  In this paper, we guarantee the existence and uniqueness (in the almost
everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation
with gradient constraint and a partial integro-differential operator whose
L\'evy measure has bounded variation. This type of equation arises in a
singular control problem, where the state process is a multidimensional
jump-diffusion with jumps of finite variation and infinite activity. We verify,
by means of {\epsilon}-penalized controls, that the value function associated
with this problem satisfies the aforementioned HJB equation.
