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

Abstract

We consider the ordinary differential equation $x^2{u}^{\prime \prime }=ax{u}^{\prime }+bu-c(u^{\prime }-1{)}^2,x\in (0,x_0)$, 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 solutions exist, whereas if $a+b\ge 0$ then there are infinitely many 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 $x_0=\infty$ which is such that $0\le u(x)\le x$ for all $x>0$, and that this solution is strictly increasing and concave.