2013/07/11 by Ghaliah Alhamzi, Alhamzi, Ghaliah, Edwin Beggs +3
Mathematics · #46L87 #81S25 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Probability (math.PR) #Quantum Algebra (math.QA) #math.PR #math.QA #msc:46L87 #msc:81S25
paper · pdf · doi:10.48550/arxiv.1307.3119
This is an attempt to relate stochastic calculus to deformations of differential graded algebras. Any comments very welcome, as there are quite likely many improvements that could be made, or more references that should be included
arxiv created 2013/07/11 · openalex publication_date 2013/07/11 · arxiv updated 2013/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We begin with a deformation of a differential graded algebra by adding time and using a homotopy. It is shown that the standard formulae of Itô calculus are an example, with four caveats: First, it says nothing about probability. Second, it assumes smooth functions. Third, it deforms all orders of forms, not just first order. Fourth, it also deforms the product of the DGA. An isomorphism between the deformed and original DGAs may be interpreted as the transformation rule between the Stratonovich and classical calculus (again no probability). The isomorphism can be used to construct covariant derivatives with the deformed calculus. We apply the deformation in noncommutative geometry, to the Podleś sphere S2q. This involves the Hodge theory of S2q.