An alternative approach to Privaultʹs discrete-time chaotic calculus

In this paper, we present an alternative approach to Privaultʹs discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l 2 ( Γ ) the space of square summable functions defined on Γ (the finite power set of N ). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l 2 ( Γ ) . And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l 2 ( Γ ) . We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability.