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

DOI10.1016/J.AMC.2022.127321zbMath1510.65120arXiv2206.10224WikidataQ113880709 ScholiaQ113880709MaRDI QIDQ6402695

Publication date: 21 June 2022

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.

Mathematics Subject Classification ID

Numerical methods involving duality (49M29) Numerical optimization and variational techniques (65K10) Variational inequalities (49J40) Optimal stochastic control (93E20) Numerical solutions to stochastic differential and integral equations (65C30) Numerical methods of relaxation type (49M20) Numerical methods for variational inequalities and related problems (65K15) Actuarial mathematics (91G05)

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