外文翻译--多层面现金管理系统的最优脉冲控制与广义费用函数

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1、外文翻译 Optimal impulse control for a multidimensional cash management system with generalized cost functions（节选） Auditor： Stefano Baccarin Nationality： I-10122 Turin, Italy Derivation： European Journal of Operational Research ， NO.196 ， 2009， PP.198206 Abstract We consider the optimal control of a mul

2、tidimensional cash management system where the cash balances uctuate as a homogeneous diffusion process. We formulate the model as an impulse control problem on an unbounded domain with unbounded cost functions. Under general assumptions we characterize the value function as a weak solution ofa quas

3、i-variational inequality in a weighted Sobolev space and we show the existence of an optimal policy. Moreover we prove the local uniform convergence of a nite element scheme to compute numerically the value function and the optimal cost. We compute the solution of the model in two-dimensions with li

4、near and distance cost functions, showing what are the shapes of the optimal policies in these two simple cases. Finally our third numerical experiment computes the solution in the realistic case of the cash concentration of two bank accounts made by a centralized treasury. Introduction The cash man

5、agement problem has been considered in the mathematical fnance literature only in dimension one. From the pioneer works of Baumol (1952), Tobin (1956), Miller and Orr(1966), it originated the main mathematical model used by the theory of the transactions and precautionary demand for money. The decis

6、ive improvement in solving the model was the use of the optimal control technique of impulse control”, introduced by Bensoussan and Lions (1973, 1975). Assuming linear holding/penalty costs,fixed plus proportional costs of control and the cash stock dynamics described by a generalized Brownian motio

7、n, Constantinides and Richard (1978)first showed the existence of a simple optimal impulse policy of a control band type. Harrison et al.(1983), proved a similar result assuming a nonnegative constraint for the cash balance which implies that the lowest barrier is equal to zero and they also gave a

8、simple numerical procedure to compute the three remaining optimal barriers. The assumption of a nonnegative minimum level for the cash balance is common to many papers (see Frenkel and Jovanovic, 1980； Chang, 1999； BarIlan et al., 2004), but it is basically due to technical difficulties in obtaining

9、 closed-form solutions for the more general case. However it is an essential feature of liquidity management to allow a negative cash balance (at some penalty rate) and as pointed out in Bar-Ilan (1990), the transaction demand for money should have to consider the possibility of overdraft cash facil

10、ities. In Baccarin (2002) the existence results of Constantinides and Richard(1978) are extended to the case of holding-penalty costs which have also a quadratic term .In that paper it is shown that as long as the penalty costs are nite the lowest barrier of the optimal policy is always below zero a

11、nd a simple numerical algorithm is given to compute the four critical numbers. In a recent paper, Bar-Ilane al.(2004) model the changes in the money stock by a superposition of a Brownian motion and a compound Poisson process Assuming the optimality of a control band policy they derive the relevant

12、discounted costs as function of the barriers by using renewal and martingale techniques. This can allow to approximate the optimal control parameters numerically. Conclusions In this paper we have presented a general method to solve a multidimensional cash management problem. Our solution allows the

13、 transaction costs and the holding/penalty costs to be nonlinear functions. Furthermore the cash stocks dynamics may be correlated and they may have drift and diffusion coefcients which can depend upon the state of the system. These general assumptions are very important to deal with realistic appli

14、cations. Using the functional analysis techniques of Bensoussan and Lions we have shown that there always exists an optimal policy for our multidimensional cash management system. Moreover ,in order to compute the solution, we have proved the convergence of a numerical scheme which has a strong foun

15、dation in the theory of quasi-variational inequalities and impulse control. Although we have solved the problem in some two-dimensional examples the algorithm remains the same for greater dimensions, requiring, of course, a growing computational cost. There are several directions in which our variat

16、ional approach can be further investigated .One can think ofa more general process describing the state dynamics, a jump-diffusion process, and the set of controls could be generalized considering also a continuous control of the drift or introducing temporal lags in implementing the decisions. If w

