Abstract
A dependent insurance risk model with surrender and investment under the thinning process is discussed, where the arrival of the policies follows a compound Poisson-Geometric process, and the occurrences of the claim and surrender happen as the p-thinning process and the q-thinning process of the arrival process, respectively. By the martingale theory, the properties of the surplus process, adjustment coefficient equation, the upper bound of ruin probability, and explicit expression of ruin probability are obtained. Moreover, we also get the Laplace transformation, the expectation, and the variance of the time when the surplus reaches a given level for the first time. Finally, various trends of the upper bound of ruin probability and the expectation and the variance of the time when the surplus reaches a given level for the first time are simulated analytically along with changing the investment size, investment interest rates, claim rate, and surrender rate.
Highlights
In the classical ruin theory, compound Poisson risk model, N(t)U (t) = u + ct − ∑ Xi for t ≥ 0, (1)i=1 is the main research object [1, 2], where u ≥ 0 is the initial reserve, c is the premium rate, and {N(t), t ≥ 0} is a Poisson process with intensity λ > 0, representing the number of claims up to time t
Many studies in literature discuss the dependent relationship among the premium income, interclaim arrivals, and the claim size
Jiang et al [5] investigate some uniform asymptotic estimates for finitetime ruin probabilities when the claim size vector and its interarrival time are subject to certain general dependence structure
Summary
I=1 is the main research object [1, 2], where u ≥ 0 is the initial reserve, c is the premium rate, and {N(t), t ≥ 0} is a Poisson process with intensity λ > 0, representing the number of claims up to time t. Zhang and Yang [6], Shi et al [7], and Zou et al [8] consider a compound Poisson risk model and a dependence structure of the claim size and interclaim time modeled by a Farlie-Gumbel-Morgenstern copula. An important reason for this phenomenon is that insurance companies have adopted risk aversion mechanism, such as franchise system and no-claim discount system [9]. This makes the policy holder weighs the interests which may not claim for compensation in the event of an accident; it will cause the claim number to be less than the number of accidents.
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