Existence of solutions to the Poisson equation in L/sub p/ spaces

This paper presents two approaches to the problem of existence of solutions to the Poisson equation (PE) for a (Markov) transition probability. The first approach is an extension of the successive approximations algorithm of stochastic control theory. The second one is an algebraic approach based on the generalized Farkas theorem and it gives necessary and sufficient conditions for the existence of solutions to the PE in L/sub p/ spaces.