An $L^p$-Variational Formula on Wiener Space

RESUMO

We establish a stochastic control representation for $L^p$-norms on Wiener space. For every $p\ge1$ and every non-negative universally measurable functional $\varphi$, we show that $$\|\varphi(W)\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\|a_t\|^2\,dt} \varphi\Big( W+\sqrt{p-1}\int_0^\cdot a_t\,dt \Big) \Big],$$ 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.