A.N. Shiryaev and Contemporary Probability Theory

Mixed Gaussian processes: a filtering approach

30 nov. 2015, 15:20

40m

Angers - France

Orateur

Prof. Marina Kleptsyna (Université du Maine)

Description

We present a new approach to the analysis of mixed processes: for $t\in [0,T]$ $X_t=B_t+G_t,$ 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 $K(s,t)=\frac{{\partial }^2}{\partial s\partial t}E{G}_t{G}_s,s\ne t$ 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.