We study a piece-wise deterministic Markov process having jumps of i.i.d. sizes with a constant intensity and decaying at a constant rate (a special case of a storage process with a general release rule). Necessary and sufficient conditions for the process to be ergodic are found, its stationary...

A firm-value model similar to the one proposed by Black and Cox (1976) is considered. Instead of assuming a constant and known default boundary, the default boundary is an unobserved stochastic process. This process has a Brownian component, reflecting the influence of uncertain effects on the...

We prove some maximal inequalities for fractional Brownian motions. These extend the Burkholder-Davis-Gundy inequalities for fractional Brownian motions. The methods are based on the integral representations of fractional Brownian motions with respect to a certain Gaussian martingale in terms of...

We present some comments on moment inequalities and identities for martingales in the context of the paper of Langovoy (2011).

We prove two martingale identities which involve exit times of Lévy-driven Ornstein-Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Lévy process are exponentially distributed.