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Optimal Dividends and Capital Injections in the Dual Model with Diffusion

Published online by Cambridge University Press:  09 August 2013

Benjamin Avanzi ,
Jonathan Shen  and
Bernard Wong
Jonathan Shen
Affiliation:
School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney NSW 2052, Australia, E-mail: j.shen@unsw.edu.au
Bernard Wong
Affiliation:
School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney NSW 2052, Australia, E-mail: bernard.wong@unsw.edu.au
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Abstract

The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, Avanzi and Gerber (2008) showed how to determine the expected present value of dividends, if a barrier strategy is followed. In this paper, we further include capital injections and allow for (proportional) transaction costs both on dividends and capital injections.

We determine the optimal dividend and (unconstrained) capital injection strategy (among all possible strategies) when jumps are hyperexponential. This strategy happens to be either a dividend barrier strategy without capital injections, or another dividend barrier strategy with forced injections when the surplus is null to prevent ruin. The latter is also shown to be the optimal dividend and capital injection strategy, if ruin is not allowed to occur. Both the choice to inject capital or not and the level of the optimal barrier depend on the parameters of the model.

In all cases, we determine the optimal dividend barrier and show its existence and uniqueness. We also provide closed form representations of the value functions when the optimal strategy is applied. Results are illustrated.

Keywords

Dual modeldiffusiondividendscapital injectionsHJB equation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 2 , November 2011 , pp. 611 - 644
Copyright
Copyright © International Actuarial Association 2011

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