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Classical Ornstein–Uhlenbeck (ou) processes are Lévy-driven linear stochastic models with exponentially decaying autocorrelation functions which do not always fit more slowly decaying real time series data. A superposition of ou processes (known as a supou process) is proposed to overcome this issue for application to river discharge time series data. The discharge data has a sub-exponential autocorrelation function and this is captured by the supou process based on the mean reversion speed generated by a Gamma distribution. All the parameters of the supou process are identified by matching the autocorrelation and the first to fourth statistical moments of the discharge data. The empirical and modelled histograms of the discharge data are comparable with each other. References O. E. Barndorff-Nielsen. Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45.2 (2001), pp. 175–194. doi: 10.1137/S0040585X97978166 O. E. Barndorff-Nielsen, F. E. Benth, and A. E. D. Veraart. Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes. Bernoulli 19.3 (2013), pp. 803–845. doi: 10.3150/12-BEJ476 O. E. Barndorff-Nielsen and N. N. Leonenko. Burgers’ turbulence problem with linear or quadratic external potential. J. Appl. Prob. 42.2 (2001), pp. 550–565. url: http://www.jstor.org/stable/30040809 J. Beran, Y. Feng, S. Ghosh, and R. Kulik. Long-Memory Processes. Springer-Verlag, Berlin, Heidelberg, 2016. doi: 10.1007/978-3-642-35512-7 F. Fuchs and R. Stelzer. Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model. ESAIM: Prob. Stat. 17 (2013), pp. 455–471. doi: 10.1051/ps/2011158 Y. Kabanov and S. Pergamenshchikov. Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process. Fin. Stoch. 24.1 (2020), pp. 39–69. doi: 10.1007/s00780-019-00413-3 R. Kawai and H. Masuda. On simulation of tempered stable random variates. J. Comput. Appl. Math. 235.8 (2011), pp. 2873–2887. doi: 10.1016/j.cam.2010.12.014 S. Pelacani and F. G. Schmitt. Scaling properties of the turbidity and streamflow time series at two different locations of an intra-Apennine stream: Case study. J. Hydro. 603.B (2021), p. 126943. doi: 10.1016/j.jhydrol.2021.126943 R. Stelzer, T. Tosstorff, and M. Wittlinger. Moment based estimation of supOU processes and a related stochastic volatility model. Stat. Risk Model. 32.1 (2015), pp. 1–24. doi: 10.1515/strm-2012-1152 S. Suweis, E. Bertuzzo, G. Botter, A. Porporato, I. Rodriguez-Iturbe, and A. Rinaldo. Impact of stochastic fluctuations in storage-discharge relations on streamflow distributions. Water Resource. Res. 46.3 (2010), W03517. doi: 10.1029/2009WR008038 M. Tamborrino and P. Lansky. Shot noise, weak convergence and diffusion approximations. Physica D: Nonlinear Phenomena 418 (2021), p. 132845. doi: 10.1016/j.physd.2021.132845 E. Taufer and N. Leonenko. Simulation of Lévy-driven Ornstein–Uhlenbeck processes with given marginal distribution. In: Comput. Stat. Data Anal. 53.6 (2009), pp. 2427–2437. doi: 10.1016/j.csda.2008.02.026 C. Van Den Broeck. On the relation between white shot noise, Gaussian white noise, and the dichotomic Markov process. J. Stat. Phys. 31 (1983), pp. 467–483. doi: 10.1007/BF01019494 H. Yoshioka and Y. Yoshioka. Designing cost-efficient inspection schemes for stochastic streamflow environment using an effective Hamiltonian approach. Opt. Eng. (2021), pp. 1–33. doi: 10.1007/s11081-021-09655-7

We study a version of the Ornstein-Uhlenbeck bridge driven by a spectrally-positive subordinator. Our formulation is based on a Linear-Quadratic control subject to a singular terminal condition. The Ornstein-Uhlenbeck bridge, we develop, is written as a limit of the obtained optimally controlled processes, and is shown to admit an explicit expression. Its extension with self-excitement is also considered. The terminal condition is confirmed to be satisfied by the obtained process both analytically and numerically. The methods are also applied to a streamflow regulation problem using a real-life dataset., Comment: This is a preprint version of the article with the same title to be published in "IEEE Control Systems Letters" (L-CSS)

