The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions.

In this paper, we consider discounted penalty functions, also called Gerber-Shiu functions, in a Markovian shot-noise environment. At first, we exploit the underlying structure of piecewise-deterministic Markov processes (PDMPs) to show that these penalty functions solve certain partial integro-differential equations (PIDEs). Since these equations cannot be solved exactly, we develop a numerical scheme that allows us to determine an approximation of such functions. These numerical solutions can be identified with penalty functions of continuous-time Markov chains with finite state space. Finally, we show the convergence of the corresponding generators over suitable sets of functions to prove that these Markov chains converge weakly against the original PDMP. That gives us that the numerical approximations converge to the discounted penalty functions of the original Markovian shot-noise environment.

On a dividend problem with random funding.

We consider a modification of the dividend maximization problem from ruin theory. Based on a classical risk process we maximize the difference of expected cumulated discounted dividends and total expected discounted additional funding (subject to some proportional transaction costs). For modelling dividends we use the common approach whereas for the funding opportunity we use the jump times of another independent Poisson process at which we choose an appropriate funding height. In case of exponentially distributed claims we are able to determine an explicit solution to the problem and derive an optimal strategy whose nature heavily depends on the size of the transaction costs. Furthermore, the optimal strategy identifies unfavourable surplus positions prior to ruin at which refunding is highly recommended.

Approximation methods for piecewise deterministic Markov processes and their costs.

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.