Optimal Dividends and Capital Injections in the Dual Model with Diffusion

28 Pages Posted: 15 Nov 2010  

Benjamin Avanzi

UNSW Australia Business School, School of Risk and Actuarial Studies; University of Montreal - Department of Mathematics and Statistics

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: November 15, 2010

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) transactions costs both on dividends and capital injections.

We determine the optimal dividend and (unconstrained) capital injection strategy 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. These results can be used to approximate the optimal strategies' parameters and value functions when jumps follow an arbitrary distribution. Examples are provided.

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 (November 15, 2010). UNSW Australian School of Business Research Paper No. 2010ACTL15. Available at SSRN: https://ssrn.com/abstract=1709174 or http://dx.doi.org/10.2139/ssrn.1709174

Benjamin Avanzi (Contact Author)

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

UNSW Sydney, NSW 2052
Australia

University of Montreal - Department of Mathematics and Statistics ( email )

Montreal, Quebec H3C 3J7
Canada
+1 514 343 6695 (Phone)

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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