We investigate in this paper submodular properties of the value function arrizing in complex Dynamic programming (DPs). We consider in particular DPs that include concatenation and linear combinations of standard DP operators, as well as combination of maximizations and minimizations. These DPs have many applications and interpretations, both in stochastic control (and stochastic zero-sum games as well as in the analysis of (non-controlled) discrete-event dynamic systems. The submodularity implies the monotonicity of the selectors appearing in the DP equations, which translates, in the context of stochastic control and stochastic games, to monotone optimal policies. Our work is based on the score-space approach of Glasserman and Yao.
