Speaker
Prof.
Marina Kleptsyna
(Université du Maine)
Description
We present a new approach to the analysis of mixed processes: for $t\in [0,T]$
$
\hspace{6cm} 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
$
\hspace{5cm} K(s,t) = \frac{\partial^2}{\partial s\partial t} E G_t G_s,\quad 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.
Primary author
Prof.
Marina Kleptsyna
(Université du Maine)
