Smooth pasting conditions for optimal investment with transactions costs

Question

I'm reading this paper relating to optimal investment with transaction costs where some value function $F(x)$ is optimized. At some boundary $x=u$ it will be optimal to pay a proportional cost $C$ which gives the boundary condition

\begin{equation} F'(u) = -C \end{equation}

The author argues that optimality also implies a boundary condition for the second derivative

\begin{equation} F''(u) = 0 \end{equation}

but I'm struggling to understand why this is the case. Any hints that will help me understand the intuition behind this condition?

Answer 1

I think that they are saying that, at special point $u$, we have:

$$ F'(u) = -\rho. $$

Also, that in a neighborhood from the left, we have:

$$ F'(u- dU) = -\rho $$

for any small positive $dU$.

Then, with Taylor on the left:

$$ F'(u) - F''(u)dU = - \rho $$

Hence: $$ F''(u) = 0 $$

Basically, if the first derivative of a function is constant in a neighborhood of a point, then its second derivative must be null at that point.