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, us...

We show that the general stable convergence results proved in Peccati and Taqqu (2007) for generalized adapted stochastic integrals can be used to obtain limit theorems for multiple stochastic integrals with respect to independently scattered random measures. Several applications are developed in a companion paper (see Peccati and Taqqu, 2008a), where we prove central limit results involving si...

Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O’Brien [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 169–181] for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit m-sphere are used to produce stochastic rules for these integrals. The two types of rules are combine...

We show how a technique from signal processing known as zero-delay convolution can be used to develop more efficient dynamic programming algorithms for a broad class of stochastic optimization problems. This class includes several variants of discrete stochastic shortest path, scheduling, and knapsack problems, all of which involve making a series of decisions over time that have stochastic con...

Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C∗-hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C∗-bialgebra. Stochastic (or Markovian) cocycles on operator algebras are basic objects of interest in qu...

We study stochastic integrals driven by a general subordinator and establish zero-one law for the finiteness of resulting integral as well moment estimates. As an application, we use these results to obtain structural properties SPDEs multiplicative pure jump noise, which include (1) maximal inequality convolution <mml:math xmlns:mml="http://www.w3.or...

In [10] a “direct” stochastic transfer principle was introduced, which represented multiple integrals with respect to fractional Brownian motion in terms of multiple integrals with respect to standard Brownian motion. The method employed in [10] involved an operator Γ (n) H , mapping a class of functions LH to L 2. However, the operator does not map LH onto L 2. Hence Γ (n) H is not invertible....

Stochastic diierential equations in R n with random coeecients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W 2 for some > 1=2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution proced...

In this work, we obtain quantitative convergence of moderately interacting particle systems towards solutions nonlinear Fokker-Planck equations with singular kernels. addition, prove the well-posedness for McKean-Vlasov SDEs involving these kernels and trajectorial propagation chaos associated systems. Our results only require very weak regularity on interaction kernel, including Biot-Savart at...