CONVERGENCE RATE OF AN EXPLICIT FINITE DIFFERENCE SCHEME FOR A CREDIT RATING MIGRATION PROBLEM

In this paper, we present a rigorous proof for the convergence and error estimates of a free boundary problem modeling a credit rating migration. The high and low rating regions are characterized by different volatilities sigma(H) and sigma(L), respectively. The high and low rating regions are separated by a free boundary. The changes in volatilities across the free boundary result in a discountinuity of the leading order coefficients, which implies that the solution will have a discontinuous second order spatial derivative across the free boundary; it is a challenge to establish convergence of a finite difference scheme for this type of solution. Through an approximation by smooth coefficients, energy estimates, delicate construction of super- and subsolutions, and establishment of a comparison principle, we prove L-infinity convergence of the explicit finite difference scheme. Furthermore, the convergence rate is of 1/4 order temporal accuracy and of 1/2 order spatial accuracy. To our best knowledge, this is the first work on convergence of a finite difference scheme for a rating migration problem.