Parameter identification for deterministic and stochastic differential equations using the "collage method" for fixed point equations

A number of inverse problems may be viewed in terms of the approximation of a target element x in a complete metric space (X,d) by the fixed point x* of a contraction function T : X -> X. In practice, from a family of contraction functions T(a) one wishes to find the parameter a for which the approximation error d(x,x*(a)) is as small as possible. Thanks to a simple consequence of Banach's fixed point theorem known as the Collage Theorem, most practical methods of solving the inverse problem for fixed point equations seek to find an operator T(a) for which the so called collage distance d(x,T(a)x) is as small as possible. We first show how to solve inverse problems for deterministic and random differential equations and then we switch to the analysis of stochastic differential equations. Here inverse problems can be solved by minimizing the collage distance in an appropriate metric space. At the end we show an application of this approach to a system of coupled stochastic differential equations which describes the interaction between particles in a physical system