Intrinsic expansions for averaged diffusion processes

TL;DR

This paper demonstrates that asymptotic expansion convergence rates are higher for degenerate diffusion processes than for elliptic ones, using intrinsic functional spaces and with applications in finance such as path-dependent derivatives.

Contribution

It provides a general analysis of convergence rates for degenerate diffusions employing hypoelliptic operator theory, extending previous Malliavin calculus results.

Findings

Higher convergence rates in degenerate diffusions

Application to path-dependent derivatives in finance

Use of hypoelliptic Kolmogorov operators

Abstract

We show that the rate of convergence of asymptotic expansions for solutions of SDEs is generally higher in the case of degenerate (or partial) diffusion compared to the elliptic case, i.e. it is higher when the Brownian motion directly acts only on some components of the diffusion. In the scalar case, this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin calculus techniques. In this paper, we provide a general and detailed analysis by employing the recent study of intrinsic functional spaces related to hypoelliptic Kolmogorov operators in (Pagliarani et al. 2016). Relevant applications to finance are discussed, in particular in the study of path-dependent derivatives (e.g. Asian options) and in models incorporating dependence on past information.