For a finite state Markov process and a finite collection ${ Gamma_k, k in K }$ of subsets of its state space, let $tau_k$ be the first time the process visits the set $Gamma_k$. We derive explicit/recursive formulas for the joint density and tail pr
obabilities of the stopping times ${ tau_k, k in K}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.
In this Note we study a class of BSDEs which admits a particular singularity in their driver. More precisely, we assume that the driver is not integrable and degenerates when approaching to the terminal time of the equation.
This work is concerned with the theory of initial and progressive enlargements of a reference filtration F with a random time {tau}. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alte
rnative proofs to results concerning canonical decomposition of an F-martingale in the enlarged filtrations. Also, we address martingales characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable representation theorems in the enlarged filtrations.
