Recent Advances and Applications of the Theory of Stochastic Convexity. Application to Complex Bio-Inspired and Evolution Models

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(θ)|θ ∈ T } over a probability space (Ω, ℑ, Pr), where the parameter θ usually represents some distribution moments. Regular, sample-path, and strong stochastic convexity notions have been defined to intuitively describe how the random objects X(θ) 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.