Itô Formula for Free Stochastic Integrals

The subject of this paper is stochastic integration in the context of free probability. Noncommutative stochastic processes with freely independent increments, especially the free Brownian motion, have been investigated in a number of sources, see [BS98], [Ans00] and their references. In the latter paper, we started the analysis of such processes, which we also call free stochastic measures, using the combinatorial machinery inspired by the work of Rota and Wallstrom [RW97]. Starting with a free stochastic measure X(t), in that paper we defined a family of multi-dimensional free stochastic measures derived from it, indexed by set partitions. In particular, we defined the family of higher diagonal stochastic measures ∆k(t), which give a precise meaning to the heuristic expression d∆k = (dX) . In this paper we define integrals with respect to free stochastic measures, and investigate their properties. We restrict the analysis to free stochastic measures consisting of bounded operators. Note that this has no analog in the classical stochastic integration theory: there are no (non-trivial) compactly supported infinitely divisible distributions, while the class of compactly supported freely infinitely divisible distributions is dense, and includes the free analogs of the normal and the Poisson distribution. After completing this paper, we learned from Steen Thornbjørnsen about a recent preprint [BNTr00]. In it, using a remarkable bijection between the free and classical infinitely divisible distributions, the authors define stochastic integrals with respect to any stationary stochastic process with free increments, with the Riemann sums defining the integrals converging in probability. On the other hand, since we are dealing with bounded operators, we can achieve convergence in the operator norm, both for the integrals and the limits defining the higher diagonal measures. This requires a definition of a family of “mixed-p” norms on the integrands, with the integrands which are bounded in the ∞-norm giving stochastic integrals which are bounded in the operator norm. Boundedness of the integrators also allows us to define the integrals for a significantly wider (not necessarily scalar, not necessarily continuous) class of integrands. Most importantly, we prove the functional Itô formula for such integrals, which involves the integration with respect to the original free stochastic measure as well as its higher diagonal measures. The importance of free Itô formulas is indicated by recent applications of the Itô formula for the free Brownian motion to fine properties of random Gaussian matrices in