2. Conditional transition probabilities

To state the conditional embedding problem, we first introduce the concept of conditional transition probability.

Consider a continuous-time homogeneous Markov chain (‘CTHMC’) $ (X_t)_{t\geq 0}$ on a probability triple $ (\Omega,\mathscr{F},\mathbb{P})$ with state space $ \mathcal{S}=\{1,2,\ldots,n\}$ .

Definition 2.1. Let $ t\geq 0$ and $ \mathcal{E}_t\in\mathscr{F}$ . We call the matrix $ \mathbf{P}^{\mathcal{E}_t}(t)$ with elements

\[ p^{\mathcal{E}_t}_{ij}(t)=\mathbb{P}(X_{t+1}=j \mid X_{t}=i ,\,\mathcal{E}_t),\quad i,j\in\mathcal{S},\]

the conditional transition probability matrix from time t to time $ t+1$ given the event $ \mathcal{E}_t$ of the CTHMC.

In this paper, for given t, the event of interest is the occurrence of at most one jump of the CTHMC in the interval $ (t,t+1]$ , i.e. $ \mathcal{E}_t=\{N_{t+1}-N_{t}\leq 1\}$ , where $ N_s$ represents the number of state changes (‘jumps’) of the CTHMC up to and including time $ s\geq 0$ . For simplicity, we denote $ \mathbf{P}^{\unicode{x1D7D9}}(t)\,:\!=\, \mathbf{P}^{\mathcal{E}_t}(t)$ .

For a CTHMC with rate matrix $ \mathbf{Q}$ , the connection between $ \mathbf{P}^{\unicode{x1D7D9}}(t)=({p}^{\unicode{x1D7D9}}_{ij}(t))$ and $ \mathbf{Q}$ is established by the following proposition.

Proposition 2.1. For a CTHMC with rate matrix $ \mathbf{Q}=(q_{ij})$ , it holds that

\[ {p}^{\unicode{x1D7D9}}_{ij}(t) = \dfrac{p_{ij}^{*}}{\sum_{k\in\mathcal{S}} p_{ik}^{*}}\quad for\ all\ i,j\ \in \mathcal{S}\ and\ t \geq 0, \]

where

(2.1) \begin{align} p_{ij}^{*} = \begin{cases}q_{ij}\,\tau(q_{ii},q_{jj}) & if\ i\neq j,\\[3pt] \tau(q_{ii},q_{ii}) & if\ i=j,\end{cases} \end{align}

and where the function $ \tau \colon \mathbb{R}^2\to\mathbb{R}$ is defined as

(2.2) \begin{align}\tau(x,y)=\int_{0}^{1} \mathrm{e}^{ux+(1-u)y}\,\mathrm{d}{u}=\begin{cases}\dfrac{\mathrm{e}^{x}-\mathrm{e}^{y}}{x-y} & if\ x\neq y,\\[3pt]\mathrm{e}^{x} & if\ x=y.\end{cases}\end{align}

Proof. Using the definition of conditional probability, we have

\[\mathbb{P}(A \mid B\cap C)=\dfrac{\mathbb{P}(A\cap B \mid C)}{\mathbb{P}(B \mid C)},\quad\text{if}\ \mathbb{P}(B \mid C) \gt 0, \]

whence

\begin{equation*}{p}^{\unicode{x1D7D9}}_{ij}(t) = \mathbb{P}(X_{t+1}=j \mid X_{t}=i,\,\mathcal{E}_t)=\dfrac{\mathbb{P}(X_{t+1}=j,\,\mathcal{E}_t \mid X_{t}=i)}{\mathbb{P}(\mathcal{E}_t \mid X_{t}=i)}.\end{equation*}

Let us denote

(2.3) \begin{align} p^{*}_{ij}(t)\,:\!=\, \mathbb{P}(X_{t+1}=j,\,\mathcal{E}_t \mid X_{t}=i).\end{align}

By the sum rule for disjoint events, we have

\[{p}^{\unicode{x1D7D9}}_{ij}(t)=\dfrac{p_{ij}^{*}(t)}{\sum_{k \in \mathcal{S}} p_{ik}^{*}(t)},\]

so it remains to be shown that $ p_{ij}^{*}(t)=p^{*}_{ij}$ for all $ i,j\in\mathcal{S}$ , where $ p^{*}_{ij}$ is given by (2.1).

