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

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 x 0 =∞ which is such that 0≤ u ( x )≤ x for all x >0, and that this solution is strictly increasing and concave. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 68 (2) – Oct 1, 2013
20 pages

/lp/springer_journal/a-singular-differential-equation-stemming-from-an-optimal-control-GSCgMzNbKo
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9205-5
Publisher site
See Article on Publisher Site

### 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2013

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