Robust equilibrium investment and reinsurance strategy with bounded memory and common shock dependence

Sheng Li

doi:10.1051/ro/2021182

Abstract

In this paper, we consider the robust investment and reinsurance problem with bounded memory and risk co-shocks under a jump-diffusion risk model. The insurer is assumed to be ambiguity-averse and make the optimal decision under the mean-variance criterion. The insurance market is described by two-dimensional dependent claims while the risky asset is depicted by the jump-diffusion model. By introducing the performance in the past, we derive the wealth process depicted by a stochastic delay differential equation (SDDE). Applying the stochastic control theory under the game-theoretic framework, together with stochastic control theory with delay, the robust equilibrium investment-reinsurance strategy and the corresponding robust equilibrium value function are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment and reinsurance strategy.

Highlights

In the insurance market, insurers improve the economic efficiency of the system by spreading individual risks
We assume that the insurer can invest his/her wealth in a risk-free asset and a risky asset satisfying jump-diffusion process
We provide a numerical example to analyze the effects of delay parameters and risk dependent parameter on robust equilibrium strategy and explain why such effects occur

Summary

Introduction

Insurers improve the economic efficiency of the system by spreading individual risks. There has been an increased interest in finding a time-consistent equilibrium strategy for the mean-variance investment and reinsurance problem. We formulate a robust optimization problem with alternative models and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. We derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function. The main contributions of this paper are as follows: (1)We incorporate past performance into the robust investment-reinsurance problem where the insurer’s optimal decision in the worst-case model is based on a weighting of past and present information.

General formulation

The solution to the optimization problem

Sensitivity analysis

Conclusion