Banach Contraction Principle and ruin probabilities in regime-switching models

We apply Banach Contraction Principle to approximate a vector Ψ of ruin probabilities in regime-switching models. A Markov chain is interpreted as a 'switch' that changes the amount and/or wait time distributions of claims. The insurer has a possibility to adapt the premium rates in response. An associated risk operator L is proven to be a contraction on a properly chosen complete metric space while Ψ is shown to be the unique fixed point of L within this space. Thus, by iterating L on any of its points, we can simultaneously approximate Ψ and control the error of approximation. Numerical examples confirm high accuracy of the resulting procedure.