On sharp rate of convergence for discretisation of integrals driven by fractional Brownian motions and related processes with discontinuous integrands

TL;DR

This paper establishes the exact rate of convergence for discretized stochastic integrals driven by fractional Brownian motions with discontinuous integrands, showing a rate proportional to n^{1-2H} and introducing a novel analytical approach.

Contribution

It provides the first precise rate of convergence for such integrals with discontinuous integrands, using a new method involving change of variables and convex analysis.

Findings

Convergence rate is proportional to n^{1-2H}.

Rate is twice better than previous results for discontinuous integrands.

Method applies change of variables and convex functions to compute expectations explicitly.

Abstract

We consider equidistant approximations of stochastic integrals driven by H\"older continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.