Optimal Dividends and Capital Injections in the Dual Model with Diffusion

ASTIN Bulletin, Vol. 41, No. 2, pp. 611-644

25 Pages Posted: 25 Aug 2011 Last revised: 6 Jan 2012

See all articles by Benjamin Avanzi

Benjamin Avanzi

UNSW Australia Business School, School of Risk and Actuarial Studies

Jonathan Shen

University of New South Wales (UNSW) - School of Actuarial Studies

Bernard Wong

UNSW Australia Business School, School of Risk & Actuarial Studies

Multiple version iconThere are 2 versions of this paper

Date Written: August 24, 2011

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 model, diffusion, dividends, capital injections, HJB equation

JEL Classification: C44, C61, G24, G32, G35

Suggested Citation

Avanzi, Benjamin and Shen, Jonathan and Wong, Bernard, Optimal Dividends and Capital Injections in the Dual Model with Diffusion (August 24, 2011). ASTIN Bulletin, Vol. 41, No. 2, pp. 611-644. Available at SSRN: https://ssrn.com/abstract=1915711

Benjamin Avanzi (Contact Author)

UNSW Australia Business School, School of Risk and Actuarial Studies ( email )

UNSW Sydney, NSW 2052
Australia

Jonathan Shen

University of New South Wales (UNSW) - School of Actuarial Studies ( email )

Sydney, NSW 2052
Australia

Bernard Wong

UNSW Australia Business School, School of Risk & Actuarial Studies ( email )

Room 2058 South Wing 2nd Floor
Quadrangle building, Kensington Campus
Sydney, NSW 2052
Australia

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