# American Institute of Mathematical Sciences

October  2015, 11(4): 1247-1262. doi: 10.3934/jimo.2015.11.1247

## Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs

 1 School of Statistics, Qufu Normal University, Shandong 273165, China 2 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  April 2014 Revised  February 2015 Published  March 2015

In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.
Citation: Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247
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