STOCHASTIC DIFFERENTIAL-CALCULUS, THE MOYAL ASTERISK-PRODUCT, AND NONCOMMUTATIVE GEOMETRY

A reformulation of the Ito calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d2 = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Ito calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.