  We consider the stochastic control problem of the shallow lake and continue
the work of G. T. Kossioris, Loulakis, and Souganidis (2019) in three
directions. First, we generalise the characterisation of the value function as
the viscosity solution of a well-posed problem to include more general
recycling rates. Then, we prove approximate optimality under bounded controls
and we establish quantitative estimates. Finally, we implement a convergent and
stable numerical scheme for the computation of the value function to
investigate properties of the optimally controlled stochastic shallow lake.
This approach permits to derive tail asymptotics for the invariant distribution
and to extend results of Grass, Kiseleva, and Wagener (2015) beyond the small
noise limit.