Denote $ \mathcal{S}$ as the disjoint union of $ \mathcal{S}^{\text{T}}\,:\!=\, \{i\in\mathcal{S} \colon q_{ii} \lt 0\}$ (the subset of transient states) and $ \mathcal{S}^{\text{A}}\,:\!=\, \{i\in\mathcal{S} \colon q_{ii}=0\}$ (the subset of absorbing states). Let $ Y_t\,:\!=\, \inf\{s \gt 0 \colon X_{t+s}\neq X_t\}$ be the residual time in state $ X_t$ until the first jump after t. In a CTHMC, the holding times are memoryless, hence the conditional distribution of $ Y_t$ , given state $ X_t$ , does not depend on time t. For $ k\in\mathcal{S}^{\text{T}}$ , let $ f_k$ denote the conditional probability density function of $ Y_t$ given $ X_t=k$ and $ F_k$ its associated cumulative distribution function. Lastly, we shall write $ s_{ij}$ to denote the transition probability from state i to state j conditional on transitioning out of state i.

We start with the case $ i,j\in\mathcal{S}^{\text{T}}$ , $ i\neq j$ . Then (2.3) entails

\[p^{*}_{ij}(t)=\mathbb{P}(Y_t\leq 1,X_{t+Y_t}=j,Y_{t+Y_t} \gt 1-Y_t \mid X_t=i),\]

and hence, by marginalizing on $ Y_t$ , we obtain

\begin{align*}p^{*}_{ij}(t)&=\int_{0}^{1}\mathbb{P}(X_{t+u}=j,Y_{t+u} \gt 1-u \mid Y_t=u,X_t=i)f_i(u)\,\mathrm{d}{u} \\[3pt]&=\int_{0}^{1}\mathbb{P}(Y_{t+u} \gt 1-u \mid X_{t+u}=j,Y_t=u,X_t=i)\, \mathbb{P}(X_{t+u}=j \mid Y_t=u,X_t=i)f_i(u)\,\mathrm{d}{u} \\[3pt]&=\int_{0}^{1}(1-F_j(1-u))\,s_{ij}\,f_i(u)\,\mathrm{d}{u}.\end{align*}

Since both residual and holding times in a transient state k have an exponential distribution with rate parameter $ -q_{kk}$ and since $ s_{ij}={{q_{ij}}/{-q_{ii}}}$ , we arrive at

\[p^{*}_{ij}(t)=\int_{0}^{1} \mathrm{e}^{q_{jj}(1-u)} \,\dfrac{q_{ij}}{-q_{ii}}\,({-}q_{ii})\mathrm{e}^{q_{ii}u}\,\mathrm{d}{u}= q_{ij}\int_{0}^{1}\mathrm{e}^{q_{ii}u+q_{jj}(1-u)}\,\mathrm{d}{u}=q_{ij}\,\tau(q_{ii},q_{jj})=p^\ast_{ij},\]

where $ p^\ast_{ij}$ and $ \tau$ are given by (2.1) and (2.2).

We now turn to the case $ i\in\mathcal{S}^{\text{T}}$ and $ j\in\mathcal{S}^{\text{A}}$ . Then, using (2.3) and marginalizing on $ Y_t$ , we get

\begin{align*}p^{*}_{ij}(t)&=\mathbb{P}(Y_t\leq 1,X_{t+Y_t}=j \mid X_t=i)\\[3pt]&=\int_{0}^{1}\mathbb{P}(X_{t+u}=j \mid Y_t=u,X_t=i)f_i(u)\,\mathrm{d}{u} \\[3pt]&=\int_{0}^{1}s_{ij}\,f_i(u)\,\mathrm{d}{u}\\[3pt]&=\int_{0}^{1} \dfrac{q_{ij}}{-q_{ii}}\,({-}q_{ii})\mathrm{e}^{q_{ii}u}\,\mathrm{d}{u}\\[3pt]&=q_{ij}\int_{0}^{1} \mathrm{e}^{q_{ii}u}\,\mathrm{d}{u}=q_{ij}\tau(q_{ii},0).\end{align*}

Consequently, $ p^\ast_{ij}(t)=q_{ij}\tau(q_{ii},q_{jj})=p^\ast_{ij}$ , as $ q_{jj}=0$ .

