On finite-time ruin probabilities in a generalized dual risk model with dependence

In this paper, we study the finite-time ruin probability in a reasonably generalized dual risK model, where we assume any non-negative non-decreasing cumulative operational cost function and arbitrary capital gains arrival process. Establishing an enlightening link between this dual risk model and its corresponding insurance risk model, explicit expressions for the finite-time survival probability in the dual risk model are obtained under various general assumptions for the distribution of the capital gains. In order to make the model more realistic and general, different dependence structures among capital gains and inter-arrival times and between both are also introduced and corresponding ruin probability expressions are also given. The concept of alarm time, as introduced in Das and Kratz (2012), is applied to the dual risk model within the context of risk capital allocation. Extensive numerical illustrations are provided

On the optimality of joint periodic and extraordinary dividend strategies

In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion. Owners can take cash out of the surplus in the form of "dividends", subject to transaction costs. However, if the surplus hits 0 then ruin occurs and the business cannot operate any more. We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) extraordinary dividend payments that can be made immediately at any time (that is, the dividend decision time space is continuous and matches that of the surplus process). Both types of dividends attract proportional transaction costs, and extraordinary distributions also attracts fixed transaction costs, a realistic feature. A dividend strategy that involves both types of distributions (periodic and extraordinary) is qualified as "hybrid". We determine which strategies (either periodic, immediate, or hybrid) are optimal, that is, we show which are the strategies that maximise the expected present value of dividends paid until ruin, net of transaction costs. Sometimes, a liquidation strategy (which pays out all monies and stops the process) is optimal. Which strategy is optimal depends on the profitability of the business, and the level of (proportional and fixed) transaction costs. Results are illustrated

Dependent Risk Modelling and Ruin Probability: Numerical Computation and Applications

In this thesis, we are concerned with the finite-time ruin probabilities in two alternative dependent risk models, the insurance risk model and the dual risk model, including the numerical evaluation of the explicit expressions for these quantities and the application of the probabilistic results obtained. We first investigate the numerical properties of the formulas for the finite-time ruin probability derived by Ignatov and Kaishev (2000, 2004) and Ignatov et al. (2001) for a generalized insurance risk model allowing dependence. Efficient numerical algorithms are proposed for computing the ruin probability with a prescribed accuracy in order to facilitate the following studies. We then propose a new definition of alarm time in the insurance risk model, which generalizes that of Das and Kratz (2012), expressed in terms of the joint distribution of the time to ruin and the deficit at ruin. The alarm time is devised to warn that the future ruin probability within a finite-time window has reached a pre-specified critical level and capital injection is required. Due to our definition, the implementation of the alarm time highly relies on the computation of the finite-time ruin probability, which utilizes the previous results on computing the ruin probability with a prescribed accuracy. The results of the ruin probability and the alarm time are then transferred nicely to a generalized dual risk model, whose name stems from its duality to the insurance risk model, through an enlightening link established between the two risk models. Finally, based on the two alternative risk models, we introduce a framework for analyzing the risk of systems failure based on estimating the failure probability, and illustrate how the probabilistic models and results obtained can be applied as risk analytic tools in various practical risk assessment situations, such as systems reliability, inventory management, flood control via dam management, infection disease spread and financial insolvency