of Technology Cottbus–Senftenberg; University of Konstanz（弗赖堡大学; 勃兰登堡工业大学科特布斯-森夫滕贝格分校; 康斯坦茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Wiener空间上建立了$L^p$-范数的随机控制表示，作为Boué–Dupuis公式的乘法类比，并推广至路径依赖随机微分方程，用于推导小噪声极限的定量估计与精确指数衰减率。
AI 中文摘要
我们在Wiener空间上建立了$L^p$-范数的随机控制表示。对于每个$p\ge1$和每个非负的泛函$\varphi$（关于泛函的可测性要求为泛函可测），我们证明$$\\|\varphi(W)\\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\\|a_t\\|^2\\,dt} \varphi\Big( W+\sqrt{p-1}\int_0^\cdot a_t\\,dt \Big) \Big],$$其中$W$是$d$维布朗运动，上确界取遍所有满足$\int_0^T\\|a_t\\|^2\\,dt<\infty$（几乎必然）的循序可测控制$a$。该恒等式可视为Boué–Dupuis变分公式的乘法$L^p$-类比。我们还将其推广到具有路径依赖系数的随机微分方程的强解情形。基于该变分公式，我们讨论了具有路径依赖系数的随机微分方程小噪声极限的有限$p$估计，包括定量的Freidlin-Wentzell界、泛函高斯近似和集中不等式，以及中等偏差界。此外，我们讨论了由停时确定的泛函的相关推论，特别得到了在优化控制后识别出精确指数衰减率的概率界。该变分公式的证明结合了概率论论证与凸对偶方法、Hamilton-Jacobi-Bellman方程的粘性理论以及动态规划原理。
英文摘要
We establish a stochastic control representation for $L^p$-norms on Wiener space. For every $p\ge1$ and every non-negative universally measurable functional $φ$, we show that $$\|φ(W)\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\|a_t\|^2\,dt} φ\Big( W+\sqrt{p-1}\int_0^\cdot a_t\,dt \Big) \Big],$$ where $W$ is a $d$-dimensional Brownian motion and the supremum is taken over all progressively measurable controls $a$ satisfying $\int_0^T\|a_t\|^2\,dt<\infty$ almost surely. This identity can be viewed as a multiplicative $L^p$-analogue of the Boué--Dupuis variational formula. We also provide an extension to strong solutions of stochastic differential equations with path-dependent coefficients. Building on the variational formula, we discuss finite-$p$ estimates for small-noise limits of stochastic differential equations with path-dependent coefficients. These include quantitative Freidlin-Wentzell bounds, a functional Gaussian approximation and concentration estimates, and moderate-deviation bounds. Moreover, we discuss consequences for functionals determined up to a stopping time, obtaining in particular probability bounds that identify the sharp exponential decay rate after optimization over controls. The proof of the variational formula combines probabilistic arguments with convex duality methods, viscosity theory for Hamilton-Jacobi-Bellman equations, and a dynamic programming principle.
