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

TL;DR

This paper analyzes a singular differential equation from an optimal control problem in financial economics, establishing existence, uniqueness, and asymptotic behavior of solutions related to asset disposal with liquidity effects.

Contribution

It characterizes solution existence and uniqueness for a specific singular ODE arising in financial economics, including asymptotic analysis and conditions for solutions.

Findings

No solutions if a+b<0

Infinitely many solutions if a+b≥0 with specific asymptotics

Unique solution for x0=∞ that is increasing and concave

Abstract

We consider the ordinary differential equation $x^2 u'' = axu'+bu-c(u'-1)^2, x\in (0,x_0)$, 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 $a+b\ge0$ 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.