Stochastic skiba sets: an example from models of illicit drug consumption

Authors:
Roswitha Bultmann;Gustav Feichtinger;Gernot Tragler
Affiliations:
Institute for Mathematical Methods in Economics (IWM), Research Unit for Operations Research and Control Systems (ORCOS), Vienna University of Technology (TU Wien), Wien, Austria;Institute for Mathematical Methods in Economics (IWM), Research Unit for Operations Research and Control Systems (ORCOS), Vienna University of Technology (TU Wien), Wien, Austria;Institute for Mathematical Methods in Economics (IWM), Research Unit for Operations Research and Control Systems (ORCOS), Vienna University of Technology (TU Wien), Wien, Austria
Venue:
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
Year:
2009

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Abstract

Skiba or DNSS sets are an important feature of many deterministic (usually convex or at least non-concave) optimal control models For stochastic models they have hardly been studied Using a classical discretization scheme, we consider a time-discrete stochastic reformulation of a well-known optimal control model of illicit drug consumption, in which the retail drug price is influenced by exogenous random forces We assume that these exogenous forces are described by i.i.d random variables on a finite probability space Having set up the model in this way, we use techniques from dynamic programming to determine the optimal solution and describe the transient sets that constitute the stochastic Skiba/DNSS sets We also show that the DNSS sets expand with the variance, and the optimal policy becomes a continuous function of the state for sufficiently high levels of the variance.