# American Institute of Mathematical Sciences

July  2018, 14(3): 1179-1201. doi: 10.3934/jimo.2018005

## A threshold-based risk process with a waiting period to pay dividends

 1 Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada 2 School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Australia 3 Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

* Corresponding author: Jae-Kyung Woo

Received  April 2016 Revised  August 2017 Published  January 2018

Fund Project: This work has been supported by the Natural Sciences and Engineering Research Council of Canada, through the Discovery grant (#238675-2010-RGPIN) of Dr. Drekic.

In this paper, a modified dividend strategy is proposed by delaying dividend payments until the insurer's surplus level remains at or above a threshold level b for a predetermined period of time h. We consider two cases depending on whether the period of time sustained at or above level b is counted either consecutively or accumulatively (referred to as standard or cumulative waiting period). In both cases, we develop a recursive computational procedure to calculate the expected total discounted dividend payments made prior to ruin for a discrete-time Sparre Andersen renewal risk process. By varying the values of b and h, a numerical study of the trade-off effects between finite-time ruin probabilities and expected total discounted dividend payments is investigated under a variety of scenarios. Finally, a generalized threshold-based strategy with a delayed dividend payment rule is studied under the compound binomial model.

Citation: Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005
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##### References:
Illustration of the threshold-based dividend strategy: Standard waiting period
Illustration of the threshold-based dividend strategy: Cumulative waiting period
Layering of the recursive algorithm for $V_i(u, m)$
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 1
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 2
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 3
Plot of $V(10, 80)$ against $(h, b)$ under Distribution 2
Plot of $V(10, 80)$ against $\psi(10, 80)$ under Distribution 2
Convergence of $V(10, m)$ under Distribution 2
Illustration of the generalized threshold-based dividend strategy
Plot of $V(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
Plot of $\psi(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
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