17、e consider the same model with a nite time horizon we obtain an evolutionary problem where we have to study and interpret the solutions of aparabolic quasi-variational inequality. 多层面现金管理系统的最优脉冲控制与广义费用函数 (节选 ) 作者 ： Stefano Baccarin 国籍： I-10122 Turin, Italy 原文出处： European Journal of Operational Resea

18、rch ， NO.196 ， 2009， P.198206 摘要 我们认为，一个多层面的现金管理系统的最优控制在于其中现金余额波动均匀并且平衡扩散的状态。我们制定的成本函数是用来解决无界域脉冲控制问题的模型。在一般假设情况下，我们认为作为一个弱解 OFA 的准变分不等式的加权 索伯列夫 空间的价值功能与我们展示的最优策略是一个很大的特点。此外，我们证明了有限元方法计算数值的价值功能理论与成本的最佳局部一致收敛。我们还计算了在两线性尺寸和距离成本函数模型的解，显示了在这两个简单的情况下，最优政策的表 现形态形状。最后，我们的第三个数值试验计算制定出实际案例解决方案其中包括了中的两个银行账户的现金

19、国库集中浓度。 导言 在本文中，我们提出了一个通用的方法用来解决多层面的现金管理问题。我们的解决方案是使交易成本和持有 /惩罚陈本为线性函数。此外，该现货库存动态可能是相关的，他们可以在依赖系统的状态下漂移和扩散。这些一般的假设在实际应用处理中非常重要。利用Bensoussan and Lions 的功能分析技术，我们已经发现，始终存在对于我们现金管理系统的多维最优策略。此外，为了计算出解决方案，至今我们已经证明了一 个数值格式的收敛性，它有一个准变分不等式及脉冲控制理论的坚实基础。虽然我们已经解决一些二维例子的问题，而且计算方法适用于更大的尺度，但是当然需要越来越多的计算成本。利用我们的变分

20、方法，我们可以向另外几个方向进一步的研究。更多的人像利用 OFA 的一般过程描述动态力学和跳跃扩散过程。控件的设置可以被认为是连续控制或者推出时间滞后的执行决定。如果我们在有限时间范围内考虑同一个模型，可以得到一个进步问题，同时我们必须要研究和寻找变分不等式的解决。 结论 现金管理问题一直被认是数学文献里财务均衡的一部分。从对鲍莫尔，托 宾，米勒和奥尔的工作中可以得出，它起源于主要的数学模型所使用的理论的交易和预防性货币需求。对解决模型起决定性作用的是使用改善 “脉冲控制的 ” 最优控制技术，这些理论都是由Bensoussan 和 lions（ 1973， 1975）介绍的。假设线性持有 /惩

21、罚成本，再加上固定比例的成本控制和现金股票是由广义布朗运动描述的动态， Constantinides 和 Richard (1978)首次指出一个简单的最优脉冲带型政策的存在。 Harrison et al(1983)证明了一个类似的结果其假设是一个非负约束的现金余额。 这意味着，最低的障碍是零，同时他们也给了一个简单的数学计算程序用来计算其余三个障碍。非负的现金余额最低水平的假设经常出现在一些论文中（如 Frenkel and Jovanovic, 1980； Chang, 1999； BarIlan et al., 2004）。但是由于它技术上的困难基本上只能取得一般情况下封闭的形势的方案。然而，它是一种流动性管理的基本功能， 但是 Bar-Ilan (1990)指出允许负现金余额（在一些惩罚性利率下）。对货币的交易需求应该考虑透支现金设施的可能性。在巴卡林， Constantinides 和 Richard (1978)的存在性结果在罚款方面也推广到二次项。在该文件中表明，只要罚款成本低于最优政策那么总是会低于零，一个简单的数值计算方法考虑到四个重要的数字。在最近的一篇文章中， Bar-Ilane l.(2004)模型的一个布朗运动的叠加和复合的假设是一个控制带的政策，他们获得有关函数