Markov Process
Updated 2017-11-21
Note: this page is quite empty, visit the tutorial page on Markov Processes and review problem for more in-depth explanation and practice of Markov chains.
Example: Rating migration of bonds
\[\mathbb P(X_{n+2}=\text{default}\vert X_n=\text{AAA})\\ =\sum_{i\in\mathcal S}\mathbb P(X_{n+2}=\text{default},X_{n+1}=i\vert X_n=\text{AAA})\\ =\sum_{i\in\mathcal S}\mathbb P(X_{n+2}=\text{default}\vert X_{n+1}=i,X_n=\text{AAA})\cdot\mathbb P(X_{n+1}=i\vert X_n=\text{AAA})\]Where \(\mathcal S\) is the state space: \(\{\text{AAA. AA,}\dotsc,\text{default}\}\).
Since \(\mathbb P(A\vert C)=\sum_B \mathbb P(A,B\vert C)\) and \(\mathbb P(A,B\vert C)=\mathbb P(A\vert B,C)\mathbb P(B\vert C)\),
Thus
\[=\sum_{i\in\mathcal S}\mathbb P(X_{n+2}=\text{default}\vert X_{n+1}=i)\cdot\mathbb P(X_{n+1}=i\vert X_n=\text{AAA})\]Now we may list the probabilities and add them
| \(i\) | Probability from \(i\) to default (in second year): \(\mathbb P(X_{n+2}=\text{default}\vert X_{n+1}=i)\) | Probability from AAA to \(i\): \(\mathbb P(X_{n+1}=i\vert X_n=\text{AAA})\) |
|---|---|---|
| AAA | 0 | 0.9366 |
| AA | 0.0002 | 0.0583 |
| A | 0.0004 | 0.0040 |
| \(\vdots\) | \(\vdots\) | \(\vdots\) |
| default | 1 | 0 |