Publication

On the (non)stationary density of fractional-driven stochastic differential equations

2023

Abstract

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 nonstationary 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 (2022) 3964–4003) 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.

Official source

https://infoscience.epfl.ch/entities/publication/5aa063c6-ea0c-4fd2-9eaa-8901b7dd0edc

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5aa063c6-ea0c-4fd2-9eaa-8901b7dd0edc

About this result

Related concepts (32)

Itô calculus

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale.

Hurst exponent

The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time.

Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution.