The Itô and Stratonovich integrals for stochastic differential equations with poisson white noise

The relationship between the Itô and the Stratonovich integrals used for solving stochastic differential equations with Gaussian white noise is well known. However, this relationship seems to be less clear when dealing with stochastic differential equations driven by Poisson white noise. It is shown that there is no difference between the Itô and the Stratonovich integrals used to define the solution of stochastic differential equations with Poisson white noise. This result is in disagreement with findings of some previous publications but in agreement with the classical definition of the Itô and Stratonovich integrals. Intuitive considerations, arguments based on the theory of stochastic integrals with semimartingales, and examples are used to prove and demonstrate the claimed equality of the Itô and Stratonovich integrals.