Statistical analysis of the mixed fractional Ornstein--Uhlenbeck process

This paper addresses the problem of estimating drift parameter of the Ornstein - Uhlenbeck type process, driven by the sum of independent standard and fractional Brownian motions. The maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit, using some recent results on the canonical representation and spectral structure of mixed processes.Comment: to appear in Theory of Probability and its Application

Mixed Gaussian processes: A filtering approach

This paper presents a new approach to the analysis of mixed processes $$X_t=B_t+G_t,\qquad t\in[0,T],$$ where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel $$K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne t$$ has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional "fractional" structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon-Nikodym densities.Comment: Published at http://dx.doi.org/10.1214/15-AOP1041 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exponential transform of quadratic functional and multiplicative ergodicity of a Gauss-Markov process

The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced

Linear filtering with fractional Brownian motion in the signal and observation processes

Integral equations for the mean-square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h∈(3/4,1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process. AMS subject classifications: 93E11, 60G20, 60G35

Asymptotically optimal filtering in linear systems with fractional Brownian noises

In this paper, the filtering problem is revisited in the basic Gaussian homogeneous linear system driven by fractional Brownian motions. We exhibit a simple approximate filter which is asymptotically optimal in the sense that, when the observation time tends to infinity, the variance of the corresponding filtering error converges to the same limit as for the exact optimal filter

Averaging of Non-Self Adjoint Parabolic Equations with Random Evolution (Dynamics)

The averaging problem for convection-diffusion non-stationary parabolic operator with rapidly oscillating coefficients is studied. Under the assumptio- n that the coefficients are periodic in spatial variables and random stationar- y in time and that they possess certain mixing properties, we show that in appropriate moving coordinates the measures generated by the solutions of original problems converge weakly to a solution of limit stochastic PDE. The homogenized problem is well-posed and defines the limit measure uniquely