# Local characteristics and tangency of vector-valued martingales

## TL;DR

This paper investigates tangent martingales in Banach spaces, introducing local characteristics, estimates, and constructions that extend previous results to infinite dimensions under the UMD condition.

## Contribution

It provides a comprehensive framework for tangent martingales in Banach spaces, including new estimates and constructions that generalize prior real-valued and vector-valued results.

## Key findings

- Established new $L^p$ and $\
- $\\phi$-estimates for tangent martingales.
- Constructed decoupled tangent martingales in infinite-dimensional Banach spaces.

## Abstract

This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $\phi$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapie\'{n}, McConnell, and Woyczy\'{n}ski. The vast majority of the assertions presented in the paper is done under the sufficient and necessary UMD assumption on the corresponding Banach space.