Differentiability of Product Measures

B. Heidergott and H. Leahu
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B. Heidergott: Vrije Universiteit Amsterdam, Faculteit der Economische Wetenschappen en Econometrie (Free University Amsterdam, Faculty of Economics Sciences, Business Administration and Economitrics

No 5, Serie Research Memoranda from VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics

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.

Date: 2008
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