精算数学中求解积分方程的逐次逼近方法
Method of successive approximations for solving integral equations of actuarial mathematics
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中文总结 AI 辅助
该论文针对精算数学中推广的风险过程的积分方程，提出并验证了逐次逼近方法，可提升破产概率计算精度，用于与Monte Carlo方法等对比验证。
中文摘要 AI 辅助
本文研究了描述保险公司资本随机演化的经典风险过程（Cramer Lundberg模型）的多种推广形式，包括带有可变确定或随机保费的过程、非泊松型保费与索赔流的过程，以及随机马尔可夫环境下的过程。针对这些推广形式，推导了作为初始资本函数的不破产概率的积分方程；对于随机马尔可夫环境中的过程，得到了对应不同初始状态的不破产概率的积分方程组。建立了解存在性与唯一性的一般充要条件及具体充分条件。对该积分方程的数值与解析解的逐次逼近方法进行了理论与实践验证，确定了其一致收敛性与收敛速率；通过构造精确解的上下近似，开发了近似解的精度估计技术。将该方法在数值算例上进行测试，并与Monte Carlo方法及已知的近似解进行比较。所开发的方法提升了精算计算的精度：它可按任意规定精度计算破产概率，迭代验证并改进经验近似，还可估计并修正Monte Carlo模拟的精度与参数。
英文摘要
The dissertation considers various generalizations of the classical risk process describing the stochastic evolution of the capital of an insurance company (Cramer Lundberg model), including processes with variable deterministic or random premiums, non Poisson flows of premiums and claims, and a stochastic Markovian environment. Integral equations for the probability of nonruin as a function of initial capital are derived for these generalizations. For processes in a stochastic Markovian environment, systems of integral equations for nonruin probabilities corresponding to different initial states are obtained. General necessary and sufficient, as well as specific sufficient, conditions for the existence and uniqueness of solutions are established. A successive approximation method for numerical and analytical solution of the integral equations is theoretically and practically validated; its uniform convergence and rate of convergence are established. A technique for estimating the accuracy of approximate solutions is developed by constructing upper and lower approximations to the exact solution. The method is tested on numerical examples and compared with the Monte Carlo method and known solution approximations. The developed method increases the accuracy of actuarial calculations: it allows the probability of ruin to be calculated with any prescribed accuracy, empirical approximations to be verified and improved iteratively, and the accuracy and parameters of Monte Carlo simulations to be estimated and corrected.
发表机构
- National Academy of Sciences of Ukraine（乌克兰国家科学院）
- V.M. Glushkov Institute of Cybernetics（V.M.格鲁什科夫控制论研究所）
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