A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics
We consider the ordinary differential equation <##> $$$$ x^2 u''=axu'+bu-c /bigl(u'-1/bigr)^2, /quad x/in(0,x{sub 0}), $$$$x{sup 2}u{sup ″}=axu{sup ′}+bu−c(u{sup ′}−1){sup 2}, x∈(0,x{sub 0}), with <##> $$ a/in/mathbb{R}, b/in/mathbb{R} $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{sub 0}=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.
- OSTI ID:
- 22210462
- Journal Information:
- Applied Mathematics and Optimization, Vol. 68, Issue 2; Other Information: Copyright (c) 2013 Springer Science+Business Media New York; http://www.springer-ny.com; This record replaces 45031388; Country of input: International Atomic Energy Agency (IAEA); ISSN 0095-4616
- Country of Publication:
- United States
- Language:
- English
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