Generalized Grassmann algebras and applications to stochastic processes

TL;DR

This paper develops a new stochastic calculus framework using Z3-graded algebras, establishing a ternary Fock space and stochastic distributions for advanced stochastic process analysis.

Contribution

It introduces a generalized algebraic structure for stochastic calculus, extending white noise analysis to supersymmetric settings with Z3 grading.

Findings

Established a ternary Fock space for Z3-graded algebras

Developed a strong algebra of stochastic distributions

Applied the framework to analyze stochastic processes in supersymmetric contexts

Abstract

In this paper we present the groundwork for an It\^o/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmentry with Z3-graded algebras. To this end we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.