Uncertain insurance risk process with single premium and multiple classes of claims
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ORIGINAL RESEARCH
Uncertain insurance risk process with single premium and multiple classes of claims Zhe Liu1 · Xiangfeng Yang2 Received: 26 January 2020 / Accepted: 17 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Traditionally an insurance risk process is considered under the framework of probability theory with a prerequisite that the estimated distribution function is close enough to the real frequency. However, due to economic or technological reasons, sometimes data are unavailable or difficult to obtain, such as when we consider a new insurance product or insurance for valuable weapons. Under these situations, reimbursement policies are based on experts’ belief degree, which has a much wider range than the real frequency. As a result, we should employ uncertain insurance risk models to better deal with human uncertainty in running an insurance company. Noticing the fact that an insurance company pays for different kinds of risks and the uncertainty of the customer’s arrivals and payments, we investigate an uncertain insurance risk process with multiple classes of claims where the premium process follows an uncertain renewal process. Then we derive expressions for the ruin index and the uncertainty distribution of the ruin time. Some numerical examples and a real data example are performed to capture more insights. Keywords Uncertain insurance model · Uncertain premium · Ruin index · Ruin time · Uncertainty theory
1 Introduction Risk theory, which is the main part of actuarial science, analyses all kinds of risk mathematically. As a core of risk theory, ruin theory indicates the robustness of insurance companies through some criteria such as ruin index, ruin time, and deficit. The theoretical foundation of ruin theory is the Cramer–Lundberg model, also known as the classical risk model (Delbaen and Haezendonck 1987), where premiums are proportional to time with a constant rate, and claims arrive according to a Poisson process. Then Andersen (1957) extended the classical model by considering arbitrary distributed claim inter-arrival times. Later, the risk process with multiple classes of claims was originated by Sundt (1999) and investigated by Chan et al. (2003) and Yuen et al. (2002). For generalizations of premium charges, Temnov * Xiangfeng Yang [email protected] 1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China
2
(2004) derived the representation of ruin probability for the random premium process and compared it with that of the classical risk process. A defective renewal equation for the discounted penalty function in a random premium process was provided by Bao (2006). Following that, more research interests (Yang and Yuen 2016; Yang et al. 2019) were drawn to study and extend this topic. These stochastic risk models aforementioned work well when there are enough data to estimate probability distri
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