Worst-Case Investment and Reinsurance Optimization for an Insurer under Model Uncertainty

Xiangbo Meng,Ziping Du,Ximin Rong,Lidong Zhang

doi:10.1155/2016/9693419

Abstract

In this paper, we study optimal investment-reinsurance strategies for an insurer who faces model uncertainty. The insurer is allowed to acquire new business and invest into a financial market which consists of one risk-free asset and one risky asset whose price process is modeled by a Geometric Brownian motion. Minimizing the expected quadratic distance of the terminal wealth to a given benchmark under the “worst-case” scenario, we obtain the closed-form expressions of optimal strategies and the corresponding value function by solving the Hamilton-Jacobi-Bellman (HJB) equation. Numerical examples are presented to show the impact of model parameters on the optimal strategies.

Highlights

In recent years, insurance companies are playing the more active role in the financial market
Throughout this paper, we work on a filtered complete probability space (Ω, F, P, {Ft}0≤t≤T), where P is a reference probability measure from which a family of real-world probability measures absolutely continuous with respect to P are generated; {Ft}0≤t≤T is the filtration generated by all Brownian motions standing for the information available up to time t
We find that (i) under model uncertainty the optimal policy θ∗ = (π∗, a∗) depends on the volatility of both the insurer’s surplus and the risky asset; (ii) if we do not take account of the model risk, the corresponding optimal investment-reinsurance strategies are different: the volatility of the insurer’s surplus has no influence on the investment strategy π(t) (see Appendix (A.11)), and the volatility of the risky asset has no influence on the reinsurance strategy ã(t) (see Appendix (A.12))

Summary

Introduction

Insurance companies are playing the more active role in the financial market. Lin et al [17] discussed an optimal portfolio selection problem of an insurer who faces model uncertainty and obtained closed-form solutions to the game problems in both the jump diffusion risk process and its diffusion approximation for the case of an exponential utility by using techniques of stochastic linear-quadratic control. The insurer’s objective is to find an optimal investment-reinsurance strategy under the criterion of minimizing the expected quadratic distance of the terminal wealth to the given benchmark in the “worst-case” scenario. Different from those in Zhang and Siu [16], we propose here a mean-variance portfolio optimization problem under model uncertainty.

Model Formulation

Numerical Results

Conclusion