# American Institute of Mathematical Sciences

• Previous Article
A numerical scheme for pricing American options with transaction costs under a jump diffusion process
• JIMO Home
• This Issue
• Next Article
Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy
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:

show all references

##### 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 $x=0.5$
 [1] Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369 [2] Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021130 [3] Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161 [4] Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021046 [5] Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251 [6] Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021137 [7] Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020143 [8] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [9] Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 [10] Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021120 [11] Shan Liu, Hui Zhao, Ximin Rong. Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021015 [12] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080 [13] Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223 [14] Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 [15] Fengjun Wang, Qingling Zhang, Bin Li, Wanquan Liu. Optimal investment strategy on advertisement in duopoly. Journal of Industrial & Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625 [16] Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 [17] Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315 [18] Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461 [19] Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 [20] María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207

2020 Impact Factor: 1.801