A note on stochastic calculus in vector bundles

TL;DR

This paper explores the relationship between covariant stochastic integration in vector bundles and the classical Stratonovich calculus, using the connector to incorporate connection dependence, thereby bridging different stochastic calculus frameworks.

Contribution

It introduces a method to relate covariant stochastic integration in vector bundles with standard Stratonovich calculus through the connector, clarifying the role of connection dependence.

Findings

Established a link between covariant stochastic calculus and Stratonovich calculus

Clarified the role of the connector in stochastic integration in vector bundles

Provided a framework for understanding connection dependence in stochastic calculus

Abstract

The aim of these notes is to relate covariant stochastic integration in a vector bundle $E$ (as in Norris \cite{Norris}) with the usual Stratonovich calculus via the connector $\K:TE \rightarrow E$ (cf. e.g. Paterson \cite{Paterson} or Poor \cite{Poor}) which carries the connection dependence.