  In the present work we employ, for the first time, backward stochastic
differential equations (BSDEs) to study the optimal control of semi-Markov
processes on finite horizon, with general state and action spaces. More
precisely, we prove that the value function and the optimal control law can be
represented by means of the solution of a class of BSDEs driven by a
semi-Markov process or, equivalently, by the associated random measure. The
peculiarity of the semi-Markov framework, with respect to the pure jump Markov
case, consists in the proof of the relation between BSDE and optimal control
problem. This is done, as usual, via the Hamilton-Jacobi-Bellman (HJB)
equation, which however in the semi-Markov case is characterized by an
additional differential term $\partial_a$. Taking into account the particular
structure of semi-Markov processes we rewrite the HJB equation in a suitable
integral form which involves a directional derivative operator $D$ related to
$\partial_a$. Then, using a formula of Ito type tailor-made for semi-Markov
processes and the operator $D$, we are able to prove that the BSDE provides the
unique classical solution to the HJB equation, which is shown to be the value
function of our control problem.
