Stochastic integration in Riemannian manifolds from a functional-analytic point of view

TL;DR

This paper develops a functional-analytic framework for stochastic integration on Riemannian manifolds, revealing infinitely many integrals and unifying Stratonovich and Itô integrals as special cases.

Contribution

It introduces a new functional-analytic approach to stochastic integration on manifolds, showing the existence of infinitely many integrals and their interrelations.

Findings

Existence of infinitely many stochastic integrals on Riemannian manifolds.

Stratonovich and Itô integrals are specific instances of the general framework.

A simple formula relates any two stochastic integrals in this setting.

Abstract

This article presents a construction of the concept of stochastic integration in Riemannian manifolds from a purely functional-analytic point of view. We show that there are infinitely many such integrals, and that any two of them are related by a simple formula. We also find that the Stratonovich and It\^o integrals known to probability theorists are two instances of the general concept constructed herein.