Originating in part from the concerns of the Victorian British upper classes that aristocratic surnames were becoming extinct, the theory of branching processes now forms a cornerstone of classical applied probability. Some of the earliest work on branching processes dates back to Galton and Watson in 1874, [25]. However, approximately 100 years later, it was discovered in [13] that the less we...

In many application domains such as autonomous avionics, power electronics and process systems engineering there exist discretely controlled continuous processes (DCCPs) which constitute a special subclass of hybrid dynamical systems. We introduce a novel simulation-based approach for DDCPs optimization under uncertainty using Reinforcement Learning with Gaussian Process models to learn the tra...

In this section we discuss a simple feature of all Lévy processes which follows as a direct consequence of stationary independent increments. That is, when the path of a Lévy process over a finite time horizon is time reversed (in an appropriate sense) the new path is equal in law to the process reflected about the origin. This property will prove to be of crucial importance in a number of fluc...

Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process. A real-valued process B = {B t : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold: (i) The paths of B are P-almost surely continuous. (ii) P(B 0 = 0) = 1. (iii) For 0 ≤ s ≤ t, B t − B s is equal in distribution to B t−s. (iv) For 0 ≤ ...