Stochastic analysis for vector-valued generalized grey Brownian motion

TL;DR

This paper introduces a new vector-valued generalized grey Brownian motion (vggBm), analyzes its properties, and demonstrates its applications in stochastic differential equations, extending the understanding of complex stochastic processes.

Contribution

The paper develops a vector-valued extension of ggBm with independent components, characterizes its distributions, and applies it to solve linear SDEs driven by vggBm noise.

Findings

vggBm has independent components iff it is a fractional Brownian motion

Existence of local times and self-intersection local times for vggBm

Solution of linear SDEs driven by vggBm noise

Abstract

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d-dimensional Donsker's delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d dimensions.