A scale function based approach for solving integral-differential equations in insurance risk models

Abstract

In risk theory, the resolutions of many interesting problems are reduced to solving some integro-differential equations (IDE), see [4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 29, 31], and references therein. Meanwhile, due to the recent advances made on Lévy processes, explicit analytical expressions of the scale functions associated with Lévy processes become on offer (see [1, 2, 11, 16, 17], Chapter 8 of [19], [20], etc). This paper aims at bridging together the scale functions and the IDEs by presenting a unified scale function based approach for solving IDEs that arise in risk theory. In particular, to demonstrate the effectiveness of this approach, a dividend and capital injection problem is considered under a jump-diffusion risk model. We first derive the IDEs satisfied by the expected accumulated discounted difference between the net dividends and the costs of capital injections, and then solve the IDEs with its solution being expressed in compact and transparent forms.