Malliavin calculus for backward stochastic differential equations and application to numerical solutions

In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the $L^p$-H\"{o}lder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained $L^p$-H\"{o}lder continuity results. The main tool is the Malliavin calculus.Comment: Published in at http://dx.doi.org/10.1214/11-AAP762 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Geometry-Aware Face Completion and Editing

Face completion is a challenging generation task because it requires generating visually pleasing new pixels that are semantically consistent with the unmasked face region. This paper proposes a geometry-aware Face Completion and Editing NETwork (FCENet) by systematically studying facial geometry from the unmasked region. Firstly, a facial geometry estimator is learned to estimate facial landmark heatmaps and parsing maps from the unmasked face image. Then, an encoder-decoder structure generator serves to complete a face image and disentangle its mask areas conditioned on both the masked face image and the estimated facial geometry images. Besides, since low-rank property exists in manually labeled masks, a low-rank regularization term is imposed on the disentangled masks, enforcing our completion network to manage occlusion area with various shape and size. Furthermore, our network can generate diverse results from the same masked input by modifying estimated facial geometry, which provides a flexible mean to edit the completed face appearance. Extensive experimental results qualitatively and quantitatively demonstrate that our network is able to generate visually pleasing face completion results and edit face attributes as well

Feynman-Kac formula for heat equation driven by fractional white noise

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the H\"{o}lder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.Comment: Published in at http://dx.doi.org/10.1214/10-AOP547 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org