  We study the following backward stochastic differential equation on finite
time horizon driven by an integer-valued random measure $\mu$ on $\mathbb
R_+\times E$, where $E$ is a Lusin space, with compensator
$\nu(dt,dx)=dA_t\,\phi_t(dx)$: \[ Y_t = \xi + \int_{(t,T]}
f(s,Y_{s-},Z_s(\cdot))\, d A_s - \int_{(t,T]} \int_E Z_s(x) \,
(\mu-\nu)(ds,dx),\qquad 0\leq t\leq T. \] The generator $f$ satisfies, as
usual, a uniform Lipschitz condition with respect to its last two arguments. In
the literature, the existence and uniqueness for the above equation in the
present general setting has only been established when $A$ is continuous or
deterministic. The general case, i.e. $A$ is a right-continuous nondecreasing
predictable process, is addressed in this paper. These results are relevant,
for example, in the study of control problems related to Piecewise
Deterministic Markov Processes (PDMPs). Indeed, when $\mu$ is the jump measure
of a PDMP, then $A$ is predictable (but not deterministic) and discontinuous,
with jumps of size equal to 1.
