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.
"Stochastic analysis of Bernoulli processes." Probab. Surveys 5 435 - 483, 2008. https://doi.org/10.1214/08-PS139