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Our problem is motivated by an exchange rate control problem, where the control is composed of a direct impulsive intervention and an indirect, continuously acting intervention given by the control of the domestic interest rate. Similarly to Cadenillas and Zapatero (Math Financ 10:141–156, 2000) we formulate it as a mixed classical-impulse control problem. Analogously to Cadenillas and Zapatero (Math Financ 10:141–156, 2000), our approach builds on a quasi-variational inequality, which we consider here in a weakened version, and we too start by conjecturing the optimal solution to have a specific structure. While in Cadenillas and Zapatero (Math Financ 10:141–156, 2000) the horizon is infinite thus leading to a time-homogeneous solution and the value function is supposed to be of class $$\mathcal{C}^1$$ C 1 throughout, we have a finite horizon T and the value function is allowed not to be $$\mathcal{C}^1$$ C 1 at the boundaries of the continuation region. By suitably restricting the class of impulse controls, we obtain a fully analytical solution.
Applied Mathematics and Optimization – Springer Journals
Published: Nov 2, 2015
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