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We derive conditional Gaussian type identities of the form E [ exp ⁡ ( i ∫ 0 T u t d B t ) | ∫ 0 T | u t | 2 d t ] = exp ⁡ ( − 1 2 ∫ 0 T | u t | 2 d t ) , for Brownian stochastic integrals, under conditions on the process ( u t ) t ∈ [ 0 , T ] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral ∫ 0 t A B s d B s under the matrix condition A † A 2 = 0 , using a characterization of Yor [6] . [ABSTRACT FROM AUTHOR]

Based on a new multiplication formula for discrete multiple stochastic integrals with respect to non-symmetric Bernoulli random walks, we extend the results of Nourdin et al. (2010) on the Gaussian approximation of symmetric Rademacher sequences to the setting of possibly non-identically distributed independent Bernoulli sequences. We also provide Poisson approximation results for these sequences, by following the method of Peccati (2011). Our arguments use covariance identities obtained from the Clark-Ocone representation formula in addition to those usually based on the inverse of the Ornstein-Uhlenbeck operator. [ABSTRACT FROM AUTHOR]