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

The Transport-based Mesh-free Method (TMM) and its applications in finance: a review