2022/04/13 by Xue-Mei Li, Li, Xue-Mei, Fabien Panloup +3 · 3 citations
Economics, Econometrics and Finance · #37A25 #60G22 #60H10 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2204.06329
openalex publication_date 2022/04/13 · openalex created_date 2022/04/19 · openalex updated_date 2026/07/28
We investigate the stationary measure π of SDEs driven by additive fractional noise with any Hurst parameter and establish that π admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this we revisit the fractional averaging principle of Li and Sieber [Ann. Appl. Probab. 32(5) (2022)] and remove an ad-hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements.