Stochastic successive approximation method for assessing the insolvency risk of an insurance company
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STOCHASTIC SUCCESSIVE APPROXIMATION METHOD FOR ASSESSING THE INSOLVENCY RISK OF AN INSURANCE COMPANY
UDC 519.21
B. V. Norkin
A stochastic successive approximation method is analyzed with a view to solving risk assessment problems that are reduced to a renewal integral equation and, in particular, to assessing the insolvency risk of an insurance company. Integrals in the equation are evaluated approximately, for example, by the Monte Carlo method. Iterations of the method are proved to converge uniformly with probability one. Theoretical results are illustrated by numeral computations. Keywords: risk process, ruin probability, terminating renewal process, integral renewal equation, method of successive approximations, Monte-Carlo method. INTRODUCTION This work is devoted to methods of analytical-statistical estimation (modifications of the Monte-Carlo method) of ruin (termination) probability of the so-called risk process [1–8] that describes the stochastic evolution of the capital of an insurance company. In the direct Monte-Carlo method, trajectories of a risk process are modelled and the portion of trajectories leading to ruin [2] is calculated. This method is general but does not provide sufficient accuracy in the case of small probabilities of ruin. Analytical methods and estimates [1–8] of ruin probability provide exact or good approximate values but are applicable to a very limited number of cases. Analytical-statistical methods combine the Monte-Carlo method and some analytical (formulary) information or other on a process [9–11]. In the present article, two such approaches are mainly considered. The first approach is based on the Polachek–Khinchin ruin probability formula [1, 5, 6] in which the ruin probability is represented as a function of the opening capital in the form of an infinite sum with decreasing coefficients of convolutions of some auxiliary distribution function. In this method, the convolutions are statistically estimated and then are summarized with the coefficients of the analytical representation. To obtain a point ruin probability estimate (for a fixed opening capital), a similar method was applied in [12–15], and earlier it was used in [16] for reliability estimation. We note that, as distinct from [13], the explicit form of requirement distribution was not used in [12–15] but its nonparametric (empirical) estimate was applied. In point estimation (for fixed values of the opening capital), the convergence of the method follows with probability one from the law of large numbers. Of course, this is also true for the collection of any finite number of estimation points. In the present article, this result is somewhat strengthened, namely, the uniform (for a continuum of values of the opening capital) convergence with probability one is proved for the method. To prove the convergence, the Glivenko–Kantelli theorem [17] on the uniform convergence of empirical distribution functions is used, and the generalized Glivenko–Kantelli theorem [17, Appendix 1] on the uniform law of large numbers o
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