When $ i\in\mathcal{S}^{\text{A}}$ and $ j\in\mathcal{S}$ , $ j\neq i$ , we obviously have $ p^\ast_{ij}(t)=0$ . Hence

\[p^\ast_{ij}(t)=q_{ij}\tau(q_{ii},q_{jj})=p^\ast_{ij},\]

as $ q_{ij}=0$ under these conditions.

Finally, if $ i\in\mathcal{S}^{\text{T}}$ , (2.3) yields

\[p^{*}_{ii}(t)=\mathbb{P}(Y_t \gt 1 \mid X_t=i)=1-F_i(1)=\mathrm{e}^{q_{ii}}=\tau(q_{ii},q_{ii})=p^\ast_{ii} ,\]

and if $ i\in\mathcal{S}^{\text{A}}$ , then

\begin{equation*}p^{*}_{ii}(t)=1=\tau(0,0)=\tau(q_{ii},q_{ii})=p^\ast_{ii}. \end{equation*}

We note that Minin and Suchard [Reference Minin and Suchard26] arrive at the same formula for $ p^\ast_{ij}$ , using a matrix differential equation involving the matrix with elements $ q_{ij}(k,t)$ representing the joint probability of transitioning in k jumps from state i to state j within a time-interval of length t.

According to Proposition 2.1, the matrix $ \mathbf{P}^{\unicode{x1D7D9}}(t)$ does not depend on t. We will therefore write $ \mathbf{P}^{\unicode{x1D7D9}}\,:\!=\, \mathbf{P}^{\unicode{x1D7D9}}(t)$ and refer to it as a conditional one-jump transition matrix.

According to Proposition 2.1, the conditional one-jump transition matrix $ \mathbf{P}^{\unicode{x1D7D9}}$ is fully specified by the rate matrix $ \mathbf{Q}$ of the CTHMC. In what follows, and when needed, we explicitly indicate this dependence using the notation $ \mathbf{P}^{\unicode{x1D7D9}}(\mathbf{Q})$ .

3. Conditional embedding problem

The following question then naturally arises. Given a stochastic matrix $ \mathbf{P}$ , is there a rate matrix $ \mathbf{Q}$ such that $ \mathbf{P}^{\unicode{x1D7D9}}(\mathbf{Q})=\mathbf{P}$ ? And if so, is this rate matrix $ \mathbf{Q}$ unique?

It will be helpful to introduce some terminology before proceeding.

For a transition matrix $ \mathbf{P}$ that is not embeddable, a ${\unicode{x1D7D9}}$ -generator can be seen as a solution to a conditional version of the embedding problem where the rate matrix $ \mathbf{Q}$ satisfies $ \mathbf{P}^{\unicode{x1D7D9}}(\mathbf{Q})=\mathbf{P}$ . Corollary 2.1 establishes that a ${\unicode{x1D7D9}}$ -embeddable stochastic matrix must not contain a zero on its main diagonal. Consequently, our subsequent analysis will focus on stochastic matrices $ \mathbf{P}=(p_{ij})$ where $ p_{ii} \gt 0$ for all i. Note that this necessary condition for ${\unicode{x1D7D9}}$ -embeddability is not more restrictive than the necessary embedding condition $ \prod_{i=1}^{n} p_{ii} \geq \det{P} \gt 0$ as formulated by Goodman in [Reference Goodman16].

