A stochastic differential game for optimal investment of an insurer with regime switching

Abstract

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.

Keywords:

• Optimal investment
• Insurance company
• Markov regime-switching models
• Model uncertainty
• Insurance claim process
• Stochastic differential games
• Exponential utility
• Survival probability
• Dynamic programming
• HJB equations

Acknowledgements

The authors wish to thank the two anonymous referees for their comments and suggestions. We would also like to thank the ARC and the NSERC for support.

Notes

†Here the market is interpreted in a broad sense to include both financial and insurance markets. It is a ‘fictitious’ player in the games.

‡The liquidity of an asset may change when the economic situation changes. For example, it may be more difficult to liquidate an asset in a ‘Bad’ economy than in a ‘Good’ economy. Therefore, it may be interesting to incorporate the presence of liquidity risk in a regime-switching environment, where the state of the economy is allowed to vary over time. This represents an interesting topic for further research.

†A random measure γ(·, ·) is a family {γ(·, ·, ω) | ω ∈ Ω} of σ-finite measures on (𝒯 × 𝒵, ℬ(𝒯) × ℬ(𝒵)) (see, for example, Elliott (1982, chapter 15)), where ℬ(A) is the Borel σ-field generated by open subsets A of ℜ. To simplify the notation, we suppress ω and write γ(·, ·) for γ(·, ·, ω).

†The notion of a conditional Poisson process given the sample path of a Markov chain was considered by Elliott (1990).