Numerical Computation of Ruin Probability Using Hybrid Extreme Learning Machine And Whale Optimization Algorithm

Abstract

Accurate estimation of ruin probabilities remains a fundamental challenge in actuarial risk theory, particularly when claim-size distributions do not admit closed form solutions. In such cases, numerical and approximation techniques are essential for evaluating infinite-time ruin probabilities in classical surplus processes. In this study, we propose a hybrid learning framework that integrates an Improved Extreme Learning Machine (IELM) with the Whale Optimization Algorithm (WOA). While ELM-based models offer fast training, their performance is sensitive to random initialization of hidden-layer parameters and network configuration. Incorporating WOA provides a structured global search to optimize hidden-layer selection and parameter initialization, thereby enhancing stability and predictive accuracy. The ruin probability satisfies a renewal-type integro– differential equation, which is approximated using a trigonometric neural network representation. Convolution terms in the renewal structure are computed via numerical quadrature, allowing the framework to handle general claim size distributions without restrictive assumptions. Optimization minimizes the mean squared approximation error, guiding WOA to identify network configurations that yield accurate and stable estimates. We validate the approach through numerical experiments under exponential, Weibull, and Pareto claim-size distributions. Across all scenarios, the hybrid ELM–WOA model consistently outperforms the standard ELM with random initialization, achieving lower mean absolute error (MAE) and root mean square error (RMSE) while maintaining computational efficiency. These results demonstrate that coupling neural-network approximation with metaheuristic optimization offers a robust and practical alternative for computing ruin probabilities in complex actuarial risk models, particularly where analytical solutions are unavailable.