Robust optimal investment and reinsurance of an insurer under Jump-diffusion models

Xin Zhang; Hui Meng; Jie Xiong; Yang Shen

doi:10.3934/mcrf.2019003

Article Contents

2019, Volume 9, Issue 1: 59-76. Doi: 10.3934/mcrf.2019003

This issue Previous Article Decay rates for stabilization of linear continuous-time systems with random switching Next Article On Algebraic condition for null controllability of some coupled degenerate systems

Robust optimal investment and reinsurance of an insurer under Jump-diffusion models

1.
School of Mathematics, Southeast University, Nanjing, Jiangsu Province, 211189, China
2.
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China
3.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong Province, 518055, China
4.
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

^* Corresponding author: Xin Zhang
^* Corresponding author: Xin Zhang

Received: July 31, 2017

Revised: March 31, 2018

Published: January 2019

X. Zhang is supported by the National Natural Science Foundation of China (grant nos. 11771079, 11371020). H. Meng is supported by the National Natural Science Foundation of China (grant no. 11771465), the Program for Innovation Research in Central University of Finance and Economics, and the 111 Project (grant no. B17050). J. Xiong is supported by Southern University of Science and Technology startup fund (grant No. 28/Y01286120). Y. Shen is supported by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2016-05677).

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Abstract

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

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Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 91A15.

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Figure 1. Effects of $\lambda$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$

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Figure 2. Effects of $q$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$

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Figure 3. Effects of $\beta_1$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$

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Figure 4. Effects of $\beta_2$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$

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