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Applied Mathematics & Optimization

, Volume 68, Issue 2, pp 255–274 | Cite as

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

  • Pavol Brunovský
  • Aleš Černý
  • Michael Winkler
Article

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.

Keywords

Singular ODE Initial value problem Supersolution Subsolution Nonuniqueness 

Notes

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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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Pavol Brunovský
    • 1
  • Aleš Černý
    • 2
  • Michael Winkler
    • 3
  1. 1.Department of Applied Mathematics and StatisticsComenius University BratislavaBratislavaSlovakia
  2. 2.Cass Business SchoolCity University LondonLondonUK
  3. 3.Institut für MathematikUniversität PaderbornPaderbornGermany

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