Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment

Shan Liu; Hui Zhao; Ximin Rong

doi:10.3934/jimo.2021015

Article Contents

2022, Volume 18, Issue 2: 1185-1222. Doi: 10.3934/jimo.2021015

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Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment

School of Mathematics, Tianjin University, Tianjin 300350, China

^* Corresponding author: Ximin Rong
^* Corresponding author: Ximin Rong

Received: May 2020

Revised: November 2020

Early access: December 2020

Published: March 2022

This research was supported by National Natural Science Foundation of China (Grant Nos. 11771329, 11871052, 11971301), and Nature Science foundation of Tianjin city (20JCYBJC01160)

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Abstract

This paper considers an investment and reinsurance problem with a defaultable security for an insurer in an environment with parameter uncertainties. Suppose that the insurer is ambiguous about the insurance claims. Specifically, the insurance claim is exponentially distributed and the rate parameter is uncertain. The insurer is allowed to invest in a financial market consisting of a risk-free bond, a stock whose price process satisfies the Heston's SV model and a defaultable bond. Moreover, the insurer is allowed to purchase proportional reinsurance and aims to maximize the smooth ambiguity utility proposed in Klibanoff et al. [15]. By applying stochastic control approach, we establish the extended HJB system and derive the time-consistent investment-reinsurance strategy for the post-default case and the pre-default case, respectively. Finally, a sensitivity analysis is provided to illustrate the effects of model parameters on the equilibrium reinsurance-investment strategy under the smooth ambiguity.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 91G80, 91G10.

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Figure 1. Effects of $ k $ and $ \alpha $ on $ a^{*} $

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Figure 2. Effects of $ \nu_1 $ and $ q^{\nu}_{1} $ on $ a^{*} $

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Figure 3. Effects of $ \lambda $ and $ \theta $ on $ a^{*} $

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Figure 4. Effects of $ k $ and $ \xi $ on $ \pi_{s}^{*} $

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Figure 5. The Effect of the $ \rho $ on $ \pi_{s}^{*} $

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Figure 6. The Effect of parameter $ \gamma $ on $ \pi_{s}^{*} $

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Figure 7. The Effect of parameter $ \sigma $ on $ \pi_{s}^{*} $

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Figure 8. Effects of parameters $ \alpha, k, h^{\mathrm{P}}, \zeta $ and $ \delta $ on $ \pi^{*}_{p} $

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