free boundary and its applications in finance math

Abstract

This paper is a book which is written based on the contents of references [1] and [2]. First of all, we study the obstacle problem, and define the free boundary of the obstacle problem in chapter 1. In chapter 2, we prove that the existence of solutions of the obstacle problem as three methods

energy minimizer, singular perturbation method, and viscosity method. And in chapter 3, we study that the $C^{1,1}$ regularity of solutions of the obstacle problem and in the free boundary using the AFC monotonicity formula. Last of all, we show that the regularity of solutions of the American put option in finance math using the previous chapters.