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Differentiability of Product Measures

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  • Heidergott, B.

    (Vrije Universiteit Amsterdam, Faculteit der Economische Wetenschappen en Econometrie (Free University Amsterdam, Faculty of Economics Sciences, Business Administration and Economitrics)

  • Leahu, H.

Abstract

In this paper, we study cost functions over a finite collection of random variables. For this type of models, a calculus of differentiation is developed that allows to obtain a closed-form expression for derivatives, where “differentiation” has to be understood in the weak sense. The techniques for establishing the results is new and establish an interesting link between functional analysis and gradient estimation. By establishing a product rule of weak analyticity, Taylor series approximations of finite products can be established. In particular, from characteristics of the individual probability measures a lower bound, i.e., domain of convergence can be established for the set of parameter values for which the Taylor series converges to the true value. Applications of our theory to the ruin problem from insurance mathematics and to stochastic activity networks arising in project evaluation review technique are provided.

Suggested Citation

  • Heidergott, B. & Leahu, H., 2008. "Differentiability of Product Measures," Serie Research Memoranda 0005, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    • Handle: RePEc:vua:wpaper:2008-5
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    File URL: http://degree.ubvu.vu.nl/repec/vua/wpaper/pdf/20080005.pdf
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    References listed on IDEAS

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    1. Rommert Dekker & Arie Hordijk, 1988. "Average, Sensitive and Blackwell Optimal Policies in Denumerable Markov Decision Chains with Unbounded Rewards," Mathematics of Operations Research, INFORMS, vol. 13(3), pages 395-420, August.
    2. Arie Hordijk & Alexander A. Yushkevich, 1999. "Blackwell optimality in the class of all policies in Markov decision chains with a Borel state space and unbounded rewards," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(3), pages 421-448, December.
    3. Arie Hordijk & Alexander A. Yushkevich, 1999. "Blackwell optimality in the class of stationary policies in Markov decision chains with a Borel state space and unbounded rewards," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(1), pages 1-39, March.
    4. Luc P. Devroye, 1979. "Inequalities for the Completion Times of Stochastic PERT Networks," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 441-447, November.
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    Cited by:

    1. Bernd Heidergott & Arie Hordijk & Haralambie Leahu, 2009. "Strong bounds on perturbations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(1), pages 99-127, August.

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