Mathematical Sciences: Applications of Stochastic Control toConsumption/Investment Decisions, Equilibrium Analysis and Production

这个项目包括三个子领域的研究，其中前两个领域紧密相连，而最后一个领域略有脱节。在第一个领域，普遍的问题是，货币必须持续投资或消费，以使总消费的净收益达到最大。这些投资可以分为固定回报率的债券和几只高风险的股票。借贷和卖空(投资负额)等变化是可能的。在以前的一些工作中，作者在各种收益率是随机的而不是固定的情况下取得了一些进展。同样在最近的一项工作中，通过度量变化论证，他们将反馈形式的最优消费率与消费函数的导数的泛函逆联系起来。作者建议在无限版本的问题、借款和卖空受到限制的情况下以及涉及破产偿付的情况下进行扩展。第二个领域与第一个领域密切相关，涉及多个代理人，他们都在追求各自的最优消费和投资政策。其目的是研究净投资、财富创造等都为零的均衡情况，并证明现货价格的存在。与上面略有脱节的最后一个领域涉及在m台机器上生产n种商品的制造系统中的生产控制。目标是最大限度地减少库存成本和生产系统，以满足特定的需求，即使机器容易出现故障。作者建议研究故障率与机器利用率成正比的情况。他们还建议在存在机器连续体的情况下进行扩展，但这似乎不那么令人信服。这种类型的研究对经济理论和库存管理都很有兴趣，同时也是当前控制研究的主流，被视为数学的一部分。

This project consists of research suggested in three sub-areas, of which the first two are closely linked, while the last is slightly disjoint. In the first area, the general problem is one where money has to be continuously invested or consumed so that the net benefit of the total consumption is a maximum. The investments can be divided between a bond, with a fixed rate of return, and several risky stocks. Variations such as borrowing and short selling (investing negative amounts) are possible. In some previous work the authors have made some progress on the case where the various rates of return are random rather than fixed. Also in a recent work, by means of measure change argument, they have linked the optimal consumption rate in feedback form to the functional inverse of the derivative of the consumption function. The authors propose extensions in infinite versions of the problem, situations where borrowing and short selling are restricted, and cases where a bankruptcy payment is involved. The second area, which is closely related to the first, deals with a multiplicity of agents all pursuing their individually optimal consumption and investment policies. The goal is to investigate the equilibrium situation where net investment, and creation of wealth are all zero etc. and show the existence of spot prices. The last area, which is slightly disjoint from the above deals with the control of production in a manufacturing system producing n commodities on m machines. The goal is to minimize holding inventory costs and production systems so as to meet specified demands, even when machines are liable to failure. The authors propose to investigate the case where the failure rate is proportional to machine utilization. They also suggest an extension where there is a continuum of machines, but this appears less convincing. The research of this type is of interest to economic theory and stock management, while at the same time being in the mainstream of current research efforts in control seen as part of mathematics.