Il giorno venerdì 22 marzo alle ore 14:30 nell'Aula Seminario 2 del Dipartimento di Matematica dell'Università di Bologna, piazza Porta S. Donato 5, il prof. Pietro Rigo del Dipartimento di Matematica dell'Università di Pavia terrà un seminario dal titolo:
On the existence of continuous processes with given one-dimensional distribution
Abstract:
Let $\mathcal{P}$ be the collection of Borel probability measures on $\mathbb{R}$, equipped with the weak* topology, and let $\mu:[0,1]\rightarrow\mathcal{P}$ be a continuous map. Say that $\mu$ is presentable if $X_t\sim\mu_t$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu$ fails to be presentable. Conditions for presentability are given in this note. For instance, $\mu$ is presentable if $\mu_t$ is supported by an interval for all but countably many $t$. In addition, assuming $\mu$ presentable, we investigate whether there is a continuous process $X$ with the same finite dimensional distributions as the quantile process $Q$ induced by $\mu$. The latter is defined, on the probability space $((0,1),\mathcal{B}(0,1),,$Lebesgue measure$)$, by \begin{gather*} Q_t(\alpha)=\inf,\bigl{x\in\mathbb{R}:\mu_t(-\infty,x]\ge\alpha\bigl}\quad\quad\text{for all }t\in [0,1]\text{ and }\alpha\in (0,1). \end{gather*} Various open problems are stated as well.