Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme

2022/05/03 by Peng Chen, Chen, Peng, Chang‐Song Deng +5 · 5 citations
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2205.01342

openalex publication_date 2022/05/03 · openalex created_date 2022/05/08 · openalex updated_date 2026/07/28

Abstract

We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an α-stable Lévy process (1<α<2): an approximation scheme with the α-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-1 distance are in the order of η^1-ε and η^\frac2α-1, respectively, where ε∈ (0,1) is arbitrary and η is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck α-stable process shows that the rate η^\frac2α-1 cannot be improved.

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• Optimal Rates for Ergodic SDEs Driven by Multiplicative α-Stable Processes in Wasserstein-1 distance
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