On Inverse Problems and Machine Learning

Abstract

This document is related to Ill-Posed and Inverse problems particularly focused on economicmeasurements. In 2015, I proposed to myself to work both analytically and numerically on a very fresh and surprising idea: to predict prices of stock options using the famous Black- Scholes equation. In mathematical finance, the Black–Scholes equation is a parabolic partial differential equation in both time and space that models the price of common financial assets. This equation when solved forwards in time to forecast prices of stock options is an ill-posed inverse problem. Note that standard techniques which were known at that time, do not provide any tools for predictions of prices. Besides, solving the Black-Scholes equation forwards in time is an ill-posed problem, which complicates the matter tremendously. On the other hand, it is intuitively clear that an accurate prediction of prices even for the next couple business days is exactly what the market dreams about.