We show how to solve Merton optimal investment stochastic control problem for Hawkesbased models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R(t)...

The aim of this paper is to introduce a general bootstrap by exchangeable weight random variables for empirical estimators of the semi-Markov kernels and of the conditional transition probabilities for semi-Markov processes with countable state space. Asymptotic properties of these generalized...

We study the time evolution of an increasing stochastic process governed by a first-order stochastic differential system. This defines a particular piecewise deterministic Markov process (PDMP). We consider a Markov renewal process (MRP) associated to the PDMP and its Markov renewal equation...

This paper investigates the rate of occurrence of failures (ROCOF) of finite state semi-Markov systems. Firstly, a formula for evaluating the ROCOF for semi-Markov systems is derived. As a consequence of this result, we derive the ROCOF of Markov systems as well as the ROCOF of the alternating...