Asymmetric Jump-Telegraph Processes

The concepts and results of Chap. 2 are generalised, first, to asymmetric processes equipped with jumps (of deterministic amplitudes) and having different alternating speeds and switching intensities, and then to other versions of piecewise deterministic processes. Processes with sequentially taken arbitrary states (self-exciting and self-fading processes) are studied. Further, various memory modes are considered, such as the case where the renewable state depends on the time elapsed in the previous state, or fully resumable restart. We introduce several more complicated modes of distribution of time intervals between successive switchings, which can be considered as a kind of self-exciting Hawkes process. First, there are double telegraph processes based on a doubly stochastic Poisson process, that is, on a point process with the switching intensity accumulated by the telegraph process. Second, we study piecewise linear processes, the regimes of which are switched at Poisson-modulated exponential time intervals. Finally, we present a process that is defined in an arbitrary topological space and follows two alternating patterns. A self-similar telegraph process is also built. With applications in mind, we supply all models with various types of jump components. Martingality and the Girsanov transform are also studied.