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A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

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An Erratum to this article was published on 28 December 2016

Abstract

We consider the ordinary differential equation

$$x^2 u''=axu'+bu-c \bigl(u'-1\bigr)^2, \quad x\in(0,x_0), $$

with \(a\in\mathbb{R}, b\in\mathbb{R}\), c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b<0 then no continuous solutions exist, whereas if a+b>0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x 0=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.

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Acknowledgements

We would like to thank an anonymous referee for very detailed and thoughtful comments. This research was undertaken while Aleš Černý visited Comenius University in Bratislava in spring 2012. The financial support of the VÚB Foundation under the “Visiting professor 2011” funding scheme is gratefully acknowledged. Pavol Brunovský gratefully acknowledges support by the grants VEGA 1/0711/12 and VEGA 1/2429/12.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Comenius University Bratislava, 84248, Bratislava, Slovakia

    Pavol Brunovský

  2. Cass Business School, City University London, 106 Bunhill Row, London, EC1Y 8TZ, UK

    Aleš Černý

  3. Institut für Mathematik, Universität Paderborn, 33098, Paderborn, Germany

    Michael Winkler

Authors
  1. Pavol Brunovský
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  2. Aleš Černý
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  3. Michael Winkler
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Corresponding author

Correspondence to Pavol Brunovský.

Additional information

Communicated by Alain Bensoussan.

An erratum to this article is available at http://dx.doi.org/10.1007/s00245-016-9398-5.

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Brunovský, P., Černý, A. & Winkler, M. A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics. Appl Math Optim 68, 255–274 (2013). https://doi.org/10.1007/s00245-013-9205-5

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  • DOI: https://doi.org/10.1007/s00245-013-9205-5

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