Mixed Gaussian processes: A filtering approach

Abstract

This paper presents a new approach to the analysis of mixed processeswhere Bt is a Brownian motion and Gt is an independent centered Gaussian process. We obtain a new canonical innovation representation of X, using linear filtering theory. When the kernelhas 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.

Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process

“…Further, a mixed fractional Brownian motion and related models were comprehensively considered by Mishura [12]. Finally, the results of recent works of Cai, Kleptsyna, and Chigansky [4] and Chigansky and Kleptsyna [6] concerning the new canonical representation of mixed fractional Brownian motion present a great value for the purposes of this paper.…”

“…An interesting change in properties of a mixed fractional Brownian motion B occurs depending on the value of H. In particular, it was shown (see [5]) that B is a semimartingale in its own filtration if and only if either H = 1 2 or H ∈ ( 3 4 , 1]. The main contribution of paper [4] is a novel approach to the analysis of mixed fractional Brownian motion based on the filtering theory of Gaussian processes. The core of this method is a new canonical representation of B.…”

Mixed fractional Brownian motion: A spectral take

“…Another interesting feature is revealed by its canonical representation from [5], based on the martingale…”

Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions

“…Similar problems for the model with linear drift driven by fractional Brownian motion were studied in [5,16,26,35]. The mixed Brownian -fractional Brownian model was treated in [7]. In [4,39] the nonparametric functional estimation of the drift of a Gaussian processes was considered (such estimators for fractional and subfractional Brownian motions were studied in [13] and [43] respectively).…”