Differentiability of Product Measures
AbstractIn 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.
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Bibliographic InfoPaper provided by VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics in its series Serie Research Memoranda with number 0005.
Date of creation: 2008
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- Bernd Heidergott & Arie Hordijk & Miranda van Uitert, 2005. "Series Expansions for Finite-State Markov Chains," Tinbergen Institute Discussion Papers 05-086/4, Tinbergen Institute.
- Felisa J. Vazquez-Abad & Bernd Heidergott, 2003. "Gradient Estimation for a Class of Systems with Bulk Services: A Problem in Public Transportation," Tinbergen Institute Discussion Papers 03-057/4, Tinbergen Institute.
- Heidergott, Bernd & Vazquez-Abad, Felisa J. & Volk-Makarewicz, Warren, 2008. "Sensitivity estimation for Gaussian systems," European Journal of Operational Research, Elsevier, vol. 187(1), pages 193-207, May.
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