Optimal Harvesting When the Exchange Rate Is a Semimartingale

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process , where is a semimartingale, and we ask the following question: What harvesting strategy and the value function maximize the expected total income of an investment? We formulate a singular stochastic control problem and give sufficient conditions for the existence of an optimal strategy. We found that, if the value function is not too sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and ？ksendal. However, the general solution of this problem still remains elusive. 1. Introduction This paper is concerned with an optimal harvesting strategy in the Black-Scholes Quanto Market when the exchange rate is being driven by a general semimartingale. Specifically, it is proposed that the optimal harvesting strategy can be found under certain conditions. The paper aims to make a contribution by deriving the general formula for an optimal harvesting strategy when the exchange rate is a semimartingale. This study could shed light on the application of general semimartingales in optimization of harvests from investments. Optimal harvesting is one of the crucial areas in finance because investment into stocks and bonds can be used as a source of revenue to expand business. Therefore, making investors happy through an optimal harvesting strategy could lead to more investments and consequently to further expansion of business. This study will make reference to dividend policy to illustrate an optimal harvesting strategy. Indeed a suboptimal dividend policy can result in destruction of shareholder confidence. How much to payout and still maintain growth of investments has been a challenge. For example, Miller and Modigilliani [1] claimed that a dividend policy was irrelevant in perfect markets because it had no impact on firm value. However, research on dividend policy that followed Miller and Modigilliani [1] has further examined various market imperfections and have identified the relevance of dividend policy. A number of stochastic models for optimal dividend policy can also be found in Taksar [2] and the references therein. A number of these models ([1, 3], etc.) have developed an optimal harvesting strategy as optimal stochastic control problems, and that is our approach in this paper. For example, Asmussen and Taksar [4] applied the theory of singular control in their study of a company’s optimal dividend policy that tries to maximize expected value of the total (discounted) payments to the shareholders. Asmussen