In this paper we develop the theory of the so-called W and Z scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form of the generating function of W and its connection with the occupation mass formula, which provides the tools for deriving semi-explicit expressions for corresponding exit problems for the upward-skip free process and it's reflections, in terms the scale matrices.

In this paper, we propose a generalisation to the Cramér-Lundberg risk model, by allowing for a delayed receipt of the required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of an insurance firm.

In this paper, we generalise the classic compound Poisson risk model, by the introduction of ordered capital levels, to model the solvency of an insurance firm. A breach of the higher capital level, the magnitude of which does not cause further breaches of either the lower level or the so-called intermediate confidence level (of the shareholders), requires a capital injection to restore the surplus to a solvent position. On the other hand, if the confidence level is breached capital injections are no longer a viable method of recapitalisation. Instead, the company can borrow money from a third party, subject to a constant interest rate, which is paid back until the surplus returns to the confidence level and subsequently can be restored to a fully solvent position by a capital injection. If at any point the surplus breaches the lower capital level, the company is considered 'insolvent' and is forced to cease trading. For the aforementioned risk model, we derive an explicit expression for the 'probability of insolvency' in terms of the ruin quantities of the classical risk model. Under the assumption of exponentially distributed claim sizes, we show that the probability of insolvency is in fact directly proportional to the classical ruin function. It is shown that this result also holds for the asymptotic behaviour of the insolvency probability, with a general claim size distribution. Explicit expressions are also derived for the moment generating function of the accumulated capital injections up to the time of insolvency and finally, in order to better capture the reality, dividend payments to the companies shareholders are considered, along with the capital constraint levels, and explicit expressions for the probability of insolvency, under this modification, are obtained.

Please cite this article as: L. Ramsden, A.D. Papaioannou, Asymptotic results for a Markov-modulated risk process with stochastic investment, Journal of Computational and Applied Mathematics (2016), http://dx.Abstract 4 In this paper we consider a Markov-modulated risk model, where the premium 5 rates, claim frequency and the distribution of the claim sizes vary depending on the 6 state of an external Markov chain. The free reserves of the insurer are invested in a 7 risky asset whose prices are modelled by a geometric Brownian motion, with param-8 eters that are also influenced according to the external Markov process. A system of 9 integro-differential equations for the ruin probabilities and for the expected discounted 10 penalty function is derived. Using Laplace transforms and regular variation theory, 11 we investigate the asymptotic behaviour of both quantities for the case of light or 12 heavy tailed claim size distributions. Specifically, within this set up (where we lose the 13 strong Markov property of the risk process), we show that the ruin probabilities de-14 crease asymptotically as a power function in the case of the light tailed claims, whilst 15 for the heavy tails we show that the probabilities of ruin decay either like a power 16 function, depending on the parameters of the investment, or behave asymptotically 17 like the tails of the claim size distributions.18

In this paper we propose a generalisation to the Markov Arrival Process (MAP) risk model, by allowing for a delayed receipt of required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of a firm. We introduce a MAP risk model that allows for capital injections to be received instantaneously, or with a random delay, depending on the amount of deficit experienced by the firm. For this model, we derive a system of Fredholm integral equations of the second kind for the Gerber-Shiu function and obtain an explicit expression (in matrix form) in terms of the Gerber-Shiu function of the MAP risk model without capital injections. In addition, we show that the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, to illustrate the applicability of our results, numerical examples are given. Keywords Delayed capital injections • Markov Arrival Process • Systems of Fredholm integral equations • Gerber-Shiu function • Expected discounted-accumulated capital injections until ruin Mathematics Subject Classification (2010) 45B05 • 60J25 • 91B05