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Markov processes conditioned to never exit a subspace of the state space

Palmowski Z., Rolski T.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

339--353

In this paper we study Markov processes never exiting (NE) a subspace A of the state space E or, in other words, Markov processes conditioned to stay in the subspace A. We show how the knowledge of the exact asymptotics of the tail distribution of the exit time helps to find the suitable exponential martingale, which, in turn, serves for the change of measure. Under the new probability measure the process is the sought for never exiting one the subspace A. We also find its extended generator and study relationships between the invariant measure (INE) and the quasi-stationary (QS) distribution. We analyze in detail the PDMP processes.

Tail probabilities for a risk process with subexponential jumps in a regenerative and diffusion environment

Palmowski Z.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

381-405

In this paper we find a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity proces.