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October  2017, 13(4): 1771-1791. doi: 10.3934/jimo.2017018

## Minimizing expected time to reach a given capital level before ruin

 1 School of Sciences, Hebei University of Technology, Tianjin 300401, China 2 School of Mathematical Sciences, Nankai University, Tianjin 300071, China

* Corresponding author: Lihua Bai

Received  November 2015 Published  December 2016

Fund Project: The first author is supported by the National Natural Science Foundation of China (11571189),the project RARE -318984 (an FP7 Marie Curie IRSES) and High School National Science Foundation of Hebei Province (QN2016176), and the second author is supported by the National Natural Science Foundation of China (11471171).

In this paper, we consider the optimal investment and reinsurance problem for an insurance company where the claim process follows a Brownian motion with drift. The insurer can purchase proportional reinsurance and invest its surplus in one risky asset and one risk-free asset. The goal of the insurance company is to minimize the expected time to reach a given capital level before ruin. By using the Hamilton-Jacobi-Bellman equation approach, we obtain explicit expressions for the value function and the optimal strategy. We also provide some numerical examples to illustrate the results obtained in this paper, and analyze the sensitivity of the parameters.

Citation: Xiaoqing Liang, Lihua Bai. Minimizing expected time to reach a given capital level before ruin. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1771-1791. doi: 10.3934/jimo.2017018
##### References:

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##### References:
The minimal expected time and the associated optimal strategies for $\sigma=0.1$.
The minimal expected time and the associated optimal strategies for $\sigma=0.01$.
The minimal expected time and the associated optimal strategies for $b=0.03$.
The minimal expected time and the associated optimal strategies for $b=0.3$.
Expected time vs goal for $x=0.5$
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