Credit Risk Assessment Using Fuzzy Inhomogeneous Markov Chains Within a Fuzzy Market

In the present study, we model the migration process and the changes in the market environment. The migration process is being modeled as an $\mathcal{F}$-inhomogeneous semi-Markov process with fuzzy states. The evolution of the migration process takes place within a stochastic market environment with fuzzy states, the transitions of which are being modeled as an $\mathcal{F}$-inhomogeneous semi-Markov process. We prove a recursive relation from which we could find the survival probabilities of the bonds or debts as functions of the basic parameters of the two $\mathcal{F}$-inhomogeneous semi-Markov processes. The asymptotic behavior of the survival probabilities is being found under certain easily met conditions in closed analytic form. Finally, we provide maximum likelihood estimators for the basic parameters of the proposed models.