Algebraic Stochastic Calculus

TL;DR

This paper introduces Algebraic Stochastic Calculus, a categorical framework that reinterprets probability theory and stochastic differential equations using sheaf theory and deformation problems, aiming to replace traditional stochastic calculus.

Contribution

It develops a categorical, sheaf-theoretic foundation for stochastic calculus, replacing differential equations with deformation theory in a purely formal setting.

Findings

Reinterprets probability spaces as Grothendieck sites

Defines Brownian motions via sheaves in symmetric monoidal infinity-categories

Proposes a time-independent, categorical formalism for stochastic processes

Abstract

We develop the foundations of Algebraic Stochastic Calculus, with an aim to replacing what is typically referred to as Stochastic Calculus by a purely categorical version thereof. We first give a sheaf theoretic reinterpretation of Probability Theory. We regard probability spaces (X, F, P) as Grothendieck sites (F, J_P) on which Brownian motions are defined via sheaves in symmetric monoidal infinity-categories. Due to the complex nature of such a formalism we are naturally led to considering a purely categorical, time independent formalism in which stochastic differential equations are replaced by studying problems in deformation theory.