  Environmental management and restoration should be designed such that the
risk and uncertainty owing to nonlinear stochastic systems can be successfully
addressed. We apply the robustified dynamic Orlicz risk to the modeling and
analysis of environmental management and restoration to consider both the risk
and uncertainty within a unified theory. We focus on the control of a
jump-driven hybrid stochastic system that represents macrophyte dynamics. The
dynamic programming equation based on the Orlicz risk is first obtained
heuristically, from which the associated Hamilton-Jacobi-Bellman (HJB) equation
is derived. In the proposed Orlicz risk, the risk aversion of the
decision-maker is represented by a power coefficient that resembles a certainty
equivalence, whereas the uncertainty aversion is represented by the
Kullback-Leibler divergence, in which the risk and uncertainty are handled
consistently and separately. The HJB equation includes a new state-dependent
discount factor that arises from the uncertainty aversion, which leads to a
unique, nonlinear, and nonlocal term. The link between the proposed and
classical stochastic control problems is discussed with a focus on
control-dependent discount rates. We propose a finite difference method for
computing the HJB equation. Finally, the proposed model is applied to an
optimal harvesting problem for macrophytes in a brackish lake that contains
both growing and drifting populations.
