# American Institute of Mathematical Sciences

September  2020, 16(5): 2563-2579. doi: 10.3934/jimo.2019070

## Optimal investment and risk control problems with delay for an insurer in defaultable market

 1 School of Finance, Guangdong University of Foreign Studies, 510006, Guangzhou, China 2 Institute of Big Data and Internet Innovation, Hunan University of Commerce, 410205, Changsha, China 3 Business School, Central South University, 410012, Changsha, China

* Corresponding author: Yan Chen

Received  July 2017 Revised  March 2019 Published  September 2020 Early access  July 2019

This paper addresses a investment and risk control problem with a delay for an insurer in the defaultable market. Suppose that an insurer can invest in a risk-free bank account, a risky stock and a defaultable bond. Taking into account the history of the insurer's wealth performance, the controlled wealth process is governed by a stochastic delay differential equation. The insurer's goal is to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. We decompose the original optimization problem into two subproblems: a pre-default case and a post-default case. The explicit solutions in a finite dimensional space are derived for a illustrative situation, and numerical illustrations and sensitivity analysis for our results are provided.

Citation: Chao Deng, Haixiang Yao, Yan Chen. Optimal investment and risk control problems with delay for an insurer in defaultable market. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2563-2579. doi: 10.3934/jimo.2019070
##### References:

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##### References:
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $k^{*}(t)$
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $\gamma^{*}(t)$
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal risk control $l^{*}(t)$
Value functions with respect to $x$
Effect of delay parameters $\beta$ on the pre-default value function
Effect of the default parameters $1/\Delta$ and $\zeta$ on the pre-default value function
Model parameter values
 Symbol Value Symbol Value $\alpha$ $0.1$ $\nu$ $1$ $u$ $5$ $\lambda$ $0.3$ $\beta$ $0.3$ $\theta$ $0.1$ $r$ $0.05$ $\eta$ $0.4$ $\zeta$ $0.5$ $p$ $1$ $\Delta$ $0.25$ $c$ $0.5$ $\mu$ $0.15$ $\sigma$ $0.2$
 Symbol Value Symbol Value $\alpha$ $0.1$ $\nu$ $1$ $u$ $5$ $\lambda$ $0.3$ $\beta$ $0.3$ $\theta$ $0.1$ $r$ $0.05$ $\eta$ $0.4$ $\zeta$ $0.5$ $p$ $1$ $\Delta$ $0.25$ $c$ $0.5$ $\mu$ $0.15$ $\sigma$ $0.2$
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