Martingales in Reductive Homogeneous spaces

TL;DR

This paper investigates martingales within reductive homogeneous spaces using principal fiber bundle perspectives, stochastic logarithms, and specific examples like spheres and special linear groups.

Contribution

It introduces a novel approach to study martingales on homogeneous spaces via horizontal martingales and stochastic logarithms, extending understanding in geometric stochastic analysis.

Findings

Characterization of martingales using stochastic logarithm.

Analysis of martingales on spheres and certain Lie group quotients.

New methods linking principal fiber bundles and stochastic processes.

Abstract

The subject of this work is to study martingales in a reductive homogeneous space with respect to a symmetric connection. Our basic idea is to view homogenous spaces as principal fiber bundles and, thus, to study martingales on homogeneous space with aid of horizontal martingales on Lie group. Furthermore, using the stochastic logarithm we give a characterization of martingales on homogenous space. To end, we study the martingales in spheres $S^{n}$ and $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, $n \geq 2$.