  In this paper, we propose a finite element method to study the problem
ofcredit rating migration problem narrowed to a free boundary problem.
Freeboundary indeed separates the high and low rating region for a firm
andcauses some difficulties including discontinuity of second order derivative
ofthe problem. Exploiting the weak formulation of the problem utilized in
theGalerkin method, the discontinuity of second order derivative is averted.
Inthis investigation we prove optimal convergence and stability of the
proposedmethod. Numerical results illustrate how derived convergence results
areconsistent into practice ones.
