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International Journal of Stochastic Analysis
Volume 2014, Article ID 746815, 8 pages
http://dx.doi.org/10.1155/2014/746815
Research Article

Optimal Foreign Exchange Rate Intervention in Lévy Markets

1Department of Applied Mathematics, National University of Science and Technology, P.O. Box AC 939, Ascot, Bulawayo, Zimbabwe
2Department of Mathematics and Computer Science, Great Zimbabwe University, P.O. Box 1235, Masvingo, Zimbabwe

Received 27 March 2014; Accepted 8 September 2014; Published 26 November 2014

Academic Editor: Agnès Sulem

Copyright © 2014 Masimba Aspinas Mutakaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers an exchange rate problem in Lévy markets, where the Central Bank has to intervene. We assume that, in the absence of control, the exchange rate evolves according to Brownian motion with a jump component. The Central Bank is allowed to intervene in order to keep the exchange rate as close as possible to a prespecified target value. The interventions by the Central Bank are associated with costs. We present the situation as an impulse control problem, where the objective of the bank is to minimize the intervention costs. In particular, the paper extends the model by Huang, 2009, to incorporate a jump component. We formulate and prove an optimal verification theorem for the impulse control. We then propose an impulse control and construct a value function and then verify that they solve the quasivariational inequalities. Our results suggest that if the expected number of jumps is high the Central Bank will intervene more frequently and with large intervention amounts hence the intervention costs will be high.