A Splitting Method for Band Control of Brownian Motion: With Application to Mutual Reserve Optimization
Alain Bensoussan, John J. Liu, Jiguang Yuan
Operations Research · 2024 · Volume 72 · Issue 6
Abstract
In this paper, we develop a splitting solution method for two-sided impulse control of Brownian motion, which leads to an expanding range of band control applications and studies, such as monetary reserves (including the previously studied cash management problem, exchange rate control in central banks, and marine mutual insurance reserves), inventory systems, and lately natural resources and energy reservation. It has been shown since earlier studies in 1970s that the optimal two-sided impulse control can be characterized by a two-band control policy of four parameters ( 𝑎 , 𝐴 , 𝐵 , 𝑏 ) ( a , A , B , b ) with 𝑎 < 𝐴 ≤ 𝐵 < 𝑏 a < A ≤ B < b , of which the dynamic programming characteristics leads to a quasi-variational inequality (QVI) with two sides. Thus far, the focus of band control problems has been on determination of optimal band policy parameters. Its solution methods, as far as we can ascertain from the current literature, have centered on finding the four parameters by solving simultaneously characteristic systems of QVI inequalities, of which analytical solutions of closed form remain unattainable and computational solutions are still largely intractable. The key contributions of this paper are (1) development of a splitting method of decomposing a general two-sided band control problem into two iterative one-sided band control problems, each iteration being reduced to a one-dimension optimization; (2) obtaining a theorem on geometrical characterization of band-splitting control, including QVI and computational analytics and characteristics of band-splitting functions and solutions; and (3) development of a band-splitting solution algorithm for the two-sided impulse control, including an effective initial-point selection method that is termed the separate approximation of geometric conditions method. Numerical comparison experiments are carried out to validate and test the effectiveness and accuracy of the splitting solution method. The method is not only computationally effective, but also useful for proving theoretical results. Funding: A. Bensoussan is supported in part by the National Science Foundation [Grant DMS-2204795]. The study was supported by the General Research Fund [Grants 5230/06E and 5225/07E], both of which J. J. Liu was the principal investigator as chair professor of maritime studies, and was based on the PhD thesis of J. Yuan, who was jointly supervised by A. Bensoussan, all at Hong Kong Polytechnic University. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2011.0427 .
Keywords
Stochastic Models, stochastic impulse control and optimization, QVI inequalities, mutual reserve optimization
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