Assessing fluctuations of long-memory environmental variables based on the robustified dynamic Orlicz risk

Environmental variables that fluctuate randomly and dynamically over time, such as water quality indices, are considered to be stochastic. They exhibit sub-exponential memory structures that should be accounted for in their modeling and analysis. Furthermore, risk assessments based on these environmental variables should consider limited data availability, which may introduce errors, e.g., model misspecifications, into their modeling. In this study, we present a pair of risk measures to determine the exponential disutility of a generic environmental variable both from below and above. The generic environmental variable is modelled as an infinite-dimensional nonlinear as well as affine stochastic differential equation and its moments and sub-exponential autocorrelations are estimated analytically. Novel risk measures, called dynamic robustified Orlicz risks, are formulated subsequently, and long, sub-exponential memory is efficiently addressed using them. The worst-case upper and lower bounds of the disutility are identified in closed form from the Hamilton-Jacobi-Bellman equations associated with the Orlicz risks. Finally, the proposed methodology is applied to weekly water quality data in a river environment in Japan.