  This paper studies the optimal extraction and taxation of nonrenewable
natural resources. It is well known that the market values of the main
strategic resources such as oil, natural gas, uranium, copper,..., etc,
fluctuate randomly following global and seasonal macroeconomic parameters,
these values are modeled using Markov switching L\'evy processes. We formulate
this problem as a differential game. The two players of this differential game
are the mining company whose aim is to maximize the revenues generated from its
extracting activities and the government agency in charge of regulating and
taxing natural resources. We prove the existence of a Nash equilibrium. The
corresponding Hamilton Jacobi Isaacs equations are completely solved and the
value functions as well as the optimal extraction and taxation rates are
derived in closed-form. A Numerical example is presented to illustrate our
findings.
