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For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages. while the use of importance sampling instead of crude Monte Carlo sampling has been recommended by Professor Peter W. Glynn. At the beginning of 1989 the author joined the Department of Operations Research at Stanford University as a visiting scholar from Vienna University of Technology, and started to work closely with Professor Dantzig on the development of techniques for solving large-scale stochastic linear programs. Progress on this topic has led to extensions of the author's visit and to his current position of Senior Research Associate at the Department of Operations Research at Stanford University. It has been an outstanding and influential experience for the author to know and to collaborate with Professor George Dantzig. The author wants to thank Professor Dantzig for everything, all his advice, his outstanding professional support and his great friendship. The author wants to thank Professor Peter Glynn for the many valuable discussions on the topic of planning under uncertainty and importance sampling and his great support. The author is grateful to Professor Michael Saunders and Dr. John Stone for the valuable discussions on the topic of large-scale systems and their important and helpful suggestions concerning previous versions of this paper. The author collaborated with Professor James K. Ho on using parallel processors. The author wishes to thank Professor Ho for this collaboration. Thanks also to David Morton for valuable comments on a previous version of this paper and to Alamuru Krishna, who assisted in preparing some of the test problems. The author is especially grateful to Professor Peter Jansen, his "Doktorvater", for initiating the author's visit to Stanford University and his continuing great support.

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The paper demonstrates how multi-period portfolio optimization problems can be efficiently solved as multi-stage stochastic linear programs. A scheme based on a blending of classical Benders decomposition techniques and a special technique, called importance sampling, is used to solve this general class of multi-stage stochastic linear programs. We discuss the case where stochastic parameters are dependent within a period as well as between periods. Initial computational results are presented.*Research and reproduction of this report were partially supported by the Office of Naval lie.earch Contract N00014-89-J-1659; the National Science Foundation Grants ECS-8906260, )\MS-89liI$ 9). the Electric Power Research Institute Contract RP 8010-09. CSA-1005335, and the Austrian Science Foundation, "Fonds zur F6rderung der wissenschaftlichen Forschung." Grant .10323-Phy. Any opinions, findings, and conclusions or recommendations expressed in this publication are thos' of the autlhur., and dIo NOT necessarily reflect the view-. ,-f the above sponsors.

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