Optimal Harvesting When the Exchange Rate Is a Semimartingale

Abstract

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process 𝐸𝑡=𝐸0exp{𝑋𝑡}, where 𝑋𝑡 is a semimartingale, and we ask the following question: What harvesting strategy 𝛾∗ and the value function Φ 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 Øksendal. However, the general solution of this problem still remains elusive.