We formulate a new two-variable river environmental restoration problem based on jump stochastic differential equations (SDEs) governing the sediment storage and nuisance benthic algae population dynamics in a dam-downstream river. Controlling the dynamics is carried out through impulsive sediment replenishment with discrete and random observation/intervention to avoid sediment depletion and thick algae growth. We consider a cost-efficient management problem of the SDEs to achieve the objectives whose resolution reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation. We also consider a Fokker-Planck (FP) equation governing the probability density function of the controlled dynamics. The HJB equation has a discontinuous solution, while the FP equation has a Dirac's delta along boundaries. We show that the value function, the optimized objective function, is governed by the HJB equation in the simplified case and further that a threshold-type control is optimal. We demonstrate that simple numerical schemes can handle these equations. Finally, we numerically analyze the optimal controls and the resulting probability density functions., Comment: Accepted manuscript

Non-smooth dynamics driven by stochastic disturbance arise in a wide variety of engineering problems. Impulsive interventions are often employed to control stochastic systems; however, the modeling and analysis subject to execution delay have been less explored. In addition, continuously receiving information of the dynamics is not always possible. In this paper, with an application to an environmental restoration problem, a continuous-time stochastic impulse control problem subject to execution delay under discrete and random observations is newly formulated and analyzed. The dynamics have a non-smooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the non-smoothness. The goal of the control problem is to find the most cost-efficient policy that can prevent the dynamics from attaining the undesirable state. We demonstrate that finding the optimal policy reduces to solving a non-standard system of degenerate elliptic equations, the optimality equation, which is rigorously and analytically verified in a simplified case. The associated Fokker-Planck equation for the controlled dynamics is derived and solved explicitly as well. The model is finally applied to numerical computation of a recent river environmental restoration problem. The optimality and Fokker-Planck equations are successfully computed, and the optimal policy and the probability density functions are numerically obtained. The impacts of execution delay are discussed to deeper analyze the model., 8 figures and 4 tables

Simulating streamflow under both high-and low-flows is required for versatile eco-hydrological modeling. Typical streamflow simulators require hydrological data such as river geometry and observed river discharge/water level as upstream/downstream boundary conditions. However, these are not always available in data-sparse regions. Furthermore, because of the potential inaccuracy of digital elevation model (DEM) data around water surfaces, this data has not generally been utilized in streamflow simulations. Therefore, this study explores the potential applicability of DEM data to extract river cross-sections, focusing on the upstream Hiikawa River, Japan. A 1-D streamflow simulation was performed using river cross-sections extracted from a 5 m LiDAR DEM and the observed dam discharge from 2018 to 2020 as the upstream boundary condition. The simulated water depths with Manning’s roughness coefficients of 0.03 to 0.05 m–1/3 s reproduce the observation results with Nash-Sutcliffe coefficients of 0.91–0.97 for the whole period and 0.60–0.97 for a flood event. The accurate results for both low and high flows were considered to reflect the reasonable representations of the river cross-section. Finally, the velocity-based suitability index for Ayu (P. altivelis) was evaluated. We demonstrate applicability and several possible limitations of DEM data for eco-hydrological modeling of data-scarce rivers.

A new stochastic control problem of a dam–reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic regime-switching inflow process. The system is controlled to balance among the operation purpose and the internal and downstream environmental conditions. Finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous Hamiltonian, which is a system of nonlinear degenerate parabolic (or hyperbolic) equations. We show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions. The model is applied to numerical computation of the operation policy of an existing dam–reservoir system using a high-order finite difference scheme. The computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river.

