Stochastic volatility model with long memory for water quantity-quality dynamics

Abstract: Water quantity and quality are vital indices for assessing fluvial environments. These indices are highly variable over time and include sub-exponential memory, where the influences of past events persist over long durations. Moreover, water quantity and quality are interdependent, with the former affecting the latter. However, this relationship has not been thoroughly studied from the perspective of long-memory processes, which this paper aims to address. We propose applying a new stochastic volatility model, a system of infinite-dimensional stochastic differential equations, to describe dynamic asset prices in finance and economics. Although the stochastic volatility model was originally developed for phenomena unrelated to the water environment, its mathematical universality allows for an interdisciplinary reinterpretation: river discharge is analogous to volatility, and water quality to asset prices. Moreover, the model's infinite-dimensional nature enables the analytical description of sub-exponential memory. The moments and autocorrelations of the model are then obtained analytically. We mathematically analyze the stochastic volatility model and investigate its applicability to the dynamics of water quantity and quality. Finally, we apply the model to real time-series data from a river in Japan, demonstrating that it effectively captures both the memory and the correlation of water quality indices to river discharge. This approach, grounded in infinite-dimensional stochastic differential equations, represents a novel contribution to the modelling and analysis of environmental systems where long memory processes play a role.