Nicolas Bouleau

Nicolas Bouleau

This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second order vector in differential geometry, emphasizing the existence of a slight ambiguity concerning the bias operator. The third part in devoted to the powerful framework of Dirichlet forms whose main feature is to apply easily to infinite dimensional models including the Wiener space (giving an interpretation of Malliavin calculus in terms of errors), the Poisson space and the Monte Carlo space. In the fourth part we show how an error in the usual mathematical sense, i.e. an approximate quantity, may yield a Dirichlet form and we introduce the four bias operators. Eventually we connect the Dirichlet form with statistics by identifying the square of field operator with the inverse of the Fisher information matrix.

Nicolas Bouleau

The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming back to the finite dimensional case, these methods give a new light on the very classical concept of 'numerical approximation' and suggest changes in the habits. We show that for some kinds of approximations only an Ito-like second order differential calculus is relevant to describe and propagate numerical errors through a mathematical model. We call these situations strongly stochastic. The main point of this work is an argument based on the arbitrary functions principle of Poincaré-Hopf showing that the errors due to measurements with graduated instruments are strongly stochastic. Eventually we discuss the consequences of this phenomenon on the specification of an approximate numerical result.