We study the pricing of American put and call options in a market with jumps. We extend and make rigorous previous work that characterizes the price as a solution of an integro-differential equation set on the whole domain. The equation closely resembles the equation for the corresponding European options, but involves an additional reaction term that depends on the American option value in a nonlinear, nonlocal and discontinuous manner. Thus standard theory for partial differential equations does not apply, and we give a proper definition of a viscosity solution of the equation. We then show that the characterization is well posed. In particular, we prove a strong comparison principle for the equation using an original approach that overcomes some problems related to the appearance of integrals with respect to unbounded measures. In short, we extend the results in  to a general class of exponential additive models. The formulation constitutes a starting point for designing and analyzing ''easy to implement'' numerical algorithms for computing the value of an American option.