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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.

Exchange rate can be described as the value of foreign nation’s currency in terms of home nation’s currency. Exchange rate policy is an important tool for the Central Bank in its quest to control volatility of the exchange rate. The Central Bank controls the volatility of the exchange rate by keeping it as close as possible to a prespecified target [

This research considers intervention by purchasing and selling reserves. The exchange rate should always be maintained within a band or interval around the target rate, determined by the country’s Central Bank [

The exchange rate will always have a tendency to move out of the target interval or target zone. The duty of the Central Bank is to come up with an optimum intervention strategy, that is, to determine the right time to intervene and the appropriate intervention size or amount. The Central Bank may experience very high costs when controlling the exchange rate. These high costs lead to failure to control the exchange rate and as a result the exchange rate may be characterised by fluctuations. Fluctuations create uncertainty in trade and arbitrage opportunities.

The theory of stochastic impulse control in controlling the exchange rate was first applied by Jeanblanc-Picque [

Korn [

The work in Mundaca and Øksendal [

Cadenillas and Zapatero [

Both Mundaca and Øksendal [

Huang [

Silva [

Perera [

From the above researches, empirical results are also disappointing regarding the ability to explain future exchange rate movements for currencies. The recent periods of turbulence in foreign exchange markets have renewed interest in the difficult task of identifying the optimal intervention times and sizes. A good example of a country with difficulties in explaining the future trends of their currency is Nigeria. Prior to the structural adjustment program (SAP), Naira enjoyed appreciable value against United States dollar, a factor that created rapid opportunity for economic growth and stability. With the introduction of new economic program, the country began to suffer unstable exchange rate that caused high degree of uncertainty in the Nigeria business environment [

This study is extending the model from Huang [

One major contribution is the formulation of quasivariational inequalities (QVI) that involve an integrodifferential equation which is not easy to solve. However, by carefully applying the method of undetermined coefficients, we managed to find an explicit solution for the impulse control problem. We also propose a new numerical scheme, which caters for a jump component in our model and some numerical analysis is done to illustrate the numerical scheme.

The rest of the paper is organised as follows. In Section

To place our discussion in a rigorous mathematical framework, we consider a probability space

Let

Motivated by the model in Huang [

Now, suppose that the Central Bank is allowed to intervene so as to control the exchange rate with the objective of keeping it as close to a prespecified target value

An impulse control

Suppose that the controlled exchange rate process with an initial value of

Define the performance functional

We take

The case

An impulse control

The problem is to find the value function

For a function

We define the Operator

One says that a function

Note that the solution

Let

This means that the Central Bank intervenes whenever

Let

Consider any admissible control

In this section we propose an impulse control of (

We now propose an optimal impulse control:

That is

The strategy indicates that the value function

Suppose that

Now to get the particular solution we consider

Note that the value function has properties (

The theorem below is used to prove the conjecture stated above.

Let

The proof is in two parts (a) and (b).

We show that

First inequality:

Second inequality:

Third inequality:

In this part we show that

In this section Newton’s method to solve the nonlinear system of ((

Let

The algorithm above gives us the results for

The authors declare that there is no conflict of interests regarding the publication of this paper.