Data Analysis and Related Applications, Volume 2

8Dynamic Optimization with Tempered Stable Subordinators for Modeling River Hydraulics

We apply a tempered stable subordinator to modeling and control of river hydraulics, such as streamflow and water quality dynamics. The streamflow dynamics follow a stochastic differential equation driven by a tempered stable subordinator with self-excitation. An entropic dynamic risk measure is employed to evaluate a flood risk under model uncertainty. The problem is solved via a Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. From the HJBI equation, we explicitly derive an optimal flood mitigation policy of a hydraulic structure, along with the worst-case probability measure of streamflow. An interesting point is that it has a bifurcation point around which the value function diverges or not. A related backward stochastic differential equation for water quality dynamics is also briefly discussed.

8.1. Introduction

Lévy processes are additive and time-homogeneous jump–diffusion processes that have independent increments (Kyprianou 2014). Lévy processes that have infinite activities serve as building blocks for modern mathematical modeling and analysis of stochastic dynamical systems, such as finance and insurance (Molina-Muñoz et al. 2020), and machine learning (Li et al. 2021). Tempered stable subordinators have recently been found to be the right candidates for describing hydraulic processes occurring in river environments (Yoshioka and Yoshioka 2021a; Yoshioka and Yoshioka 2021c). A Lévy ...