Stochastic analysis of Bernoulli processes

Abstract

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.

Originally published in at http://dx.doi.org/10.1214/08-PS139 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org) This version includes a revision of Section 9

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• The uniform measure for quantum walk on hypercube: a quantum Bernoulli noises approach
• A Peccati-Tudor type theorem for Rademacher chaoses
• A simplified second-order Gaussian Poincaré inequality in discrete setting with applications
• Stochastic analysis for obtuse random walks
• Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals
• Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises
• Spectral Integrals of Bernoulli Generalized Functionals