Interestingly, the off-diagonal elements of a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ are uniquely defined by its diagonal elements and the elements of $ \mathbf{P}$ . To express this relationship, we introduce the function $ \rho \colon \mathbb{R}_+^2\to\mathbb{R}_+$ , defined as follows:

(3.1) \begin{align} \rho(x,y)=\dfrac{\mathrm{e}}{\tau(1-\ln{x},1-\ln{y})} =\begin{cases} xy\dfrac{\ln{x}-\ln{y}}{x-y} & \text{if}\ x\neq y,\\[3pt] x & \text{if}\ x=y. \end{cases}\end{align}

Proposition 3.1. Suppose $ \mathbf{P}=(p_{ij})$ is an $ n\times n$ stochastic matrix that satisfies $ p_{ii} \gt 0$ for all i. If $ \mathbf{Q}=(q_{ij})$ is a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ , then

\begin{equation*} q_{ij}= \dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_ip_{ii}} \quad for\ all\ i \neq j, \end{equation*}

where $ \theta_i=\mathrm{e}^{1-q_{ii}}$ for all i and the function $ \rho \colon \mathbb{R}_+^2\to\mathbb{R}_+$ is given by (3.1).

Proof. Suppose $ \mathbf{Q}$ is a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}=(p_{ij})$ and let $ i\neq j$ . Then, according to Proposition 2.1, we have

\[ \dfrac{p_{ij}}{p_{ii}}=\dfrac{{p}^{\unicode{x1D7D9}}_{ij}}{{p}^{\unicode{x1D7D9}}_{ii}}=\dfrac{p^*_{ij}}{p^*_{ii}}=\dfrac{q_{ij}\tau(q_{ii},q_{jj})}{\tau(q_{ii},q_{ii})}. \]

Consequently, since $ \tau(q_{ii},q_{ii})=\mathrm{e}^{q_{ii}}=\mathrm{e}/\theta_i$ and $ \tau(q_{ii},q_{jj})=\tau(1-\ln{\theta_i},1-\ln{\theta_j})=\mathrm{e}/\rho(\theta_i,\theta_j)$ , we get

\begin{equation*} q_{ij}=\dfrac{\tau(q_{ii},q_{ii})p_{ij}}{\tau(q_{ii},q_{jj})p_{ii}}=\dfrac{(\mathrm{e}/\theta_i)p_{ij}}{(\mathrm{e}/\rho(\theta_i,\theta_j))p_{ii}}=\dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_i p_{ii}}\quad \text{for all } i \neq {j.} \end{equation*}

The outcome of Proposition 3.2 provides a requirement concerning the diagonal elements of any ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ .

Proposition 3.2. Let $ \mathbf{P}=(p_{ij})$ be an $ n\times n$ stochastic matrix that satisfies $ p_{ii} \gt 0$ for all i. If $ \mathbf{Q}=(q_{ij})$ is a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ , then the n-tuple $ (\mathrm{e}^{1-q_{11}},\ldots,\mathrm{e}^{1-q_{nn}})$ is a fixed point of the vector function $ \mathbf{T}=(T_1,\ldots,T_n) \colon \mathbb{R}_+^{n}\to\mathbb{R}_+^{n}$ , which is defined as

(3.2) \begin{align} T_i(x_1,\ldots,x_n)=\exp{W_0\Biggl(\dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(x_i,x_j)\Biggr)}\quad for\ all\ i \in\mathcal{S}, \end{align}

where $ W_0$ represents the principal branch of the Lambert W function, and where the function $ \rho \colon \mathbb{R}_+^2\to\mathbb{R}_+$ is defined as in (3.1).

Proof. Denote $ \theta_i=\mathrm{e}^{1-q_{ii}}$ for all $ i\in\mathcal{S}$ . Then $ \theta_i \gt 0$ and $ q_{ii}=1-\ln{\theta_i}$ for all i. Using Proposition 3.1 and the fact that $ \mathbf{Q}$ is a rate matrix, we then have

\[ -1+\ln{\theta_i}=-q_{ii}=\sum_{j\in\mathcal{S}\setminus\{i\}}q_{ij}=\sum_{j\in\mathcal{S}\setminus\{i\}}\dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_i p_{ii}}\quad\text{for all } i \in\mathcal{S}, \]

which can be rewritten, using the fact that $ \rho(\theta_i,\theta_i)=\theta_i$ , as

(3.3) \begin{align} \theta_i\ln{\theta_i}=\dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(\theta_i,\theta_j)\quad\text{for all } i\in\mathcal{S}. \end{align}

Using the principal branch $ W_0$ of the Lambert W function (which is the multi-valued inverse of the function $ w\mapsto w\mathrm{e}^{w}$ ( $ w\in\mathbb{C}$ ); see [Reference Corless, Gonnet, Hare, Jeffrey and Knuth9]), we find that

\[ \ln{\theta_i}=W_0\Biggl(\dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(\theta_i,\theta_j)\Biggr)\quad\text{for all } i\in\mathcal{S}, \]

which proves the result.

