# American Institute of Mathematical Sciences

April  2019, 15(2): 517-535. doi: 10.3934/jimo.2018055

## Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model

 1 College of Economics and Business Administration, Chongqing University, Chongqing 400030, China 2 Department of Mathematics, Wayne State University, MI, USA, 48202

* Corresponding author: Manman Li

Received  August 2017 Revised  October 2017 Published  April 2018

Fund Project: The research of M. Li was supported in part by MOE Project of Humanities and Social Sciences on the west and the border area (No.14XJC910001) and the Fundamental Research Funds for the Central Universities (No.106112016CDJXY100002). The research of G. Yin was supported in part by the National Science Foundation under DMS-1207667.

This paper focuses on optimal threshold strategies for a spectrally negative Lévy (SNL) risk process with capital injections and proportional transaction costs. Restricted to solvency constraint, our model requires the shareholders of dividends prevent ruin by injecting capitals. Value function of the firm is assumed to be an expected discounted total [dividends less discounted capital injection]. Under such a setup, we derive certain key identities in connection with value function of the firm of a maximum dividend rate. Under restricted dividend rates and capital injection, we give analytical description of the maximum value function of the firm and the optimal threshold strategy explicitly.

Citation: Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial & Management Optimization, 2019, 15 (2) : 517-535. doi: 10.3934/jimo.2018055
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##### References:
The modified Lévy risk process
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