A.N. Shiryaev and Contemporary Probability Theory

Inversion, duality and h-processes of self-similar Markov processes

30 nov. 2015, 17:00

40m

Angers - France

Orateur

Prof. Loïc Chaumont (Université d'Angers)

Description

We show that any ${\mathbb{R}}^d$-valued self-similar Markov process $X$ with index $\alpha >0$ absorbed at 0, can be represented as a path transformation of some Markov additive process (MAP) $(\theta ,\xi )$ in $S_{d-1}\times \mathbb{R}$. This result extends the well known Lamperti transformation. Then we prove that the same transformation of the dual MAP in the weak sense of $(\theta ,\xi )$ is itself in weak duality with $X$, with respect to the measure $\pi (x/\Vert x\Vert )\Vert x{\Vert }^{\alpha -d}dx$, if and only if $(\theta ,\xi )$ is reversible with respect to the measure $\pi (s)ds$, where $ds$ is the Lebesgue measure on $S_{d-1}$. Besides, the dual process $\hat{X}$ has the same law as the inversion $(X_{{\gamma }_t}/\Vert X_{{\gamma }_t}{\Vert }^2,t\ge 0)$ of $X$, where ${\gamma }_t$ is the inverse of $t\mapsto {\int }_0^t\Vert X{\Vert }_s^{-2\alpha }ds$. As an application, we prove that in some instances, the Kelvin transform of $X$ can be obtained as an $h$-transform of some functional of $X$. This is a joint work with Larbi Alili, Piotr Graczyk and Tomasz Zak. $$