Internal mathematics for stochastic calculus: a tripos-theoretic approach

We approach the problem of understanding the logical aspects of stochastic calculus through topos theoretic methods. In particular, we construct a tripos which encodes a higher-order logic tailor-made for a specific probability space, which we call Scott tripos. Some internal features and constructions of the associated topos are discussed. Furthermore, we study a family of adjoint modal operators arising from a filtration on a probability space. We explore whether these are related to modal operators in process logics (CTL*, PDL) and we give a negative answer.