A singular differential equation stemming from an optimal control problem in financial economics (Q2441469)

The authors consider the following initial value problem \[ x^2u''=axu'+bu-c(u'-1)^2, \;u(0)=0, \] where \(x\in (0, x_0)\), \(a, b\in \mathbb{R}\), and \(c>0\). The problem arises from a model of optimal liquidation of a large quantity of an asset whose market price is affected by its ongoing sale. The authors prove that for \(a+b>0\) the considered problem has a continuum of local solutions and one global bounded solution, while for \(a+b<0\) no solution exists. Moreover, they analyze monotonicity and convexity properties of the global solution.