Hsueh, Kun-Hung2017-08-282017-08-282017-08-282017-08-25http://hdl.handle.net/10012/12243The purpose of this thesis is to elaborate the similarities between the classical and the free probability by means of developing the chaos decomposition of stochastic integrals driven by Brownian motion and its free counterparts in a parallel manner. The work focuses on constructing an apparatus that is general enough so that these similarities are apparent, yet not too general that their distinctions are completely obscured. In particular, we employ the notion of lattice paths to bring about the moment calculation of normally distributed random variables in the non-commutative probability environment; and we exploit the structure of the lattice of partition of n-elements, which underlies the relationships between stochastic integrations defined in the Itô and in the Wiener sense, to prove both the classical and the free chaos decomposition result.enNon-commutative probabilityProbabilityCombinatoricsMaster Thesis