On the Optimal Control of a Random Walk with Jumps and Barriers

Article

Abstract

The modeling and optimal control of a class of random walks (RWs) is investigated in the framework of the Chapman-Kolmogorov (CK) and Fokker-Planck (FP) equations. This class of RWs includes jumps driven by a compound Poisson process and are subject to different barriers. A control mechanism is investigated that is included in the CK stochastic transition matrix and the purpose of the control is to track a desired discrete probability density function and attain a desired terminal density configuration. Existence and characterization of optimal controls are discussed. The proposed approach allows the derivation of a new FP model that accommodates the presence of the jumps and guarantees conservation of total probability in the case of reflecting barriers, which are modelled by appropriate operators. Results of numerical experiments are presented that successfully validate the proposed control framework.

Keywords

Random walk with jumps Optimal control Modeling of random walk Fokker-Planck equation 

Mathematics Subject Classification (2010)

93E20 60G50 34K99 60J27 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Tim Breitenbach
    • 1
  • Mario Annunziato
    • 2
  • Alfio Borzì
    • 1
  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly

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