Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law

2023/07/13 by Jorge A. Leòn, León, Jorge, Yanghui Liu +3 · 1 citation
Economics, Econometrics and Finance · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Financial Markets and Investment Strategies

paper · pdf · doi:10.48550/arxiv.2307.06759

Abstract

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter H>1/3. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size n. Then we use the integrability results to derive the convergence rate in law n^1-4H+ε for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.

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