PARAMETRIC STOCHASTIC CONVEXITY AND CONCAVITY OF STOCHASTIC-PROCESSES

A collection of random variables {X(θ), θ∈Θ} is said to be parametrically stochastically increasing and convex (concave) in θ∈Θ if X(θ) is stochastically increasing in θ, and if for any increasing convex (concave) function φ{symbol}, Eφ{symbol}(X(θ)) is increasing and convex (concave) in θ∈Θ whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {Xn(θ), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {Xn(θ), θ∈Θ}, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given. © 1990 The Institute of Statistical Mathematics.