A scale function based approach for solving integral-differential equations in insurance risk models. (English) Zbl 07701061
Summary: In risk theory, the resolutions of many interesting problems are reduced to solving some integro-differential equations (IDE), see [Zbl 1518.91214; Zbl 1259.60050; Zbl 1416.91166; JFM 56.1100.03; B. De Finetti, “Su un’impostazione alternativa della teoria collettiva del rischio, in: Proceedings of the Transactions on XV International Congress of Actuaries, New York. 433–443 (1957); Zbl 1097.91040; Zbl 0723.62065; Zbl 0193.20501; Zbl 0924.60075; Zbl 1162.91374; Zbl 1081.60550; Zbl 1411.91303; Zbl 1208.91069; Zbl 1157.91383; F. Lundberg, “Über die Theorie der Rückversicherung, Trans. VIth Int. Congr. Actuar. 1, 877–948 (1909); Zbl 1189.91075; Zbl 1163.91441; Zbl 1264.91074; Zbl 1183.91077; Zbl 1427.91080], 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 [Zbl 1461.60028; Zbl 1527.60034; Zbl 1291.60094; Zbl 1274.60148; Zbl 1475.60088], Chapter 8 of [Zbl 1384.60003], [Zbl 1447.91142], 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.
MSC:
| 91G05 | Actuarial mathematics |
| 35R09 | Integro-partial differential equations |
| 91B05 | Risk models (general) |
Citations:
Zbl 1259.60050; Zbl 1416.91166; Zbl 1097.91040; Zbl 0723.62065; Zbl 0193.20501; Zbl 0924.60075; Zbl 1162.91374; Zbl 1081.60550; Zbl 1411.91303; Zbl 1208.91069; Zbl 1157.91383; Zbl 1189.91075; Zbl 1163.91441; Zbl 1264.91074; Zbl 1183.91077; Zbl 1427.91080; Zbl 1461.60028; Zbl 1291.60094; Zbl 1274.60148; Zbl 1475.60088; Zbl 1384.60003; Zbl 1447.91142; JFM 56.1100.03; Zbl 1518.91214; Zbl 1527.60034References:
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