No, it is not possible to write a Poisson process. As you seem to be thinking about a stochastic model where shocks happen by surprise, you might be able to implement this yourself at first order, see the discussion at [Stochastic Simulation based on Non-Normal Shocks). Due to certainty equivalence at first order, it does not matter for the solution which type of shocks you feed in. The problem will be that you are working with a linear solution, which is not what the article you mentioned did.