Before revision:  These equations are the natural candidates for the probabilistic interpretation of  fully non-linear partial integro-differential equations, which is the point of our paper 18 . Finally, we give an application of second order BSDEs to the study of a robust exponential utility maximization problem under model uncertainty. The uncertainty affects both the volatility process and the jump measure compensator. We prove existence of an optimal strategy, and that the value function of the problem is the unique solution of a particular second order BSDE with jumps .
After revision:  These equations are the natural candidates for the probabilistic interpretation of fully non-linear partial integro-differential equations, which is the point of our paper 18 the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs .