Zusammenfassung: Assume the capital or surplus of an insurance company evolves randomly over time as the spectrally negative Lévy process but where in addition the company has the possibility to pay out dividends to shareholders and to inject capital at a cost from shareholders. We impose that when the resulting surplus becomes negative the company has to decide whether to inject capital to get to a positive surplus level in order for the company to survive or to let ruin occur. The objective is to find the combined dividends and capital injections strategy that maximises the expected paid out dividends minus cost of injected capital, discounted at a constant rate, until ruin. We consider the setting where the cost of capital is level-dependent in the sense that it is higher when the surplus is below 0 than when it is above 0. We investigate optimality of a 3-parameter strategy with parameters -r < 0 < c < b where dividends are paid out to keep the surplus below b, capital injections are made in order to keep the surplus above c unless capital drops below the level -r in which case the company decides to let ruin occur. The proof is based on some monotonicity properties for the solution of the renewal equations with log-convex kernel
Zusammenfassung:
In this talk, we will discuss exit problems for general upward skip-free
Markov additive chains (MACs) or Markov-modulated random walks. In particular,
we will construct and characterise a number of fundamental matrices related to
the process, namely G and the so-called W and Z scale matrices, demonstrating
how these quantities can be used to derive exit problems, as well as other
fluctuation identities, including those related to the corresponding
`reflected process'. The theory developed in this discrete setup is chosen to
echo those of the theory for continuous-time Markov additive processes (MAPs),
further emphasising the utility of the scale functions/matrices, which allow
us to identify the probabilistic construction, generating function and simple
recursion relation for these matrices, as well as their connection to the
so-called occupation mass functions.
In the second part of the talk, we will use this general fluctuation theory
developed for (MACs) to develop the Gerber-Shiu theory for the classic and
dual discrete risk processes in a regime-switching environment. In particular,
by expressing the Gerber-Shiu function in terms of potential measures of an
upward (downward) skip-free MAC, we derive closed form expressions for the
Gerber-Shiu function in terms of the aforementioned scale matrices. Finally,
we also present results for the value function of the associated constant
dividend barrier problems for both risk processes.