Authors:
Eva Maria Ortega
1
and
Jose Alonso
2
Affiliations:
1
University Miguel Hernandez Elche, Spain
;
2
Hospital Universitario Virgen de Arrixaca, Spain
Keyword(s):
Stochastic directional convexity, Stochastic ordering, Increasing directionally convex function, Variability, bounding, biologically inspired models, Evolution models, mixtures, Multiplicative process, Tree network.
Abstract:
The theory of stochastic convexity is widely recognised as a framework to analyze the stochastic behaviour
of parameterized models by different notions in both univariate and multivariate settings. These properties
have been applied in areas as diverse as engineering, biotechnology, and actuarial science. Consider a
family of parameterized univariate or multivariate random variables {X(q)|q ∈ T} over a probability space
(W,Á,Pr), where the parameter q usually represents some distribution moments. Regular, sample-path, and
strong stochastic convexity notions have been defined to intuitively describe how the random objects X(q)
grow convexly (or concavely) concerning their parameters. These notions were extended to the multivariate
case by means of directionally convex functions, yielding the concepts of stochastic directional convexity for
multivariate random vectors and multivariate parameters. We aim to explain some of the basic concepts of
stochastic convexity, to discuss how this
theory has been used into the stochastic analysis, both theoretically
and in practice, and to provide some of the recent and of the historically relevant literature on the topic. Finally,
we describe some applications to computing/communication systems based on bio-inspired models.
(More)