Optimal exchange rates management using stochastic impulse control for geometric Lévy processes

In this paper, we consider the optimal impulse control problem of a currency with exchange rate dynamics whose state follows a geometric Lévy process. The objective of the Central Bank is to keep the exchange rate as close as possible to a given target. Running costs associating with the difference between the actual exchange rate and the target are continuously incurred to the system. We suppose that, when the Central Bank intervenes in the system, it requires the fixed and proportional costs. Applying the theory of stochastic impulse controls, we find the optimal exchange rate at which interventions should be performed and the optimal sizes of the interventions, so as to minimize the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. Furthermore, numerical comparisons for the optimal intervention strategy between the geometric Lévy process model and the geometric Brownian motion model are investigated in details.