Discretisation error for stochastic integrals with respect to the fractional Brownian motion with discontinuous integrands and local times

2023/05/08 by Valentin Garino, Lauri Viitasaari, Garino, Valentin +1
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #26A33 #60G15 #60G22 #60H05 #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2305.04733

openalex publication_date 2023/05/08 · openalex created_date 2023/05/10 · openalex updated_date 2026/07/28

Abstract

We consider equidistant Riemann approximations of stochastic integrals ∫₀^T f(B^H_s)dB^H_s with respect to the fractional Brownian motion with H>\frac12, where f is an arbitrary function of locally bounded variation, hence possibly possessing discontinuities. We prove that properly normalised approximation error converge in the L²-topology to a functional of the local time, and we provide rate of convergence for this approximation. As such, our results complements some recent advances on the topic as well as provides new methods for simulation of local times.

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