A Markov additive process can be seen as a Lévy process modulated by a Markov chain with additional jumps at phase switching epochs, where all jumps are of the same sign. This talk at the beginning will focus on some basic theory for one-sided Markov additive processes (possibly with an upper and a lower barrier, where each can be either reflecting or terminating). We will consider the three fundamental matrices: the right-, the left-solution of a certain matrix equation, and the linking matrix of expected occupation times at zero. Moreover, we will construct the so-called scale (matrix-valued) functions and discuss exit problems, potential measures and Wiener-Hopf factorization. Finally, we also show possible applications in applied probability intimately related with financial mathematics, risk theory and queueing theory. In particular, we will focus on a functional discounting and Poisson observations. Prior knowledge of Lévy processes will not be assumed, but will render this talk easier to follow.

The talk is based on the following papers:

[1] H. Albrecher and J.Ivanovs (2013). A risk model with an observer in a Markov environment. Risks 1(3), 148-161.
[2] I. Czarna, A. Kaszubowski, S. Li and Z. Palmowski (2018). Fluctuation identities for omega-killed Markov additive processes and dividend problem. Submitted for publication, https://arxiv.org/abs/1806.08102.
[3] J. Ivanovs and Z. Palmowski (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stochastic Processes and their Applications 122(9), 3342-3360.
[4] J. Ivanovs (2014). Potential measures of one-sided Markov additive processes with reflecting and terminating barriers. Journal of Applied Probability 51(4), 2014, 1154-1170.
[5] J. Ivanovs (2017). Splitting and time reversal for Markov additive processes. Stochastic Processes and Their Applications 127(8), 2699-2724.
[6] J. Ivanovs and H. Albrecher (2017). Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Stochastic Processes and Their Applications 127(2), 643-656.
[7] P. Klusik and Z. Palmowski (2014). A note on Wiener-Hopf factorization for spectrally negative Markov Additive processes. Journal of Theoretical Probability 27, 202-219.