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ScenTrees.jl : A Julia Package for Generating Scenario Trees and Scenario Lattices for Multistage Stochastic Programming
No full textScenTrees v0.3.0
We present ScenTrees.jl, a open-source Julia package for generating scenario trees and scenario lattices which can be used, for example, for multistage stochastic optimization problems. It allows users to represent possible sequences of stochastic processes in form of a scenario tree in the case of a discrete time stochastic process and a scenario lattice for Markovian data processes. In extension, it also provides users with a platform for generating new and additional trajectories in case of a limited data using conditional density estimation
Subdifferential representation of risk measures
No full textMeasures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk measures (risk deviation measures) are natural generalizations of the standard deviation. While pure risk measures are typically convex, acceptability measures are typically concave. In both cases, the convexity (concavity) implies under mild conditions the existence of subgradients (supergradients). The present paper investigates the relation between the subgradient (supergradient) representation and the properties of the corresponding risk measures. In particular, we show how monotonicity properties are reflected by the subgradient representation. Once the subgradient (supergradient) representation has been established, it is extremely easy to derive these monotonicity properties. We give a list of Examples
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