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July 2016 Mixed Gaussian processes: A filtering approach
Chunhao Cai, Pavel Chigansky, Marina Kleptsyna
Ann. Probab. 44(4): 3032-3075 (July 2016). DOI: 10.1214/15-AOP1041

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.

Citation

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Chunhao Cai. Pavel Chigansky. Marina Kleptsyna. "Mixed Gaussian processes: A filtering approach." Ann. Probab. 44 (4) 3032 - 3075, July 2016. https://doi.org/10.1214/15-AOP1041

Information

Received: 1 August 2014; Revised: 1 June 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 1351.60038
MathSciNet: MR3531685
Digital Object Identifier: 10.1214/15-AOP1041

Subjects:
Primary: 60G15
Secondary: 60G22, 60G30, 60G35

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 4 • July 2016
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