## The Annals of Probability

### Mixed Gaussian processes: A filtering approach

#### Abstract

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

$K(s,t)=\frac{\partial^{2}}{\partial s\,\partial t}\mathbb{E}G_{t}G_{s},\qquad s\ne t$ 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.

#### Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 3032-3075.

Dates
Revised: June 2015
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.aop/1470139159

Digital Object Identifier
doi:10.1214/15-AOP1041

Mathematical Reviews number (MathSciNet)
MR3531685

Zentralblatt MATH identifier
1351.60038

#### Citation

Cai, Chunhao; Chigansky, Pavel; Kleptsyna, Marina. Mixed Gaussian processes: A filtering approach. Ann. Probab. 44 (2016), no. 4, 3032--3075. doi:10.1214/15-AOP1041. https://projecteuclid.org/euclid.aop/1470139159

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