Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk
For a <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-inhomogeneous semi-Markov chain and <inline-formula><math display="inline"><semanti...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-12-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/1/55 |
| _version_ | 1827699061884452864 |
|---|---|
| author | P.-C.G. Vassiliou |
| author_facet | P.-C.G. Vassiliou |
| author_sort | P.-C.G. Vassiliou |
| collection | DOAJ |
| description | For a <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-inhomogeneous semi-Markov chain and <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds. |
| first_indexed | 2024-03-10T13:41:46Z |
| format | Article |
| id | doaj.art-8622d72e98a04c4aa76377f2328cdbbf |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T13:41:46Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-8622d72e98a04c4aa76377f2328cdbbf2023-11-21T02:59:26ZengMDPI AGMathematics2227-73902020-12-01915510.3390/math9010055Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit RiskP.-C.G. Vassiliou0Department of Statistical Sciences, University College London, Gower Street, London WC1E 6BT, UKFor a <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-inhomogeneous semi-Markov chain and <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.https://www.mdpi.com/2227-7390/9/1/55credit ratingnon-homogeneous semi-Markov chainsnon-homogeneous Markov renewal processeschange of measure |
| spellingShingle | P.-C.G. Vassiliou Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk Mathematics credit rating non-homogeneous semi-Markov chains non-homogeneous Markov renewal processes change of measure |
| title | Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk |
| title_full | Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk |
| title_fullStr | Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk |
| title_full_unstemmed | Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk |
| title_short | Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk |
| title_sort | non homogeneous semi markov and markov renewal processes and change of measure in credit risk |
| topic | credit rating non-homogeneous semi-Markov chains non-homogeneous Markov renewal processes change of measure |
| url | https://www.mdpi.com/2227-7390/9/1/55 |
| work_keys_str_mv | AT pcgvassiliou nonhomogeneoussemimarkovandmarkovrenewalprocessesandchangeofmeasureincreditrisk |