Application of Microlocal Analysis to an Inverse Problem Arising from Financial Markets

TL;DR

This paper introduces a novel application of microlocal analysis to an inverse problem in financial markets, demonstrating uniqueness of solutions within a new arbitrage model based on the Black--Scholes framework.

Contribution

It is the first to apply microlocal analysis to a financial market model, establishing solution uniqueness for a new inverse problem derived from arbitrage considerations.

Findings

Proved uniqueness of the inverse problem's solution using microlocal analysis.

Developed a stable linearization and integral equation for space-dependent drift.

Applied microlocal analysis to demonstrate solution stability and uniqueness.

Abstract

One of the most interesting problems discerned when applying the Black--Scholes model to financial derivatives, is reconciling the deviation between expected and observed values. In our recent work, we derived a new model based on the Black--Scholes model and formulated a new mathematical approach to an inverse problem in financial markets. In this paper, we apply microlocal analysis to prove a uniqueness of the solution to our inverse problem. While microlocal analysis is used for various models in physics and engineering, this is the first attempt to apply it to a model in financial markets. First, we explain our model, which is a type of arbitrage model. Next we illustrate our new mathematical approach, and then for space-dependent real drift, we obtain stable linearization and an integral equation. Finally, by applying microlocal analysis to the integral equation, we prove our uniqueness of the solution to our new mathematical model in financial markets.