Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

Yaegashi, Yuta; Yoshioka, Hidekazu; Tsugihashi, Kentaro; Fujihara, Masayuki

doi:10.1016/j.crma.2019.02.007

Cited by 7 publications

(20 citation statements)

References 22 publications

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“…implying that numerical solutions are discrete analogue of the PDF. In addition, nonnegativity of the numerical solution is satisfied for sufficiently small t ∆ [12], meaning that the scheme is stable. This stability is owing to the flux discretization (21).…”

Section: N +mentioning

confidence: 99%

“…These equations are local like the conventional heat and wave equations. However, Yaegashi et al [12] theoretically showed for the perfect impulse control model that the Fokker-Planck equation has a non-local term connecting the probability flux at y x = and that at y x = , based on the physical consideration that the process is immediately and irreversibly transported y x = from to y x = . In this research, we extend their approach to the model with imperfect interventions.…”

Section: (Optimal Policy)mentioning

confidence: 99%

“…Following the fitting technique that evaluates a numerical flux from exact solutions to auxiliary two-point boundary value problems [12,16]…”

Section: N +mentioning

confidence: 99%

“…A particular emphasis is put on the fish-eating waterfowl management problem [7,8]: a serious ecological problem in many countries of Europe, America, Asia, and New Zealand. This is an engineering problem with a relatively simple dynamics following an SDE subject to impulsive harvesting [8,12]. We show that the stationary PDF of the population is governed by a 1-D Non-Local Fokker-Planck Equation (NL-FPE) with a special non-locality, which is potentially an obstacle in its analysis and computation.…”

Section: Introductionmentioning

confidence: 99%

“…A Fokker-Planck equation is a governing differential equation of a probability density function (PDF) of stochastic system dynamics [11]. Recently, Yaegashi et al [12] approached this issue through derivation and analysis of a 1-D Fokker-Planck equation of the controlled single-species population dynamics, which is the governing differential equation of the PDF of the state variable. However, they assumed that human interventions are perfect and that there exists no uncertainty in the interventions, which is not consistent with the above-mentioned reality of environmental and ecological systems management.…”

Section: Introductionmentioning

confidence: 99%

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Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization

Yoshioka

Yaegashi

Tsujimura

et al. 2019

Communications in Computer and Information Science

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Human interventions to control environmental and ecological system dynamics are efficiently described as impulsive interventions by which the system state suddenly transits. Such interventions in applications are imperfect in the sense that the state transition is not exactly controllable and thus uncertain. Mathematical description of the imperfect impulsive interventions, despite relevance in practical problems of environmental and ecological engineering, has not been addressed so far to the best of the authors' knowledge. The objectives and contributions of this research are formulation and numerical computation of single-species population dynamics controlled through imperfect impulsive interventions. We focus on a management problem of a waterfowl population as a model problem where the population dynamics follows a stochastic differential equation subject to impulsive harvesting. We show that the stationary probability density function of the population dynamics is governed by a 1-D Fokker-Planck equation with a special non-locality, which potentially becomes an obstacle in analyzing the equation. We demonstrate that the equation is analytically solvable under a simplified condition, which is validated through a Monte-Carlo simulation result. We also demonstrate that a simple finite volume scheme can approximate its solution in a stable, conservative, and super-convergent manner.

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Section: N +mentioning

confidence: 99%

Section: (Optimal Policy)mentioning

confidence: 99%

“…Following the fitting technique that evaluates a numerical flux from exact solutions to auxiliary two-point boundary value problems [12,16]…”

Section: N +mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization

Yoshioka

Yaegashi

Tsujimura

et al. 2019

Communications in Computer and Information Science

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Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

Yoshioka

Yaegashi²

2021

Optim Control Appl Methods

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Nonsmooth 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 article, 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 nonsmooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the nonsmoothness. 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 nonstandard system of degenerate elliptic equations, the Hamilton-Jacobi-Bellman equation (HJBE), which is rigorously and analytically verified in a simplified case. The associated Fokker-Planck equation (FPE) 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 HJBE and FPE 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.

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Fokker-Planck equations of jumping particles and mean field games of impulse control

Bertucci

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

Cited by 7 publications

References 22 publications

Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization

Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

Fokker-Planck equations of jumping particles and mean field games of impulse control

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