Optimal Investment and Reinsurance for Insurers with Uncertain Time-Horizon

Ailing Gu,Dezhu Ye,Bo Yi

doi:10.1155/2014/271930

Abstract

This paper considers the investment-reinsurance problems for an insurer with uncertain time-horizon in a jump-diffusion model and a diffusion-approximation model. In both models, the insurer is allowed to purchase proportional reinsurance and invest in a risky asset, whose expected return rate and volatility rate are both dependent on time and a market state. Meanwhile, the market state described by a stochastic differential equation will trigger the uncertain time-horizon. Specifically, a barrier is predefined and reinsurance and investment business would be stopped if the market state hits the barrier. The objective of the insurer is to maximize the expected discounted exponential utility of her terminal wealth. By dynamic programming approach and Feynman-Kac representation theorem, we derive the expressions for optimal value functions and optimal investment-reinsurance strategies in two special cases. Furthermore, an example is considered under the diffusion-approximation model, which shows some interesting results.

Highlights

Investment and reinsurance are two important ways for insurers to balance their profit and risk
For each t ∈ [0, τ], let b(t) be the proportion of total assets invested in the risky asset at time t
By dynamic programming approach and Feynman-Kac representation, we derive the expression for optimal investment and reinsurance strategy

Summary

Introduction

Investment and reinsurance are two important ways for insurers to balance their profit and risk. The problem of optimal investment and/or reinsurance for insurers has been extensively studied in the literature. The death of investors or some defaultable security defaults maybe terminate the horizon immediately Based on these, it is of theoretical and practical interest to study optimal investment strategies under uncertain time-horizon. Research on this subject has been literally discussed by some researchers in last decades: Karatzas and Wang [13] maximize the expected discount utility from consumption and the terminal wealth, assuming that the time-horizon is a stopping time with respect to (w.r.t.) an asset price; Kraf and Steffensen [14] study an optimal investment and consumption problem and introduce an arithmetic Brownian motion to trigger the exiting time.

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Conclusion