Propositions 3.1 and 3.2 imply that a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ defines a fixed point of the vector function $ \mathbf{T}$ . The reverse is also true, as stated in the following proposition.

Proposition 3.3. Suppose $ \mathbf{P}=(p_{ij})$ is a stochastic matrix that satisfies $ p_{ii} \gt 0$ for all $ i\in\mathcal{S}$ . Let $ \boldsymbol{\theta}=(\theta_1,\ldots,\theta_n)$ be a fixed point of the vector function $ \mathbf{T} \colon \mathbb{R}_+^n\to\mathbb{R}_+^n$ , defined in (3.2). Then the matrix $ \mathbf{Q}=(q_{ij})$ with elements

(3.4) \begin{align} q_{ii}= 1-\ln{\theta_i},\quad q_{ij}=\dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_i p_{ii}},\quad i\neq j, \end{align}

where $ \rho$ is defined by (3.1), is a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ .

Proof. Let $ \boldsymbol{\theta}=(\theta_1,\theta_2,\ldots,\theta_n)\in\mathbb{R}_+^n$ be a fixed point of $ \mathbf{T}$ and let the matrix $ \mathbf{Q}$ be defined by (3.4). We first show that $ \mathbf{Q}$ is a rate matrix. By (3.4), all off-diagonal elements of $ \mathbf{Q}$ are non-negative. Since $ T_i(\boldsymbol{\theta})=\theta_i$ , we have

\[ \ln{\theta_i}=W_0\Biggl(\frac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(\theta_i,\theta_j)\Biggr), \]

yielding

\[ \theta_i\ln{\theta_i}=\frac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(\theta_i,\theta_j), \]

by definition of the Lambert $ W_0$ function. Using (3.4) and since $ \rho(\theta_i,\theta_i)=\theta_i$ , we can rewrite this equation as

\[ \theta_i(1-q_{ii})=\theta_i+\frac{1}{p_{ii}}\sum_{j\in\mathcal{S}\setminus\{i\}}q_{ij}\theta_ip_{ii}. \]

After simplification, we get $ q_{ii}=-\sum_{j\in\mathcal{S}\setminus\{i\}}q_{ij}$ . Thus $ \mathbf{Q}$ has zero row sums. Consequently, $ \mathbf{Q}$ is a rate matrix.

It remains to be proved that $ {p}^{\unicode{x1D7D9}}_{ij}(\mathbf{Q})=p_{ij}$ for all i and j. Let $ p^\ast_{ij}$ be defined by (2.1) and (2.2). Using (3.4) and (3.1), we have

\[ p^*_{ij}=q_{ij}\tau(q_{ii},q_{jj})=\dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_i p_{ii}}\tau(1-\ln{\theta_i},1-\ln{\theta_j})=\dfrac{\mathrm{e}\, p_{ij}}{\theta_ip_{ii}}, \quad\text{if }i\neq {j,} \]

and

\[ p^*_{ii}=\tau(q_{ii},q_{ii})=\mathrm{e}^{q_{ii}}=\mathrm{e}^{1-\ln{\theta_i}}=\dfrac{\mathrm{e}}{\theta_i}\quad \text{for all} \, \textit{i}, \]

hence

\[ p^*_{ij}=\frac{\mathrm{e}\, p_{ij}}{\theta_ip_{ii}} \quad\text{for all} \, \textit{i} \, and \, \textit{j}. \]

Consequently, by Proposition 2.1 and since $ \mathbf{P}$ is a stochastic matrix, we now have

\[ {p}^{\unicode{x1D7D9}}_{ij}(\mathbf{Q})=\dfrac{p^*_{ij}}{\sum_{k\in\mathcal{S}} p^*_{ik}}=\dfrac{\mathrm{e}\,p_{ij}}{\theta_ip_{ii}}\biggm/ \dfrac{\mathrm{e}}{\theta_ip_{ii}}=p_{ij}, \]

which concludes the proof.

