Dividend problems in the dual risk model

• Published in: Insurance, mathematics & economics mathematics & economics, 2013-11, Vol.53 (3), p.906-918
• Main Authors: Afonso, Lourdes B., Cardoso, Rui M.R., Egídio dos Reis, Alfredo D.
• Format: Article
• Language: English
• Subjects: Capital returns; Classical risk model; Differential equations; Discounted dividends; Dividend amounts; Dividend probabilities; Dividends; Dual risk model; Economic activity; Insurance applications; Laplace transforms; Mathematical models; Number of dividends; Poisson distribution; Probability; Risk; Risk management; Ruin probabilities; Stock returns; Studies; Venture capital
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/j.insmatheco.2013.10.003

We consider the compound Poisson dual risk model, dual to the well known classical risk model for insurance applications, where premiums are regarded as costs and claims are viewed as profits. The surplus can be interpreted as a venture capital like the capital of an economic activity involved in research and development. Like most authors, we consider an upper dividend barrier so that we model the gains of the capital and its return to the capital holders. By establishing a proper and crucial connection between the two models we show and explain clearly the dividends process dynamics for the dual risk model, properties for different random quantities involved as well as their relations. Using our innovative approach we derive some already known results and go further by finding several new ones. We study different ruin and dividend probabilities, such as the calculation of the probability of a dividend, distribution of the number of dividends, expected and amount of dividends as well as the time of getting a dividend. We obtain integro-differential equations for some of the above results and also Laplace transforms. From there we can get analytical results for cases where solutions and/or inversions are possible, in other cases we may only get numerical ones. We present examples under the two cases. •We consider the compound Poisson dual risk model with a dividend barrier strategy.•We establish a closed connection between the dual and the classical risk models.•We present a new approach for the calculation of expected discounted dividends.•We study ruin and dividend probabilities, number of dividends and time to a dividend.•We also present the distribution for the amount of single dividends.