In this paper the authors consider the temporal stochastic convexity and concavity properties of Markov processes {X(t),t∈S∈ in discrete time (then S=𝒩Å+∈{0,1,2,...∈) or in continuous time (then S=[0,∈)). That is, they obtain conditions on the process {X(t),t∈S∈ which imply that the expectation Ef(X(t)) is a montone convex (concave) function of t whenever f is a montone convex (concave) function. The theory is illustrated through some examples. After giving some background the authors define ℱ-monotonicity and then obtain some general results concerning stochastic convexity and concavity of Markov processes by using the notion of ℱ-monotone operators. They then introduce a method for identifying ℱ-monotone operators, and the authors discuss the relationship between the present results concerning temporal convexity and other results in the literature. In particular, they center their attention on a notion of stochastic concavity. In this respect the authors show that a result of Shaked and Shanthikumar is incorrect and they prove two alternative versions of it. The approach that the authors use here is an operator-analytic approach. This approach is quite powerful, but not as intuitive as sample path approaches used in other papers. However, using it, the authors can obtain results that they could not obtain otherwise.
