  We review a numerical technique, referred to as the Transport-based Mesh-free
Method (TMM), and we discuss its applications to mathematical finance. We
recently introduced this method from a numerical standpoint and investigated
the accuracy of integration formulas based on the Monte-Carlo methodology:
quantitative error bounds were discussed and, in this short note, we outline
the main ideas of our approach. The techniques of transportation and
reproducing kernels lead us to a very efficient methodology for numerical
simulations in many practical applications, and provide some light on the
methods used by the artificial intelligence community. For applications in the
finance industry, our method allows us to compute many types of risk measures
with an accurate and fast algorithm. We propose theoretical arguments as well
as extensive numerical tests in order to justify sharp convergence rates,
leading to rather optimal computational times. Cases of direct interest in
finance support our claims and the importance of the problem of the curse of
dimensionality in finance applications is briefly discussed.
