A variational representation for random functionals on abstract Wiener spaces

Abstract

We extend to abstract Wiener spaces the variational representation \[ \mathbb{E}[e^{F}] = \exp \left( \sup_{v\in\mathcal{H}^a}\mathbf{E} \left[ F(\cdot +v) - \frac{1}{2}\|v\|^2_{\mathbb{H}} \right] \right), \] proved by Boué and Dupuis [1] on the classical Wiener space. Here $F$ is any bounded measurable function on the abstract Wiener space $(\mathbb{W},\mathbb{H},\mu)$, and $\mathcal{H}^a$ denotes the space of $\mathcal{F}_t$-adapted $\mathbb{H}$-valued random fields in the sense of Üstünel and Zakai [11]. In particular, we simplify the proof of the lower bound given in [1, 3] by using the Clark-Ocone formula. As an application, a uniformly Laplace principle is established.