A singular differential equation stemming from an optimal control problem in financial economics

Abstract

We consider the ordinary differential equation x2 u''=axu'+bu-c \bigl(u'-1\bigr)2, \quad x\in(0,x0), with a ε ℝ, b ε 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. © 2013 Springer Science+Business Media New York.