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



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 x0=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.


Singular ODE Initial value problem Supersolution Subsolution Nonuniqueness 


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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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