A Reversible Investment Problem with Capacity and Demand in Finite Horizon: Free Boundary Analysis

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-04-03 DOI:10.1137/22m1469547

Xiaoru Han, Fahuai Yi, Jianbo Zhang

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1207-1234, April 2024.
Abstract. This paper investigates a reversible investment problem with finite horizon, in which a social planner aims to determine the project’s capacity level to minimize the expected total costs. These costs depend on the demand for the good, the supply in terms of production capacity, and the proportional costs. The issue of irreversible investment has been examined by Han and Yi [Commun. Nonlinear Sci. Numer. Simul., 109 (2022), 106302]. Mathematically, the reversible investment problem can be formulated as a singular stochastic control problem. The value function satisfies a two-dimensional parabolic variational inequality subject to gradient constraint, which leads to two time-dependent free boundaries representing optimal investment and disinvestment strategies. We employ a partial differential equation approach to characterize the continuity, monotonicity, and horizontal asymptotes of free boundaries, as well as establish the [math] regularity of the value function. To the best of our knowledge, the approach to analyze the behavior of free boundaries is novel in the literature.

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有限地平线上有能力和需求的可逆投资问题：自由边界分析

SIAM 控制与优化期刊》，第 62 卷，第 2 期，第 1207-1234 页，2024 年 4 月。摘要本文研究了一个有限时间跨度的可逆投资问题，其中社会规划者的目标是确定项目的产能水平，使预期总成本最小化。这些成本取决于对商品的需求、生产能力方面的供给以及比例成本。Han 和 Yi 研究了不可逆投资问题[Commun. Nonlinear Sci. Numer. Simul.从数学上讲，可逆投资问题可以表述为奇异随机控制问题。价值函数满足二维抛物线变分不等式，并受到梯度约束，这导致了两个与时间相关的自由边界，分别代表最优投资和取消投资策略。我们采用偏微分方程的方法来描述自由边界的连续性、单调性和水平渐近线，并建立价值函数的[数学]正则性。据我们所知，分析自由边界行为的方法在文献中尚属首次。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.