# Inverse problems for random differential equations using the collage method for random contraction mappings

*Davide La Torre* (),
*Herb Kunze* and
*Ed Vrscay*

No unimi-1036, UNIMI - Research Papers in Economics, Business, and Statistics from Universitá degli Studi di Milano

**Abstract:**
Most natural phenomena or the experiments that explore them are subject to small variations in the environment within which they take place. As a result, data gathered from many runs of the same experiment may well show differences that are most suitably accounted for by a model that incorporates some randomness. Differential equations with random coefficients are one such class of useful models. In this paper we consider such equations as random fixed point equations T(w,x(w)) = x(w), where T : \Omega × X \to X is a random integral operator, \Omega is a probability space and X is a complete metric space. We consider the following inverse problem for such equations: given a set of realizations of the fixed point of T (possibly the interpolations of different observational data sets), determine the operator T or the mean value of its random components, as appropriate. We solve the inverse problem for this class of equations by using the collage theorem.

**Keywords:** random fixed point equations; random differential equations (search for similar items in EconPapers)

**Date:** 2006-09-07

**Note:** oai:cdlib1:unimi-1036

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