The Transport‐based Mesh‐free Method: A Short Review

Abstract

We introduce a new numerical strategy which we refer to as the Transport-based Mesh-free Method (TMM) and discuss its applications to mathematical finance. The proposed method enjoys good accuracy properties similar to those obtained with integration formulas based on the Monte-Carlo methodology, and in particular enjoys quantitative error bounds which have important implications in applications. In this short review, we outline the main ideas behind this new strategy which relies on techniques of transportation and reproducing kernels. It leads us to an efficient method for numerical simulations while providing some light on the techniques currently developed by the artificial intelligence community. In the applications in the finance industry, our approach provides us with an accurate and fast algorithm, allowing us to compute various types of risk measures. Theoretical arguments are also put forward in order to justify the sharp convergence rates and almost optimal computational times that we observe in our numerical tests and, in addition, typical cases arising in finance applications support our claims. The problem of the curse of dimensionality in finance is briefly discussed.