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

2022/09/14 by Ehsan Azmoodeh, Pauliina Ilmonen, Azmoodeh, Ehsan +7
Economics, Econometrics and Finance · #60G15 #60G22 #60H05 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2209.06708

openalex publication_date 2022/09/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>\frac12 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L¹-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.

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