By combining Propositions 3.1, 3.2, and 3.3, we establish a one-to-one relationship between the ${\unicode{x1D7D9}}$ -generators of $ \mathbf{P}$ and the fixed points of the vector function $ \mathbf{T}$ . The subsequent lemma describes the properties of the vector function $ \mathbf{T}$ , which are crucial to determining its number of fixed points.

Lemma 3.1. Let $ \mathbf{P}=(p_{ij})$ be an $ n\times n$ stochastic matrix. Let $ \delta=\min\{p_{ii},\,i\in\mathcal{S}\}$ and $ \Delta=\max\{p_{ii},\,i\in\mathcal{S}\}$ . Suppose $ p_{ii} \gt 0$ for all i. Consider the vector function $ \mathbf{T} \colon \mathbb{R}_+^n\to\mathbb{R}_+^n$ , defined as in (3.2), and the set

(3.5) \begin{align} \mathcal{X}=\{(x_1,\ldots,x_n)\in\mathbb{R}^n \colon \mathrm{e}^{1/\Delta}\leq x_i \leq \mathrm{e}^{1/\delta}\; \forall i\}. \end{align}

Then

(i) every fixed point of $ \mathbf{T}$ belongs to $ \mathcal{X}$ .

(ii) $ \mathbf{T}$ maps $ \mathcal{X}$ into $ \mathcal{X}$ .

Let r be an index such that $ \theta_r=m$ . Then, by Lemma A.2(iv), we have $ \rho(\theta_r,\theta_j)\geq m$ for all j. Since $ T_r(\boldsymbol{\theta})=\theta_r$ , we have

\[ \ln{\theta_r}=W_0\Biggl(\frac{1}{p_{rr}}\sum_{j\in\mathcal{S}}p_{rj}\rho(\theta_r,\theta_j)\Biggr), \]

yielding

\[ \theta_r\ln{\theta_r}=\frac{1}{p_{rr}}\sum_{j\in\mathcal{S}}p_{rj}\rho(\theta_r,\theta_j) \]

by definition of the Lambert $ W_0$ function. Using the fact that $ \sum_{j\in\mathcal{S}} p_{rj}=1$ , we then obtain

\[ m\ln{m}=\theta_r\ln{\theta_r}=\dfrac{1}{p_{rr}}\sum_{j\in\mathcal{S}}p_{rj}\rho(\theta_r,\theta_j)\geq \dfrac{m}{p_{rr}}\geq\dfrac{m}{\Delta}, \]

which implies $ \ln{m}\geq{{1}/{\Delta}}$ and thus $ m\geq \mathrm{e}^{1/\Delta}$ . To prove the second inequality, let s be an index such that $ \theta_s=M$ . Then, by Lemma A.2(iv), we have $ \rho(\theta_s,\theta_j)\leq M$ for all j. Hence, by $ T_s(\theta)=\theta_s$ and the unit row sum property of $ \mathbf{P}$ ,

\[ M\ln{M}=\theta_s\ln{\theta_s}=\dfrac{1}{p_{ss}}\sum_{j\in\mathcal{S}}p_{sj}\rho(\theta_s,\theta_j)\leq \dfrac{M}{p_{ss}} \leq\dfrac{M}{\delta}, \]

which yields $ \ln{M}\leq {{1}/{\delta}}$ and thus $ M\leq\mathrm{e}^{1/\delta}$ .

