Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes

Details Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes

2008-04-26

arxiv.org/abs/0804.4258v2

Annals of Probability 2009, Vol. 37, No. 1, 250-274

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP402 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP402

Properties of the law $\mu$ of the integral $\int_0^{\infty}c^{-N_{t-}}\,dY_t$ are studied, where $c>1$ and $\{(N_t,Y_t),t\geq0\}$ is a bivariate L\'{e}vy process such that $\{N_t\}$ and $\{Y_t\}$ are Poisson processes with parameters $a$ and $b$, respectively. This is the stationary distribution of some generalized Ornstein--Uhlenbeck process. The law $\mu$ is parametrized by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalized L\'{e}vy measure of $\{(N_t,Y_t)\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$, respectively. It is shown that, under the condition that $p>0$ and $q>0$, $\mu_{c,q,r}$ is infinitely divisible if and only if $r\leq pq$. The infinite divisibility of the symmetrization of $\mu$ is also characterized. The law $\mu$ is either continuous-singular or absolutely continuous, unless $r=1$. It is shown that if $c$ is in the set of Pisot--Vijayaraghavan numbers, which includes all integers bigger than 1, then $\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely continuous whenever $q\geq C_1p\geq C_2r$. For any $c>1$ there is a positive constant $C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and $\max\{q,r\}\leq C_3p$. Here, if $\{N_t\}$ and $\{Y_t\}$ are independent, then $r=0$ and $q=b/(a+b)$.

Affiliated with

Also associated with

No associations

Rating

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-114293