Algebraic Stochastic Calculus

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.