Fuzzy semi-Markov migration process in credit risk

We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states. We study the effects of changing the real probability measure to a forward probability measure. In addition, we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default. Finally, we discuss parameter estimation and calibration of the inhomogeneous Markov chain with fuzzy states.