(ii) Let $ (x_1,\ldots,x_n)\in \mathcal{X}$ . By Lemma A.2(iv),

\begin{equation*} \mathrm{e}^{1/\Delta}\leq\min\{x_i,x_j\}\leq\rho(x_i,x_j)\leq\max\{x_i,x_j\}\leq \mathrm{e}^{1/\delta}\quad\text{for all} \textit{i} \, and \textit{j}. \end{equation*}

Then, since $ \mathbf{P}$ has unit row sums, we have for all i

\begin{equation*} \dfrac{\mathrm{e}^{1/\Delta}}{\Delta}\leq\dfrac{\mathrm{e}^{1/\Delta}}{p_{ii}}\leq \dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(x_i,x_j)\leq \dfrac{\mathrm{e}^{1/\delta}}{p_{ii}}\leq\dfrac{\mathrm{e}^{1/\delta}}{\delta}. \end{equation*}

Now, $ W_0$ and $ \exp$ are increasing functions, and therefore

\[ \exp{W_0\biggl(\frac{1}{\Delta}\mathrm{e}^{1/\Delta}\biggr)} \leq T_i(x_1,\ldots,x_n) \leq\exp{W_0\biggl(\frac{1}{\delta}\mathrm{e}^{1/\delta}\biggr)}\quad\text{for all} \, \textit{i}. \]

The result now follows by applying the property $ W_0(x\mathrm{e}^x)=x$ for $ x \gt 0$ .

Lemma 3.1 implies that the maximal and minimal diagonal elements of $ \mathbf{P}$ impose a lower and upper bound on the diagonal elements of the ${\unicode{x1D7D9}}$ -generators of $ \mathbf{P}$ .

Proposition 3.4. Let $ \mathbf{P}=(p_{ij})$ be an $ n\times n$ stochastic matrix. Let $ \Delta=\max\{p_{ii},\,i\in\mathcal{S}\}$ and $ \delta=\min\{p_{ii},\,i\in\mathcal{S}\}$ . Suppose $ p_{ii} \gt 0$ for all i. Then, if $ \mathbf{Q}=(q_{ij})$ is a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ , we have

\[ 1-\dfrac{1}{\delta}\leq q_{ii}\leq 1-\dfrac{1}{\Delta}\quad\textit{for all i.} \]

With regard to the number of fixed points of $ \mathbf{T}$ , we now prove the following important result.

We are now ready to state and prove our main result in the following theorem.

Theorem 3.2. Suppose $ \mathbf{P}=(p_{ij})$ is an $ n\times n$ stochastic matrix that satisfies $ p_{ii} \gt 0$ for all i. Then $ \mathbf{P}$ has precisely one ${\unicode{x1D7D9}}$ -generator. Moreover, this ${\unicode{x1D7D9}}$ -generator $ \mathbf{Q}=(q_{ij})$ is given by

(3.6) \begin{align} q_{ii}= 1-\ln{\theta_i},\quad q_{ij}=\dfrac{\rho(\theta_i,\theta_j)p_{ij}}{\theta_i p_{ii}},\quad i\neq j, \end{align}

where the scalar function $ \rho \colon \mathbb{R}_+^2\to\mathbb{R}_+$ is defined by (3.1), and $ (\theta_1,\ldots,\theta_n)$ is the unique fixed point of the vector function $ \mathbf{T} \colon \mathbb{R}_+^n\to\mathbb{R}_+^n$ defined by (3.2).

Since $ p_{ii} \gt 0$ for all i is a necessary condition for embeddability (see [Reference Goodman16]), Theorem 3.2 immediately establishes a link between embeddability and ${\unicode{x1D7D9}}$ -embeddability.

According to Theorem 3.2, the unique ${\unicode{x1D7D9}}$ -generator is entirely characterized by the fixed point of the function $ \mathbf{T} \colon \mathcal{X}\to\mathcal{X}$ . When $ \mathbf{T}$ acts as a contraction on $ \mathcal{X}$ , the Banach fixed point theorem ensures the convergence of the fixed point iteration method, which can then be used to estimate $ (\theta_1,\ldots,\theta_n)$ . In the case when not all diagonal elements of $ \mathbf{P}$ are equal, Proposition 3.5 produces a necessary condition for $ \mathbf{T}$ to behave as a contraction under the infinity vector norm. This condition is based on the largest and smallest diagonal elements of the matrix $ \mathbf{P}$ .

