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Numerical and Optimization Techniques

  • Charles S. Tapiero

Abstract

By far, most problems cannot be solved by the application of analytical techniques. Numerical techniques for the analysis of stochastic processes and the solution of stochastic control and optimization problems become then essential and the only means available to study and optimize stochastic processes. These techniques are varied however, often emphasizing approaches which are quite different (Tapiero 1993, Sulem and Tapiero 1994). For example, in the solution of SCP’s, various approaches can be applied, each emphasizing a different and conceptual approach to stochastic optimization, including:
  • ◆ A probabilistic approach which transforms the stochastic process into a Markov chain and then applies Markov Decision Processes (MDP) which are based for their solutions on policy and value iteration algorithms (Kushner 1967, 1971, 1977).

  • ◆ Discretization of the problem at hand, followed by the application of various techniques for the iterative solution of partial differential equations such as finite differencing, binomial trees, the finite element numerical technique and so on.

  • ◆A third approach is based on the iterative solution of problems we know how to solve analytically (in particular, the linear quadratic problem). Such an approach is called perturbation techniques (Bensoussan 1982).

  • ◆ A fourth approach is based on techniques and algorithms such as stochastic approximations (Kiefer and Wolfowitz 1952) and other related schemes.

  • ◆Stochastic programming which transforms the stochastic optimization problems to nonlinear programming (Rockafellar and Wets, 1990, Varayia and Wets, 1989 and finally

  • ◆ Simulation (Newman and Odell 1971, Rubenstein 1981) which can be applied in many different and creative ways.

Keywords

Stochastic Differential Equation Numerical Technique Stochastic Programming Markov Decision Process Stochastic Control Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.
    This section as well as subsequent ones provide a cursory approach to stochastic control optimization. For further study, references are suggested for the motivated reader.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Charles S. Tapiero
    • 1
  1. 1.ESSECCergy PontoiseFrance

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