$ X $ a Markov chain has value in $\{1,2,3\}$ with transiton matrix $P$

Question

$ X $ a Markov chain has value in $\{1,2,3\}$ with transiton matrix $P$ the transition matrix and $u = (u_1, u_2, u_3)$ the law of $X_0$. What is $P (X_0 = 1, X_1 = 1, X_2 = 3, X_3 = 2)$ as a function of $P$ and $u$?

Answer 1

Note that $P_{ij}$ denotes the probability of transitioning from state $i$ to state $j$. Then \begin{align} P(X_{0}=1, X_{1}=1, X_{2}=3, X_{3}=2) &= P(X_{3}=2|X_{2}=3)\cdot P(X_{2}=3|X_{1}=1)\cdots \\ &\cdot P(X_{1}=1|X_{0}=1)\cdot P(X_{0}=1)\\ &=P_{32}P_{13}P_{11}u_{1} \end{align}