Abstract
For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.
Highlights
The migration process is the evolution of the credit quality of corporate bonds, corporate liabilities, corporate loans, etc
The modeling of the migration process is an important issue for risk management and pricing
We will prove the following martingale characterization theorem for a Ginhomogeneous Markov renewal theorem
Summary
The migration process is the evolution of the credit quality of corporate bonds, corporate liabilities, corporate loans, etc. The differences brought about into the new definition, they are not apparently essential, are the ones needed in order that: (i ) to state theorem 1, which establishes the change from the real world probabilities to forward probabilities in an inhomogeneous semi-Markov process; (ii ) to introduce the necessary definitions, theorems, and all the results and algorithms that follow in the present paper. The forward entrance probabilities are evaluated in a classical problem using representative data
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