Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles

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.Here the coefficients a i,j , b i , c are assumed to be continues and the matrix [a i,j (x)] is positive definite for all x ∈ Ω. Additionally we assume that the coefficients are 2010 Mathematics Subject Classification. 35R35; 35D40; 65M06; 91G80.