Сalculation of ruin probability in discrete risk model with dependent insurance and financial risks
Last decades there is a discussion between leading specialists in risk theory, financial mathematics and queuing theory (S. As-mussen, W. Whitt, V. Kalashnikov and A. Novikov) about an accuracy of asymptotic formulas in stochastic models with heavy tailed distributions. Manifold numerical experiments showed that asymptotic formulas reach high accuracy only for very large meanings of arguments. The Monte-Carlo method works efficiently only for sufficiently small arguments. As a result an area of mid arguments appears. In this area known numerical methods do not work because of their complexity and asymptotic formulas do not work because of their small accuracy. To make calculations in this area A. Novikov suggested to approximate distributions with heavy tails by finite mixtures of exponential distributions. This approach demands to calculate a probability of autoregressive sequence to exit from some domain in a finite number of steps. A. Novikov solved this problem using martingale methods. But these methods work in sufficiently narrow suggestions. In this paper, the authors discard these restrictions for discrete time risk model with financial risks and heavy tailed insurance risks dependent on them. An inflation factor is represented by Markov chain with finite set of states and insurance risk has a distribution depending of this chain. Such complication of risk model is suggested by A. Novikov and is not considered yet by specialists of risk and financial mathematics. But last events showed that a dependence between insurance and financial risks is caused by natural and anthropogenic catastrophes with sizes depending on financial factors significantly. Recurrent algorithm of ruin probability calculation in risk model under finite horizon is based on an approximation of loss distributions by finite mixtures of exponential distributions and on continuity theorems. For this aim special methods of a transformation of integral relations are constructed and a problem of small denominators connected with these methods is solved. An efficient procedure of an enumeration of vectors with fixed sums of non negative integer components is constructed. On a base of this algorithm we performed numerical experiment which showed its advantage in an efficiency as compared to Monte-Carlo method with practically identical results. Some numerical results with the chosen parameters of the model (for an example with unit capital) are represented below. Steps number Methods Ruin probability 3 Monte-Carlo 0.118499 3 Recurrent formula 0.109308 4 Monte-Carlo 0.130707 4 Recurrent formula 0.121714 5 Monte-Carlo 0.139442 5 Recurrent formula 0.130206 We compare the calculation of ruin probability by accuracy formulas and by asymptotic formulas in a case when insurance losses have subexponential distributions. Below some results of numerical experiments are represented. Initial capital Formula Ruin probability 1000 Asymptotic 0.000056666 1000 Recurrent 0.0000563073 3000 Asymptotic 0.0000151627 3000 Recurrent 0.0000151122 5000 Asymptotic 8.21407x10" <sup>6</sup> 5000 Recurrent 8.56365x10" <sup>6</sup>