Abstract
The aim of this paper is to calculate ruin probabilities using Monte Carlo method for two models: i) classical risk model with claim amounts are homogeneous Markov chains; ii) generalized risk models with premiums amounts, claim amounts are homogeneous Markov chains. The sequence of random variables in the article is considered as a series of Markov dependent random variables. The main results of this paper are Lemma 3.1, Lemma 3.2 and Lemma 3.3, which have built mathematical formulas for the simulation of the probability of insurance models considered in this paper. From those lemmas, we build algorithms to simulate ruin probability for insurance models considered in this paper. From these algorithms, we build numerical results illustrating the problems posed in the paper. These results all show that when the initial capital increases, the ruin probability will decrease, and when the time increases, the ruin probability will increase. This result is consistent with the theory of the risk problem in insurance.
Highlights
In risk theory, the premiums amount U(t) at time t: Nt (1.1)Ruin probability with infinite time, denoted (u), is defined by:(u) (u, ) lim (u, t) t (1.2)If there exists a number R > 0 satisfying eRx (1 F(x))dx r (1.3) with every u 0 we have (u) e Ru
This paper study two models: i)the claim amounts is an homogeneous Markov chain in classical model; ii) premiums amounts, claim amounts are homogeneous Markov chains in the general model does not have effects of interests
The sequence of random variables in the article is considered as a series of Markov dependent random variables
Summary
With every u 0 we have (u) e Ru. If eRx (1 F(x))dx lim eRu (u) C u (1.4). Gerber [3] and Grandell [4]) For these dependency structure models, it would often be very hard to calculate the approximation of exponential constant R (see Phung Duy Quang [5], [6]). In [1], authors considered the numerical solution to one type of integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. In [2], authors studied based on a discrete version of the Pollaczeck–Khinchine formula, a general method of calculating the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution.
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