  We are interested in stochastic control problems coming from mathematical
finance and, in particular, related to model uncertainty, where the uncertainty
affects both volatility and intensity. This kind of stochastic control problems
is associated to a fully nonlinear integro-partial differential equation, which
has the peculiarity that the measure $(\lambda(a,\cdot))_a$ characterizing the
jump part is not fixed but depends on a parameter $a$ which lives in a compact
set $A$ of some Euclidean space $\R^q$. We do not assume that the family
$(\lambda(a,\cdot))_a$ is dominated. Moreover, the diffusive part can be
degenerate. Our aim is to give a BSDE representation, known as nonlinear
Feynman-Kac formula, for the value function associated to these control
problems. For this reason, we introduce a class of backward stochastic
differential equations with jumps and partially constrained diffusive part. We
look for the minimal solution to this family of BSDEs, for which we prove
uniqueness and existence by means of a penalization argument. We then show that
the minimal solution to our BSDE provides the unique viscosity solution to our
fully nonlinear integro-partial differential equation.
