Modern Stochastics: Theory and Applications

General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.