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

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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.