Fractional Diffusion Bridges

TL;DR

This paper studies fractional diffusion processes driven by fractional Brownian motion, focusing on conditioning these processes to reach a specific endpoint, and develops a rigorous mathematical framework using quasi-sure analysis and large deviation principles.

Contribution

It introduces the concept of fractional diffusion bridges and applies quasi-sure analysis and large deviation principles to these conditioned processes.

Findings

Established a rigorous conditioning framework for fractional diffusion processes.

Proved a Freidlin-Wentzell type large deviation principle for scaled fractional diffusion bridges.

Provided mathematical tools for analyzing fractional stochastic differential equations.

Abstract

Consider ``stochastic differential equations" driven by fractional Brownian motion with Hurst parameter H (1/4 <H< 1). Their solutions are sometimes called fractional diffusion processes. The main purpose of this paper is conditioning these processes to reach a given terminal point. We call the conditioned processes fractional diffusion bridges. Our main tool for mathematically rigorous conditioning is quasi-sure analysis, which is a potential theoretic part of Malliavin calculus. We also prove a small-noise large deviation principle of Freidlin-Wentzell type for scaled fractional diffusion bridges under a mild ellipticity assumption on the coefficient vector fields.