On Additive and Multiplicative (Controlled) Poisson Equations G.B. Di Masi  L. Stettner Universit`a di Padova Institute of Mathematics Dipartimento di Matematica Polish Academy of Sciences Pura ed Applicata Sniadeckich 8, 00-956 Warsaw Via Belzoni 7, 35131 Padova and CNR-LADSEB Abstract Assuming that the Markov processes satisfy minorization property existence and properties of the solutions to additive and multiplicative Poisson equations are studied using splitting techniques. The problem is then extended to study risk sensitive and risk neutral control problems and corresponding to them Bellman equations. Key words: risk neutral and risk sensitive control, discrete time Markov processes, split- ting, Poisson equations, Bellman equations AMS subject classification: primary: 93E20 secondary: 60J05, 93C55 1 Introduction On a probability space (Ω, F ,P ) consider a Markov process X =(x n ) taking values on a complete separable metric state space E endowed with the Borel σ-algebra E . Assume that (x n ) has a transition operator P (x n , ·) at generic time n. Let c : E → R be continuous bounded and γ> 0. We would like to find constants λ and λ γ such that the functions w(x) := E x  ∞  i=0 (c(x i ) − λ)  (1) and e w γ (x) := E x  exp  ∞  i=0 γ (c(x i ) − λ γ )  (2) 1