Monte-Carlo approximation of minimum entropy measures (Q2711896)

The present note is devoted to the study of the consistency of a calibration method involved in solving a certain constrained minimization problem occurred in a financial context, using a Monte Carlo approach. For a sequence \((X_i)_{i\geq 1}\) of i.i.d. (independent, identically distributed) random variables and a probability measure \(\mu\) on a Polish space \(S\), let \(\mu_n= (1/n) \sum^n_{i=1} \delta_{X_i}\) denote the associated empirical measure. The authors give a necessary and sufficient condition for the a.s. (almost sure) existence of \(N\) such that for all \(n\geq N\) there exists a probability measure \(\nu_n\) which minimizes the relative entropy with respect to \(\mu_n\), under the constraint \(\int_S f d\nu_n= 0\), where \(f: S\to\mathbb{R}^d\) is a Borel map (\(0\) is the origin of \(\mathbb{R}^d\)). For the obtained necessary and sufficient condition, the article shows that a.s. \(\nu_n\) converges weakly to the generalized solution \(\nu\) of the minimization of the entropy with respect to \(\mu\), under the same constraint.