44 research outputs found
Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates
No full textThis 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
Distribution-valued iterated gradient and chaotic decompositions of Poisson jump times functionals
No full textWe define a class of distributions on Poisson space which allows to iterate a modification of the gradient of [1]. As an application we obtain, with relatively short calculations, a formula for the chaos expansion of functionals of jump times of the Poisson process
A Transfer Principle from Wiener to Poisson Space and Applications
No full textAbstractThe aim of this work is to construct the stochastic calculus of variations on Poisson space and some of its applications via the stochastic analysis on Wiener space. We define a new gradient operator on Wiener space, whose adjoint extends the Poisson stochastic integral. This yields a new decomposition of the Ornstein-Uhlenbeck operator and a substructure of the standard Dirichlet structure on Wiener space, with applications to stochastic analysis on Poisson space and infinite-dimensional analysis for the exponential density
Backward Stochastic Difference Equations with Finite States
No full textProgress in Probability; vol. 65We define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. Solutions exist and are unique under weaker assumptions than are needed in the continuous time setting. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are explored, including a representation result.Samuel N. Cohen and Robert J. Elliot
Chaotic Kabanov formula for the Azéma martingales
No full textWe derive the chaotic expansion of the product of n-th and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes. Key words: Az'ema martingales, multiple stochastic integrals, product formulas. Mathematics Subject Classification (1991): 60G44, 60H05, 81S25. 1 Introduction The Wiener-Ito and Poisson-Ito chaotic decompositions give an isometric isomorphism between the Fock space \Gamma(L 2 (IR + )) and the space of square-integrable functionals of the process. This somorphism is constructed by association of a symmetric function f n 2 L 2 (IR + ) ffin to its multiple stochastic integral. The isometry property comes ..
Independence Of A Class Of Multiple Stochastic Integrals
No full textWe show that two multiple stochastic integrals In (fn ), Im (gm ) with respect to the solution (M t ) t2IR + of a deterministic structure equation are independent if and only if two contractions of fn and gm , denoted as fn ffi 0 1 gm , fn ffi 1 1 gm , vanish almost everywhere. 1 Introduction This paper aims to extend the necessary and sufficient conditions for the independence of single or multiple stochastic integrals of [12], [14], [15], [16], [17], cf. also [6], [7], proving and extending results that have been partially announced in [9]. Let (M t ) t2IR + be a martingale satisfying the structure equation d[M; M ] t = dt + OE t dM t ; (1) where OE : IR + ! IR is a measurable deterministic function. Such martingales are normal in the sense of [2], i.e. d ! M;M ? t = dt, t 2 IR + and they satisfy the chaos representation property, cf. [3]. Moreover, they have independent increments, and if (B t ) t2IR + , (N t ) t2IR + are independent standard Brownian motion and Poisson proce..
