Abstract

We consider the ordinary differential equation

x2u′′=axu′+bu−c(u′−1)2,x∈(0,x0),

with

a∈R,b∈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.

Publication Type:	Article
Additional Information:	The final publication is available at Springer via http://dx.doi.org/10.1007/s00245-013-9205-5
Publisher Keywords:	Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, MATHEMATICS, APPLIED, Singular, ODE, Initial value problem, Supersolution, Subsolution, Nonuniqueness
Subjects:	H Social Sciences > HB Economic Theory Q Science > QA Mathematics
Departments:	Bayes Business School > Actuarial Science & Insurance
Related URLs:	Author

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