Before revision:  In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by appropriate Levy processes, and solve explicitly for the double exponential jump diffusion process . In particular, for the spectrally negative case, we generalize the formulation using the scale function, and obtain explicitly the optimal solutions for the exponential jump diffusion process.
After revision:  In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by appropriate Levy processes, and solve explicitly for the double exponential jump diffusion process and for a general spectrally negative Levy process.