Itô Formula for Free Stochastic Integrals

Abstract

The objects under investigation are the stochastic integrals with respect to free Lévy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itô formula, we prove the full functional Itô formula in this context. © 2002 Elsevier Science (USA)

An Introduction to Random Matrices

“…In particular, we have not mentioned free Brownian motion as defined in [Spe90], which appears as the limit of the Hermitian Brownian motion with size going to infinity [Bia97a]. We refer to [BiS98b] for a study of the related stochastic calculus, to [Bia98a] for the introduction of a wide class of processes with free increments and for the study of their Markov properties, to [Ans02] for the introduction of stochastic integrals with respect to processes with free increments, and to [BaNT02] for a thorough discussion of Lévy processes and Lévy laws. Such a stochastic calculus was used to prove central limit theorem in [Cab01], large deviations results, see the survey [Gui04], or convergence of the empirical distribution of interacting matrices [GuS08].…”

Lévy Processes on Quantum Groups and Dual Groups

“…Additive free Lévy processes where first studied in [GSS92], and more recently in [Bia98,Ans01a,Ans01b,BNT01a,BNT01b].…”

Free convolution operators and free Hall transform

“…Let X be an algebra endowed with a center-valued expectation tr, and with I specified elements (X i ) i∈I . The triplet X , tr, (X i ) i∈I possesses the universal property 1.1 for index set I if for all algebras A endowed with a centervalued expectation τ , and with I elements (A i ) i∈I , there exists a unique algebra homomorphism f from X to A such that (1) for all i ∈ I, we have f (X i ) = A i ; (2) for all X ∈ X , we have τ (f (X)) = f (tr(X)).…”

“…, s(h 2 ) = κ (s(h 1 ), s((h 2 , * ))) = κ (s((h 1 , * )), s((h2 , * ))) = κ(s(h 1 ), s(h 2 )) = h 1 , h 2 H , κ s(h 1 ),s(h 2 ) = κ c(h 1 ) + s + (h 1 ), c(h 2 ) + s + (h 2 ) = κ s + (h 1 ), s + (h 2 ) = h 1 , h 2 H , κ s(h 1 ),s((h 2 , * )) = κ c(h 1 ) + s + (h 1 ), c(h 2 ) * + s − (h 2 ) = κ c(h 1 ), c(h 2 ) * = h 1 , h 2 H ,and κ s((h 1 , * )),s((h 2 , * )) = κ c(h 1 ) * + s − (h 1 ), c(h 2 ) * + s − (h 2 ) = κ s − (h 1 ), s − (h 2 ) = h 1 , h 2 H . At the end, for all h 1 , h 2 ∈ H ∪ (H × { * }), we have κ s(h 1 ), s(h 2 ) = κ s(h 1 ),s(h 2 ) .…”