Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity λ>0. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of O(λ ) for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.Ministry of Education (MOE)Accepted versionThe first author-Nicolas Privault acknowledges the financial support from the Singapore MOE Tier 2 Grant MOE2016-T2-1-036. The first author also expresses his gratitude to the hospitality of CUHK when he first discussed with the other two authors on the possibility of working out the present novel topic. The second author— Phillip Yam acknowledges the financial supports from HKGRF-14300717 with the project title: New kinds of Forward–backward Stochastic Systems with Applications, HKSAR-GRF-14301015 with title: Advance in Mean Field Theory, Direct Grant for Research 2014/15 with project code: 4053141 offered by CUHK, and the International Partnerships Development Programme 2013–14, OAL, CUHK, Hong Kong with which Nicolas and Phillip can sit together to work effectively out the present article. The last author–Zheng Zhang acknowledges the financial support from Renmin University of China with the project code 297517501221 together with the project title “Applications of Nonparametric Method in Missing Data”, and the fund for building world-class universities (disciplines) of Renmin University of China; for the purpose of his official grant acknowledgment, the last author, with the consensus of the two other authors, likes to formally declare that the present work is completed by even contribution of each of us with our authorships listed in alphabetical order of our surnames