Jevgenijs Ivanovs

Two-sided reflection of Markov-modulated Brownian motion

This article considers a Markov-modulated Brownian motion with a two-sided reflection. For this doubly-reflected process we compute the Laplace transform of the stationary distribution, as well as the average loss rates at both barriers. Our approach relies on spectral properties of the matrix polynomial associated with the generator of the free (that is, non-reflected) process. This work generalizes previous partial results allowing the spectrum of the generator to be non-semi-simple and also covers the delicate case where the asymptotic drift of the free process is zero.

Potential measures of one-sided Markov additive processes with reflecting and terminating barriers

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and non-defective processes and all possible scenarios we identify the corresponding potential measures, which generalizes a number of results for one-sided Levy processes. The resulting rather neat formulas have various applications, and in particular they lead to quasi-stationary distributions of the corresponding processes.

Another look into decomposition results

In this note, we identify a simple setup from which one may easily infer various decomposition results for queues with interruptions as well as càdlàg processes with certain secondary jump inputs. Special cases are processes with stationary or stationary and independent increments. In the Lévy process case, the decomposition holds not only in the limit but also at independent exponential times, due to the Wiener–Hopf decomposition. A similar statement holds regarding the GI/GI/1 setting with multiple vacations.

First passage of time-reversible spectrally negative Markov additive processes

We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix @L. Assuming time reversibility, we show that all the eigenvalues of @L are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of @L. Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain @L in the time-reversible case.

Zooming in on a Lévy process at its supremum

Let

A Lévy-derived process seen from its supremum and max-stable processes

M

Linking dividends and capital injections – a probabilistic approach

and

Robust bounds in multivariate extremes

\tau

Exit identities for Lévy processes observed at Poisson arrival times

be the supremum and its time of a Levy process

Occupation densities in solving exit problems for Markov additive processes and their reflections.

X