The stochastic shallow lake problem

We study the welfare function and the optimal control of the shallow lake problem with multiplicative noise. We show that the welfare function is the viscosity solution of the associated Bellman equation and we establish several properties including its asymptotic behaviour at infinity. In the presence of noise, the optimally controlled lake may have a bimodal invariant distribution and exhibit metastable transitions between oligotrophic (blue) and eutrophic (green) states.