# On Classical and Restricted Impulse Stochastic Control for the Exchange Rate

On Classical and Restricted Impulse Stochastic Control for the Exchange Rate 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# On Classical and Restricted Impulse Stochastic Control for the Exchange Rate

, Volume 74 (2) – Nov 2, 2015
32 pages

/lp/springer_journal/on-classical-and-restricted-impulse-stochastic-control-for-the-KoH1lQQHBd
Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9320-6
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Nov 2, 2015

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