scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

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.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (03) ◽  
pp. 752-756
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (2) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (3) ◽  
pp. 752-756 ◽  
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

Some Results in Markov Renewal Processes

1969 ◽  
Vol 18 (2) ◽  
pp. 61-72 ◽  
Author(s):  
A.M. Kshirsagar ◽  
Y. P. Gupta
Keyword(s):  
Generating Function ◽  
Renewal Process ◽  
First Passage Time ◽  
Probability Generating Function ◽  
Passage Time ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
First Passage ◽  
Markov Renewal Processes ◽  
Cumulative Processes

The following results are obtained in this paper: (1) The probability generating function of the simultaneous distribution of all the Nj( t)'s, where Nj( t) represents the number of times the j­th state ( j = 1, 2, ... , m) is visited in time t, in a Markov Renewal Process ; (2) the covariance between Nj( t) and Nk( t); (3) the probability generating function and moments of Nj( t)'s in a General Markov Renewal Process i.e., a Markov Renewal Process with a random origin; (4) Cumulative processes associated with a Markov Renewal Process along with its first passage time and (5) Equilibrium Markov Renewal Processes.

An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure

2004 ◽  
Vol 68 (2) ◽  
pp. 346 ◽  
Author(s):  
Keijan Wu ◽  
Naoise Nunan ◽  
John W. Crawford ◽  
Iain M. Young ◽  
Karl Ritz
Keyword(s):  
Markov Chain ◽  
Soil Structure ◽  
Markov Chain Model ◽  
Chain Model ◽  
Heterogeneous Soil

Data Mining Technique Applied To DNA Sequencing

2015 ◽  
Vol 2 (7) ◽  
pp. 18
Author(s):  
R. Jamuna
Keyword(s):  
Data Mining ◽  
Dna Methylation ◽  
Markov Chain ◽  
Human Genome ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Effective Means ◽  
Cpg Islands ◽  
Chain Model ◽  
The Given

CpG islands (CGIs) play a vital role in genome analysis as genomic markers.  Identification of the CpG pair has contributed not only to the prediction of promoters but also to the understanding of the epigenetic causes of cancer. In the human genome [1] wherever the dinucleotides CG occurs the C nucleotide (cytosine) undergoes chemical modifications. There is a relatively high probability of this modification that mutates C into a T. For biologically important reasons the mutation modification process is suppressed in short stretches of the genome, such as ‘start’ regions. In these regions [2] predominant CpG dinucleotides are found than elsewhere. Such regions are called CpG islands. DNA methylation is an effective means by which gene expression is silenced. In normal cells, DNA methylation functions to prevent the expression of imprinted and inactive X chromosome genes. In cancerous cells, DNA methylation inactivates tumor-suppressor genes, as well as DNA repair genes, can disrupt cell-cycle regulation. The most current methods for identifying CGIs suffered from various limitations and involved a lot of human interventions. This paper gives an easy searching technique with data mining of Markov Chain in genes. Markov chain model has been applied to study the probability of occurrence of C-G pair in the given   gene sequence. Maximum Likelihood estimators for the transition probabilities for each model and analgously for the  model has been developed and log odds ratio that is calculated estimates the presence or absence of CpG is lands in the given gene which brings in many  facts for the cancer detection in human genome.

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