We propose and analyze a stochastic competing two-species population dynamics model subject to jump and continuous correlated noises. Competing benthic algae population dynamics in river environment, which is an important engineering problem, motivates this new model. The model is a system of stochastic differential equations having a characteristic that the two populations are competing with each other through the environmental capacities; an increase in one population decreases the other's environmental capacity. Unique existence of the uniformly bounded strong solution is proven, and attractors of the solutions are identified depending on the parameter values. The Kolmogorov's backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. A novel uncertain correlation case is also analyzed in the framework of viscosity solutions. Numerical computation results using a finite difference scheme and a Monte-Carlo method are presented to deeper analyze the model. Our analysis results can be utilized for establishment of a foundation for modeling, analysis and control of the competing population dynamics.

Establishment of a long-term biological population management policy requires balancing its cost and utility/disutility. We approach this issue from the viewpoint of stochastic control as an effici...

Early estimation of biological growth of organisms is an indispensable task in ecology and related research areas. The biological growth is always time-continuous, while our observations of the phenomenon are time-discrete in practice. The formalism of the discrete costly observation (DCO) enables us to mathematically bridge the two qualitatively different processes. This formalism is still germinating, and its practical applications have not been carried out. This paper presents a first application of the DCO formalism to a cost-effective early estimation problem of the biological growth, and its mathematical and numerical analysis. Growth dynamics of organisms, which are fishery resources in this paper, is governed by a stochastic differential equation whose solution is observed discretely. The optimality equation to be solved for finding the most cost-effective observation policy is derived as a fixed point problem based on degenerate parabolic partial differential equations. The fixed point problem turns out to be uniquely solvable. A recursive approximation of the fixed point problem is presented and its solvability in a viscosity sense is discussed. A finite different scheme is then employed to fully-discretize the recursive equations. The present model is finally applied to a problem of Japanese smelt Plecoglossus altivelis altivelis (Ayu): an important inland fishery resource in Japan.

A mathematical framework for controlling growth rate of harmful species population is established based on the concept of stochastic control. The main problem to be addressed in this paper is to effectively suppress the population growth through manipulating its surrounding environmental conditions. The growth control is achieved through minimization of a performance index that contains the cost of interventions and a metric based on the growth rate. Solving the problem ultimately reduces to finding a solution to the associated Hamilton–Jacobi–Bellman equation. Its solution behavior is analyzed from a mathematical viewpoint, showing that the optimal control and the optimized growth rate are critically affected by the chosen metrics in the performance index. The present model is then applied to an urgent management problem of the harmful attached algae in a dam downstream river reach, in which the dam discharge is the control variable. The application results clarify environmental dependence of the optimal dam discharge and under what condition it serves as the environmental flow to suppress the algae bloom. Throughout this paper, we demonstrate how mathematical models can be applied to environmental decision-making.

A Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) for a river environmental restoration problem with wise-use of sediment is formulated and its mathematical properties are analyzed. A finite difference scheme with a penalization technique is then established for solving the HJBQVI. The scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions. A demonstrative application example of the HJBQVI is finally presented.

This paper proposes a simple mathematical model forupstream fish migration along rivers. The model describes the fish migration along a river based on a mixed optimal control approach having swimming velocity, school size, and stopping time of migration as control variables. The optimization problem reduces to a variational inequality. Its explicit “viscosity” solution is presented with the dependence of the fish migration on river environment. To prove uniqueness of the solution to the variational inequality requires a constructive argument not based on the conventional theorems. A novel finite difference scheme for solving the variational inequality is also proposed with its convergence results. An application example of the model discusses the upstream migration of Plecoglossus altivelis (Ayu) in Japan, which evaluates the dependence of the fish migration on the habitat quality and provides recommendations for managing river environment. This is an interdisciplinary research between environmental and mathematical fields.

Optimal control of Levy jump-driven stochastic differential equations plays a central role in management of resource and environment. Problems involving large Levy jumps are still challenging due to their mathematical and computational complexities. We focus on numerical control of a real-scale dam and reservoir system from the viewpoint of forward-backward stochastic differential equations (FBSDEs): a new mathematical tool in this research area. The problem itself is simple but unique, and involves key challenges common to stochastic systems driven by large Levy jumps. We firstly present an exactly-solvable linear-quadratic problem and numerically analyze convergence of different numerical schemes. Then, a more realistic problem with a hard constraint of state variables and a more complex objective function is analyzed, demonstrating that the relatively simple schemes perform well.

