  In this paper, we adapt stochastic Perron's method to analyze a stochastic
target problem with unbounded controls in a jump diffusion set-up. With this
method, we construct a viscosity sub-solution and super-solution to the
associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison
principles, uniqueness of the viscosity solutions holds and the value function
coincides with the unique solution in the parabolic interior. Since classical
control problems can be analyzed under the framework of stochastic target
problems (with unbounded controls), we use our results to generalize the
results in ArXiv:1212.2170 to problems with controlled jumps.
