ELEC 321

# 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