Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result

We show existence and uniqueness of a continuous with polynomial growth viscosity solution of a system of second order integral-partial differential equations (IPDEs for short) without assuming the usual monotonicity condition of the generator with respect to the jump component as in Barles et al.'s article (Barles et al., 1997). The Lévy measure is arbitrary and not necessarily finite. In our study the main tool we used is the notion of backward stochastic differential equations with jumps.