This thesis investigates several topics in actuarial mathematics and applied probability, including insurance valuation and reserving, regularisation of regression trees, and stochastic optimisation in an extended dividend problem. The thesis is based on four papers. 

Paper I provides a justification of the chain ladder predictor and Mack’s estimator for the prediction error within a classical compound Poisson model under large exposure, that is, when the number of contracts tends to infinity. Although the model does not satisfy the assumptions of Mack’s distribution-free chain ladder, both the predictor and the estimator are shown to arise in the large exposure limit.

Paper II studies the valuation of liability cashflows with capital requirements in a multi-period setting. Since explicit valuation is generally infeasible and Monte Carlo methods are often computationally challenging, an explicit and easily computable valuation formula is derived. The formula is obtained as a large exposure limit under a conditional weak convergence assumption on the liability cashflows.

Paper III introduces a regularisation method for regression trees based on node-wise statistical tests. At each node, a p-value is computed using a change point test, resulting in a regularised regression tree that is a deterministic function of the training data. Unlike cross-validation, the method avoids randomness from data splitting and ensures efficient use of the full dataset.

Paper IV revisits the classical dividend problem with ruin at zero by incorporating an additional default mechanism based on cumulative occupation time in a low-surplus region. This extension reflects realistic default triggers such as regulatory pressure or liquidity stress. The problem is solved explicitly, yielding closed-form expressions for both the optimal control and the value function.