Trees and asymptotic developments for fractional stochastic differential equations

TL;DR

This paper develops an elementary method to solve n-dimensional fractional SDEs driven by fractional Brownian motion with H>1/3, providing an expansion for expected functionals of the solution with improved parametrization and remainder control.

Contribution

It introduces a new approach to expand E[f(X_t)] for fractional SDEs, incorporating drifts, using tree parametrization, and achieving sharp remainder estimates.

Findings

Successfully includes drift in the expansion.

Uses tree structures for parametrization, simplifying calculations.

Provides sharp bounds on the remainder term.

Abstract

In this paper we consider a n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H>1/3. After solving this equation in a rather elementary way, following the approach of Gubinelli, we show how to obtain an expansion for E[f(X\_t)] in terms of t, where X denotes the solution to the SDE and f:R^n->R is a regular function. With respect to the work by Baudoin and Coutin, where the same kind of problem is considered, we try an improvement in three different directions: we are able to take a drift into account in the equation, we parametrize our expansion with trees (which makes it easier to use), and we obtain a sharp control of the remainder.