A variational representation for random functionals on abstract Wiener spaces
Zhang, Xicheng
J. Math. Kyoto Univ., Tome 49 (2009) no. 1, p. 475-490 / Harvested from Project Euclid
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.
Publié le : 2009-05-15
Classification:  60H07,  60F10
@article{1260975036,
     author = {Zhang, Xicheng},
     title = {A variational representation for random functionals on abstract Wiener spaces},
     journal = {J. Math. Kyoto Univ.},
     volume = {49},
     number = {1},
     year = {2009},
     pages = { 475-490},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1260975036}
}
Zhang, Xicheng. A variational representation for random functionals on abstract Wiener spaces. J. Math. Kyoto Univ., Tome 49 (2009) no. 1, pp.  475-490. http://gdmltest.u-ga.fr/item/1260975036/