Optimal Dividends and Capital Injections in the Dual Model with Diffusion

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.

[1]  Benjamin Avanzi,et al.  Optimal Dividends in the Dual Model with Diffusion , 2008 .

[2]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[3]  Christian Mazza,et al.  A link between wave governed random motions and ruin processes , 2004 .

[4]  Anja Feldmann,et al.  Fitting Mixtures of Exponentials to Long-Tail Distributions to Analyze Network , 1998, Perform. Evaluation.

[5]  K. Ortolon.  Economic survival. , 2007, Texas medicine.

[6]  Evan L. Porteus.  On Optimal Dividend, Reinvestment, and Liquidation Policies for the Firm , 1977, Oper. Res..

[7]  Benjamin Avanzi.  Strategies for Dividend Distribution: A Review , 2009 .

[8]  Lin He,et al.  Optimal financing and dividend control of the insurance company with proportional reinsurance policy , 2008 .

[9]  E. Prescott,et al.  Investment Under Uncertainty , 1971 .

[10]  Anja Feldmann,et al.  Fitting mixtures of exponentials to long-tail distributions to analyze network performance models , 1997, Proceedings of INFOCOM '97.

[11]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[12]  Mihail Zervos,et al.  Optimal dividend and issuance of equity policies in the presence of proportional costs , 2008 .

[13]  Hans U. Gerber Asa,et al.  On The Merger Of Two Companies , 2006 .

[14]  Moshe Sidi,et al.  Modeling and analysis of power-tail distributions via classical teletraffic methods , 2000, Queueing Syst. Theory Appl..

[15]  Hansjörg Albrecher,et al.  Optimality results for dividend problems in insurance , 2009 .

[16]  Harvey E. Lapan,et al.  The Mathematical Theory of Insurance. , 1975 .

[17]  Hansjörg Albrecher,et al.  A Direct Approach to the Discounted Penalty Function , 2010 .

[18]  Florin Avram,et al.  On the optimal dividend problem for a spectrally negative Lévy process , 2007, math/0702893.

[19]  K. Miyasawa.  AN ECONOMIC SURVIVAL GAME , 1961 .

[20]  Hailiang Yang,et al.  Optimal financing and dividend strategies in a dual model with proportional costs , 2010 .

[21]  Andrew Cheuk-Yin Ng,et al.  On a dual model with a dividend threshold , 2009 .

[22]  Eric C. K. Cheung,et al.  Dividend Moments in the Dual Model : Exact and Approximate Approaches , 2008 .

[23]  Erhan Bayraktar,et al.  Optimizing venture capital investments in a jump diffusion model , 2008, Math. Methods Oper. Res..

[24]  Daniel Dufresne,et al.  Fitting combinations of exponentials to probability distributions , 2007 .

[25]  Hongshuai Dai,et al.  Optimal dividend strategies in a dual model with capital injections , 2010, Math. Methods Oper. Res..

[26]  H. Gerber,et al.  Optimal dividends with incomplete information in the dual model , 2008 .

[27]  Hans U. Gerber,et al.  Games of Economic Survival with Discrete- and Continuous-Income Processes , 1972, Oper. Res..

[28]  Hanspeter Schmidli,et al.  Optimal dividend strategies in a Cramér–Lundberg model with capital injections , 2008 .