A corporate bond pricing model incorporating step-function dividends and credit rating migration

We develop a credit rating migration model with step-function dividend rates. The model incorporates two prescribed thresholds: a credit rating migration boundary and a dividend-rate adjustment boundary, which divide the firm's asset dynamics into three distinct regions. Using a corporate bond as the pricing instrument, we formulate a structural framework in which the firm continuously pays dividends, while the dividend rate changes discontinuously when the firm's asset value crosses the dividend-rate adjustment threshold. The corresponding pricing problem leads to a system of partial differential equations with discontinuous coefficients and interface conditions. The latter are derived from the continuity of the bond value and the standard Δ-hedging argument at the migration boundary. By using regularization techniques, a priori estimates, compactness arguments, and the maximum principle, we prove the existence and uniqueness of the solution, thereby establishing the mathematical well-posedness of the model. Finally, numerical experiments are presented using both a finite difference method and a Forward-Backward Stochastic Differential Equation (FBSDE) deep learning approach.