An iterative method for solving a PDE with free boundary arising from pricing corporate bond with credit rating migration

In this paper an iterative method is proposed to solve a partial differential equation (PDE) with free boundary arising from pricing corporate bond with credit grade migration risk. A iterative algorithm is designed to construct two sequences of fixed internal boundary problems, which produce two weak solution sequences. It is proved that both weak solution sequences are convergent. In each iteration step, an implicit-upwind difference scheme is used to solve the fixed internal boundary problem. It is shown that the scheme is stable and first-order convergent. Numerical experiments verify that the limit of the weak solution sequence is the solution of the free boundary problem. This method simplifies the free boundary problem solving, ensures the stability of the discrete scheme and reduces the amount of calculation.