# 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