Stochastic streamflow and dissolved silica dynamics with application to the worst-case long-run evaluation of water environment

Management of river environment requires to assess streamflows and water quality dynamics, which are often stochastic as well as not easy to model without errors. A new mathematical framework is proposed in this paper to assess an environment in a long run based on a stochastic optimization theory under model ambiguity. Focusing on the dissolved silica (DSi) load as an environmental indicator of rivers, the coupled discharge and DSi load dynamics as a two-variable continuous-state branching process with immigration is formulated. The ambiguity as a model misspecification is evaluated by a relative entropy measuring deviation between benchmark and distorted models. The model misspecification is expressed as a bound of the relative entropy. Novel stochastic optimization problems are formulated to evaluate the long-run DSi load subject to the misspecification as expectation constraints. Nonlocal degenerate elliptic Hamilton-Jacobi-Bellman (HJB) equations having Lagrangian multipliers are employed to solve these problems and their optimality are verified theoretically. The HJB equations admit closed-form solutions that can be computed efficiently. Our model is finally applied to assessing long-run DSi load and discharge in an upstream Hiikawa River, Japan.