# Numerical treatment to a non-local parabolic free boundary problem arising in financial bubbles

## TL;DR

This paper develops and analyzes an iterative numerical algorithm for solving a non-local parabolic free boundary problem related to financial bubbles, proving convergence and demonstrating computational effectiveness.

## Contribution

It introduces a novel iterative method combining parabolic obstacle problems and proves its convergence, along with a finite difference scheme for practical computation.

## Key findings

- Convergence of the iterative algorithm is rigorously proved.
- Finite difference scheme for the problem is shown to converge.
- Computational results validate the effectiveness of the proposed method.

## Abstract

In this paper we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper we present and discuss computational results.