Oscillating Gaussian processes

Pauliina Ilmonen,Soledad Torres,Lauri Viitasaari

doi:10.1007/s11203-020-09212-6

Pauliina Ilmonen, Soledad Torres + Show 1 more

Open Access

https://doi.org/10.1007/s11203-020-09212-6

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Abstract

In this article we introduce and study oscillating Gaussian processes defined by X_t = alpha _+ Y_t mathbf{1}_{Y_t >0} + alpha _- Y_tmathbf{1}_{Y_t<0}, where alpha _+,alpha _->0 are free parameters and Y is either stationary or self-similar Gaussian process. We study the basic properties of X and we consider estimation of the model parameters. In particular, we show that the moment estimators converge in L^p and are, when suitably normalised, asymptotically normal.

Highlights

In this article we introduce a class of stationary processes, called oscillating Gaussian processes, defined by
Our motivation stems partially from the connection to stochastic differential equations driven by the fractional Brownian motion with discontinuous diffusion coefficient, studied in Garzón et al (2017)
In Mishura and Nualart (2004), the authors studied a drift that is Hölder continuous except on a finite numbers of points. Another class of discontinuity in SDE driven by a fractional Brownian motion is related to adding a Poisson process to the equation

Summary

Introduction

In this article we introduce a class of stationary processes, called oscillating Gaussian processes, defined by. In Mishura and Nualart (2004), the authors studied a drift that is Hölder continuous except on a finite numbers of points Another class of discontinuity in SDE driven by a fractional Brownian motion is related to adding a Poisson process to the equation. In addition to the above mentioned links to SDEs and skew Brownian motion, we note that (1.1) could be applied in various other modelling scenarios as well, making oscillating Gaussian process an interesting object of study. In papers Dobrushin and Major (1979), Taqqu (1979, 1975) the authors studied limiting behaviour of the moment estimator in the case where the assumptions of Breuer and Major (1983)√are violated In this case, one usually needs different normalisation instead of the standard T , and the limiting distribution is not necessarily Gaussian.

Model calibration

Estimation based on discrete observations

Oscillating self-similar Gaussian processes

The case of bi-fractional Brownian motion

Discussion