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

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 x0=∞ 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 

References

  1. 1.
    Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance theory. In: Rogers, L.C.G., Talay, D. (eds.) Numerical Methods in Finance, pp. 1–21 (1997) CrossRefGoogle Scholar
  2. 2.
    Bernstein, C.: Sur certaines équations différentielles ordinaires du second ordre. C. R. Math. Acad. Sci. 138, 950–951 (1904) MATHGoogle Scholar
  3. 3.
    Černý, A.: Currency crises: introduction of spot speculators. J. Financ. Econom. 4, 75–89 (1999) Google Scholar
  4. 4.
    Cid, J.Á., López-Pouso, O., López-Pouso, R.: Existence of infinitely many solutions for second-order singular initial value problems with an application to nonlinear massive gravity. Nonlinear Anal., Real World Appl. 12, 2596–2606 (2011) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    de Coster, C., Habets, P.: Two-point boundary value problems: upper and lower solutions. In: Mathematics in Science and Engineering, vol. 205. Elsevier, Amsterdam (2006) Google Scholar
  6. 6.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) MATHGoogle Scholar
  7. 7.
    Liang, J.: A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation. J. Differ. Equ. 246, 819–844 (2009) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Nagumo, M.: Über die Differentialgleichung y″=f(x,y,y′). Proc. Phys. Math. Soc. Jpn. 19, 861–866 (1937) MATHGoogle Scholar
  9. 9.
    Wirl, F.: Energy prices and carbon taxes under uncertainty about global warming. Environ. Resour. Econ. 36, 313–340 (2007) CrossRefGoogle Scholar

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

Personalised recommendations