Matteo Capucci, University of Strathclyde
My name is stochastic calculus but everybody calls me calculus
Standard treatments of probability theory and stochastic calculus present the subject as talking about random versions of ordinary mathematical objects, and reason and manipulate them as if they were actual numbers, sets, and functions. We take this seriously and, motivated by similar endeavours in algebraic geometry (such as Blechschmidt's PhD thesis), try to explore whether probability theory and stochastic calculus are in fact elementary theories interpreted in a ‘stochastic world’, which is given by a Grothendieck topos of sheaves over the Kolmogorov probability space. We give an account of some elementary notions of stochastic calculus in this setting, showing the mathematical praxis can indeed be formally justified, and sketching how advanced notions of stochastic calculus might find a home in these renewed foundations.