How to specify an approximate numerical result

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. 1 The dichotomy of small errors. Let us begin by showing that there are two kinds of small errors which do not propagate according to the same differential calculus. Suppose two applied mathematicians A and B attempt to perform stochastic simulation rigourously. By means of the well known inversion and rejection methods, they are able to simulate any probability law provided that they can pick up a real number in the unit interval [0, 1] randomly. 1 ha l-0 07 81 41 4, v er si on 1 26 J an 2 01 3 Author manuscript, published in "RIMS Kokyuroku Bessatsu B6 (2008) 39-53"