The generator for a pure jump process with bounded jump rate is a bounded operator on the space of measurable functions. For any such process, it is simple to write a stochastic equation driven by a Poisson random measure. Uniqueness for both the stochastic equation and the corresponding martingale problem is immediate, and consequently, the martingale problem and the stochastic equation are equivalent in the sense that they uniquely characterize the same process. A variety of Markov processes, including many interacting particle models, have generators which are at least formally given by infinite sums of bounded generators. In considerable generality, we can write stochastic equations that are equivalent to these generators in the sense that every solution of the stochastic equation is a solution of the martingale problem and every solution of the martingale problem determines a weak solution of the stochastic equation. It follows that uniqueness for one approach is equivalent to uniqueness for the other.