A Splitting Method for Band Control of Brownian Motion: With Application to Mutual Reserve Optimization

As well studied in the operations research literature, optimization of a mutual reserve system (e.g., federal reserves) and a nonmutual one such as regular inventory systems requires solving simultaneous systems of quasi-variational inequalities, of which analytical solutions in closed form remain unattainable and computational solutions are still intractable. Thus far, the studies of reserve optimization are of intra-nations (e.g., central bank reserves) as opposed to inter-nations (e.g., COVID vaccine reserves of the United Nations). In this paper, we advance a method of computational analytics for mutual reserve optimization, with an international perspective in response to the intensifying challenge on global medical reserves during the COVID pandemic. A solution algorithm is developed in the context of maritime mutual insurances (a long existent international mutual reserve system) and then tested through comprehensive numerical experiments. 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 ) with 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 characteri