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

Brunovsky, P., Černý, A. & Winkler, M. (2013). A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics. Applied Mathematics & Optimization, 68(2), pp. 255-274. doi: 10.1007/s00245-013-9205-5

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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.

Item Type: Article
Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s00245-013-9205-5
Uncontrolled 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
Divisions: Cass Business School > Faculty of Actuarial Science & Insurance
Related URLs:
URI: http://openaccess.city.ac.uk/id/eprint/5938

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