Stochastic Convexity and Concavity of Markov Processes

In this paper we consider the temporal stochastic convexity and concavity properties of Markov processes {X(t), t ∈ S} in discrete time (then S = N₊ ≡ {0, 1, 2, …}) or in continuous time (then S = [0, ∞)). That is, we obtain conditions on the process {X(t), t ∈ S} which imply that the expectation Ef(X(t)) is a monotone convex (concave) function of t whenever f is a monotone convex (concave) function. The theory is illustrated through some examples. After giving some background we 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. We then introduce a method for identifying ℱ-monotone operators, and we discuss the relationship between our results concerning temporal convexity and other results in the literature. In particular, we center our attention on a notion of stochastic concavity. In this respect we show that a result of Shaked and Shanthikumar is incorrect and we prove two alternative versions of it.

The approach that we 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, we can obtain results that we could not obtain otherwise.