سود سهام مطلوب و مشکل صدور سهام با هزینه های معاملاتی متناسب و ثابت

This paper investigates the optimal dividend problem of an insurance company, which controls risk exposure by reinsurance and by issuing new equity to protect from bankruptcy. Transaction costs are incurred by these business activities: reinsurance is non-cheap, dividend is taxed and fixed costs are generated by equity issuance. The goal of the company is to maximize the expected cumulative discounted dividend minus the expected discounted costs of equity issuance. This problem is formulated as a mixed regular-singular-impulse stochastic control problem. By solving the corresponding HJB equation, we obtain the analytical solutions of the optimal return function and the optimal strategy.

Optimal dividend and reinsurance problems have attracted much attention in recent years. Two classical papers by Jeanblanc-Picqué and Shiryaev (1995) and Asmussen and Taksar (1997) studied the optimal dividend problem for an insurance company whose surplus process follows Brownian motion with drift. Højgaard and Taksar (1999) and Asmussen et al. (2000) explored the proportional reinsurance and excess-of-loss reinsurance respectively to maximize the expected present value of dividend in a diffusion approximation model. Choulli et al., 2001 and Choulli et al., 2003 investigated the optimal reinsurance and optimal dividend problems for a company with debt liability and constraints on risk reduction. When there were transaction costs for each dividend payment, Cadenillas et al. (2006) and Bai et al. (2010) studied the optimal dividend problem with proportional and excess-of-loss reinsurance, respectively. However, in most of these papers, ruin happens for an insurance company with probability 1. This is not practically interesting. It may be required by some Regulatory Authority that the insolvency company should issue equities or inject capital to protect the interests of policyholders. This idea may go back to Borch (1974, Chap. 20) and Harrison and Taylor (1978), and recent references on capital injections including Avram et al. (2007) for spectrally negative Lévy processes, Løkka and Zervos (2008), He and Liang (2009) and Meng and Siu (2011) for the diffusion model, Yao et al. (2011) for the dual model, Dai et al. (2010) and Avanzi et al. (2011) for the dual model with diffusion, and Kulenko and Schmidli (2008) and Scheer and Schmidli (2011) for the Cramér–Lundberg model. A detailed discussion of this topic on minimizing the expected discounted capital injections can be found in Eisenberg (2010).