muchen 牧辰

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 \(\mathbb P(A,B\vert C)=\mathbb P(A\vert B,C)\mathbb P(B\vert C)\),


\[=\sum_{i\in\mathcal S}\mathbb P(X_{n+2}=\text{default}\vert X_{n+1}=i)\cdot\mathbb P(X_{n+1}=i\vert X_n=\text{AAA})\]

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

Markov Chains Simulation