A problem of symmetric variational equation (Q1906617)

The author deals with a discounted cost problem in singular stochastic control. The solution to this problem leads to a differential equation with some inequality constraints: \[ {\textstyle {1\over 2}} \sigma^2 (x) \nu'' (x) + \mu(x) \nu'(x) + h(x) \geq \alpha \nu (x)) \quad \nu'' (x) \leq 0 \quad \forall x \in \mathbb{R} \] where \(\mu\), \(\nu\), \(\sigma\), \(h\) denotes real functions on \(\mathbb{R}\) which satisfy some general conditions. The author shows that under certain conditions a twice continuous derivable positive even function \(\nu(x)\) on \(\mathbb{R}\) exists, which satisfies the equation above. The practical use of the theorem is illustrated by a short example, the application of a Wiener process.