We propose an analytical framework for sustainable management of sediment-algae dynamics in dam-downstream river environment. The problem considered in this short paper is an optimization of the dynamics by sediment replenishment schemes having multiple objectives: cost-efficient sediment supply and prevention of the bloom of nuisance benthic algae. We apply an Ergodic long-run stochastic control model to this problem, and show that it can be solved analytically being different from many other multi-objective optimization models. The sediment storage and algae population dynamics follow ordinary differential equations perturbed randomly by the sediment replenishment. We show that the most cost-efficient policy, namely the optimal sediment replenishment policy, is of a threshold type that is implementable in the real world. We analytically lead to a practical sustainability criterion that is based only on the biological and physical parameters, which can be specified prior to implementing a replenishment scheme. Our model is simple, but can give useful and practical insights into sustainable river environmental management.

We present a modern stochastic control framework for dynamic optimization of river environment and ecology. We focus on fisheries in Japan and show several examples of simplified optimal control problems of stochastic differential equations modeling fishery resource dynamics, reservoir water balance dynamics, benthic algae dynamics, and sediment storage dynamics. These problems concern different phenomena with each other, but they all reduce to solving degenerate parabolic or elliptic equations. Optimal controls and value functions of these problems are computed using finite difference schemes. Finally, we present a higher-dimensional problem of controlling a dam-reservoir system using a semi-Lagrangian discretization on sparse grids. Our contribution shows the state-of-art of modeling, analysis, and computation of stochastic control in environmental engineering and science, and related research areas.

A finite difference scheme is developed for solving 1-D variational inequalities arising in stochastic control problems with several singular control variables. The scheme guarantees the uniqueness of numerical solutions. A policy iteration algorithm is then proposed to solve the discretized problem. The present approach is applied to solving variational inequalities associated to cost-effective management problems of benthic algae on the riverbed downstream of a dam: an urgent environmental problem. Accuracy of the scheme is verified to be first-order for both the solution and its free boundaries. An advanced problem that involves a max–min differential game structure is also examined. The scheme then computes reasonably accurate numerical solutions which are consistent with the theoretical asymptotic estimates.

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

Using a tractable singular stochastic control model, this study seeks a sustainable population management policy for the fish-eating waterfowl Phalacrocorax carbo (Great Cormorant), a major predator of riverine fish in Japan. The bird population dynamics around a habitat lake are described by a stochastic differential equation with a singular control variable representing human interventions to suppress the predation. The optimal population management policy maximizes a performance index that covers the cost of countermeasures, the loss of riverine fish, and the ecosystem services provided by the bird. The dynamic programming principle obtains an exactly-solvable variational inequality that governs a sustainable, threshold-type optimal management policy. In this policy, countermeasures are taken when the population reaches a certain level. The model is applied to management of P. carbo population around Lake Biwa, Japan, where predation of riverine fish is the most severe in the country. The current management policy around the lake is analyzed and verified in the context of the proposed model, focusing on the population-control threshold and the average population reduction per unit time.

We present and mathematically analyze a singular stochastic control model for cost-effective and ecologically-sound indirect population control strategy for fish-eating birds. Finding the optimal strategy of a threshold type reduces to solving a variational inequality. We prove the unique existence of its viscosity solution that is neither convex nor concave, which turns out to be a classical solution. Comparative statics on the optimal threshold is performed as well to demonstrate practical implications of the model.

Comprehending life history of migratory fish, onset of migration in particular, is a key biological and ecological research topic that still has not been clarified. In this paper, we propose a simple mathematical model for the onset of fish migration in the context of a stochastic optimal stopping theory, which is a new attempt to our knowledge. Finding the criteria of the onset of migration reduces to solving a variational inequality of a degenerate elliptic type. As a first step of the new mathematical modeling, mathematical and numerical analyses with particular emphasis on whether the model is consistent with the past observation results of fish migration are examined, demonstrating reasonable agreement between the theory and observation results. The present mathematical model thus potentially serves as a simple basis for analyzing onset of fish migration.