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Mixed Gaussian processes: a filtering approach. (English) Zbl 1351.60038
Authors’ abstract: This paper presents a new approach to the analysis of mixed processes \[ X_{t}+B_{t}+G_{t},\quad t\in \left[ 0,T\right] , \] 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 \[ K\left( s,t\right) =\frac{\partial ^{2}}{\partial s\partial t}\mathbb{E} G_{t}G_{s},\quad s\neq t \] 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.

MSC:
60G15 Gaussian processes
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion
60G35 Signal detection and filtering (aspects of stochastic processes)
60G30 Continuity and singularity of induced measures
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