Linearisation Techniques and the Dual Algorithm for a Class of Mixed Singular/Continuous Control Problems in Reinsurance. Part I: Theoretical Aspects

TL;DR

This paper develops linearisation techniques and a dual dynamic programming algorithm for mixed singular/continuous control problems in reinsurance, focusing on theoretical foundations and structural considerations for implementation.

Contribution

It introduces a novel linearisation approach for complex control problems in reinsurance and connects it with dual dynamic programming and relaxation methods.

Findings

Linearisation effectively simplifies complex control problems.

Dual dynamic programming provides a practical solution approach.

Connections to moment sum of squares and LMI relaxations are established.

Abstract

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cram{\'e}r-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding the trajectories in an appropriate family of occupation measures, we provide the linearisation of such problems in which the continuous control is given by reinsurance policies and the singular one by dividends and capital injections. The linearisation translates into a dual dynamic programming (DDP) algorithm. An important part of the paper is dedicated to structural considerations allowing reasonable implementation. We also hint connections to methods relying on moment sum of squares and LMI (linear matrix inequality)-relaxations to approximate the optimal candidates.