The rating process is modelled as a time homogeneous $K$-state Markov chain with ratings $R\in \{1,2,\dots,K\}$ where state $1$ is the best rating and $K$ is default. We may then define a $K \times K$ matrix of transition probabilities for ONE PERIOD $(t,t+1)$:

\begin{align*}
P (t,t+1)= \left(\begin{array}{cccc}p_{11} & p_{12} & \cdots & p_{1K} \\p_{21} & p_{22} & \cdots & p_{2K} \\ \vdots & \vdots &\ddots & \vdots \\p_{(K-1)1} & p_{(K-1)2} & \cdots & p_{(K-1)K}\\
0 & 0 & \cdots & 1
\end{array}\right)
\end{align*}

Default is an absorbing state. A default cannot be recovered from hence the probability of going from default $K$ to rating $1,2,\dots,(K-1)$ is zero. We therefore say that $tr_{K,R}(\tau) = 0$ for all $R \in \{1,2,\dots,(K-1)\}$

All entries must be non-negative. Each row has to sum to $1$.
$$\sum_{j=1}^K p_{ij}(t,t+1) = 1$$

Two-Period and Multi-Period, objective transition Probabilities

Two Period. The probability $p_{ij}$ that a bond rated $i$ at time $t$ can go to state $j$ at time $t+2$.
\begin{align*}
P (t,t+2)= P (t,t+1)P (t+1,t+2)
\end{align*}This is known as the discrete time or Chapman-Kolmogorov equation [ref][/ref]

Multi-Period\begin{align*}
P (t,t+n)&= P (t,t+1) P (t+1,t+2) \cdots P(t+k-1,t+n)\\
&=\prod_{j=1}^{n-1} P(t+j,t+j+1)
\end{align*}If transition probabilities are independent of time
$$P(t,t+n) = P^n$$

Risk-neutral transition Probabilities $\; \pi_{ij} (t)$

Ratings agencies provide estimates for real world $p_{ij}$ we assume is modelled by a time-homogeneous Markov Chain[ref value=’1′]Tomasz R. Bielecki & Marek Rutkowski. Credit Risk: Modelling, Valuation and Hedging. (Springer).[/ref]. Each objective measure $p_{ij}$ has a corresponding $q_{ij}$ under the risk-neutral measure
\begin{align*}
q_{ij}(t, t+1) &= \pi_{ij} (t) p_{ij} \\
&\equiv \mathbb{Q} \text{(rating $j$ at $t+1$ | rating $i$ at $t$)}\qquad i\neq j
\end{align*}where $\pi_{ij}$ is the Radon-Nikodym derivative as a function of time. It is the price of migration risk (or default) for the transition from rating $i$ to rating $j$

Conditions on $\pi_{ij}(t)$:

• $\pi_{ij}(t) >0 \; \text{  if  } \;p_{ij}>0$
• $\sum_j \pi_{ij}(t) p_{ij} = 1 \quad \text{i.e. the sum of all } q_{ij} $
• We can then infer that \begin{align*}q_{ii} = \pi_{ii} (t) p_{ii} = 1 – \sum_{j:j \neq i} \pi_{ij}(t)p_{ij}\end{align*}as long as the right hand side is positive.
• assume the price of risk $\pi_{ij} (t)$ depends only on the initial state $i$

Jarrow Lando and Turnbull (1997) Specification

\begin{align*}
q_{ij} &= \pi_{i} (t) p_{ij} \qquad j \neq i, i = 1, \cdots K-1\\
q_{Kj} &= \delta_{Kj} \qquad \text{Kronecker delta – } \; \begin{cases} 1 \; \text{ if } \; i = j \\ 0 \; \text{ if } \; i \neq j \end{cases}
\end{align*}Because default is an absorbing inescapable state, $p_{Ki}=0$ which implies $q_{iK}=0$ regardless of the value given to $\pi_{iK}(t)$. Consequently we only need to specify $\pi_i (t)$ in the JLT model.
\begin{align*}
q_{ii} &= 1 – \sum_{j:j \neq i} \pi_{i}(t)p_{ij}\\
&= 1 – \sum_{j:j \neq i} \pi_{i}(t)p_{ij}\\
& = 1 – \pi_{i}(t) \sum_{j:j \neq i}p_{ij}\\
\therefore \qquad q_{ii}& = 1 – \pi_{i}(t) (1 – p_{ii})
\end{align*}
Now introduce diagonal matrix
\begin{align*}
\Pi (t)= \left(\begin{array}{ccccc}\pi_1(t) & 0 & \cdots & 0 & 0 \\
0 & \pi_2(t) & \cdots & 0 & 0 \\
\vdots & \vdots &\ddots & \vdots & \vdots \\0 & 0 & \cdots & \pi_{(K-1)}(t) & 0\\
0 & 0 & \cdots & 0 & \pi_K(t)=1
\end{array}\right)
\end{align*}
and $I_K$ be the $K \times K$ identity matrix. then
\begin{align*}
Q (t,t+1) = I_K + \Pi (t) (P – I_K)
\end{align*}
where the right hand side is a matrix with

• off-diagonal elements $q_{ij} = \pi_{i}(t)p_{ij}$
• diagonal elements $q_{ii} = 1 + \pi_{i}(t)( p_{ii} \; – 1)$