Approximate Likelihood Construction for Rough Differential Equations

TL;DR

This paper develops an exact likelihood for rough differential equations driven by piecewise linear paths and proposes an approximate likelihood for more general rough paths, analyzing its behavior as sampling frequency increases.

Contribution

It introduces a novel likelihood construction for rough differential equations and extends it to general rough paths, providing insights into its asymptotic behavior.

Findings

Exact likelihood for piecewise linear rough differential equations

Approximate likelihood for general rough paths

Asymptotic analysis as sampling frequency increases

Abstract

The paper is split in two parts: in the first part, we construct the exact likelihood for a discretely observed rough differential equation, driven by a piecewise linear path. In the second part, we use this likelihood in order to construct an approximation of the likelihood for a discretely observed differential equation driven by a general class of rough paths. Finally, we study the behaviour of the approximate likelihood when the sampling frequency tends to infinity.