The Annals of Probability

Mixed Gaussian processes: A filtering approach

Chunhao Cai, Pavel Chigansky, and Marina Kleptsyna

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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

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

Received: August 2014
Revised: June 2015
First available in Project Euclid: 2 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G22: Fractional processes, including fractional Brownian motion 60G30: Continuity and singularity of induced measures 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Gaussian processes innovation representation linear filtering fractional processes equivalence of measures


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.

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