Proposition 3.5. Let $ \mathbf{P}=(p_{ij})$ be an $ n\times n$ stochastic matrix such that $ 0 \lt \delta \lt \Delta$ , where $ \delta=\min\{p_{ii},\,i\in\mathcal{S}\}$ and $ \Delta=\max\{p_{ii},\,i\in\mathcal{S}\}$ . Let the set $ \mathcal{X}$ be defined by (3.5) and the function $ \mathbf{T}$ as in (3.2). Then, with respect to the infinity norm, the function $ \mathbf{T}$ is Lipschitz-continuous on $ \mathcal{X}$ with Lipschitz constant

\[ K=\frac{1+({{1}/{\delta}}-1)C(\alpha)}{1+{{1}/{\Delta}}}, \]

where

\[ C(\alpha)=-1+\frac{\alpha+1}{\alpha-1}\ln{\alpha}\quad \textit{and} \quad\alpha=\mathrm{e}^{{{1}/{\delta}}-{{1}/{\Delta}}}. \]

Proof. Using Lemmas A.6 and A.7, we can prove that

\[ |T_{i}(\mathbf{x})-T_{i}(\mathbf{y})|\leq K\|\mathbf{x}-\mathbf{y}\|_{\infty}\quad\text{for all}\ \mathbf{x},\mathbf{y}\in \mathcal{X}\ \text{and for all} \, \textit{i}, \]

where

\[ K=\dfrac{1+({{1}/{\delta}}-1)C(\alpha)}{1+{{1}/{\Delta}}},\quad C(\alpha)=-1+\dfrac{\alpha+1}{\alpha-1}\ln{\alpha}\quad\text{and}\quad \alpha=\mathrm{e}^{{{1}/{\delta}}-{{1}/{\Delta}}}. \]

For more details we refer the reader to the Appendix.

If all diagonal elements of $ \mathbf{P}$ are identical, there is a closed-form expression for the unique ${\unicode{x1D7D9}}$ -generator, which is detailed in the following proposition. Such matrices occur as substitution matrices in Markov models for DNA evolution, among others. See [Reference Casanellas, Fernández-Sánchez and Roca-Lacostena6] for a discussion of the embeddability of such matrices in the context of the so-called JC69, K80, and K81 models.

Proof. Let $ \mathbf{Q}=(q_{ij})$ be a ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ . It follows from Proposition 3.4 that $ q_{ii}=1-1/p$ for all i. Moreover, if $ i\neq j$ , Theorem 3.2 and equation (3.1) imply that

\[ q_{ij}=\dfrac{\rho(\mathrm{e}^{1-q_{ii}},\mathrm{e}^{1-q_{jj}})p_{ij}}{\mathrm{e}^{1-q_{ii}}p_{ii}}=\dfrac{\rho(\mathrm{e}^{1/p},\mathrm{e}^{1/p})p_{ij}}{\mathrm{e}^{1/p}p}=\dfrac{p_{ij}}{p}. \]

In summary, we have

\[ q_{ii}=1-\frac{1}{p}=\frac{1}{p}(p_{ii}-1),\quad q_{ij}=\frac{1}{p}p_{ij},\quad i\neq j, \]

concluding the proof.

4. Comparison of alternative one-jump approaches and illustrations

Consider a continuous-time Markov chain that is discretely observed at times $ t=0, 1, 2, \ldots , T$ ( $ T\geq 2$ ). Let $ \mathbf{P}$ be the empirical transition matrix estimated from these observations. If $ \mathbf{P}$ is not embeddable, there is no Markov generator of $ \mathbf{P}$ , making it a challenge to find an approximate generator. However, if one assumes that the likelihood of more than one transition occurring between two consecutive observations is very low – a scenario referred to as a one-jump setting – it is possible to find a unique rate matrix serving as an approximate generator of $ \mathbf{P}$ .

Let us adopt the notation $ \mathbf{Q}^{\text{JLT}}$ , employed in [Reference Israel, Rosenthal and Wei17], to denote the one-jump rate matrix of Jarrow et al. [Reference Jarrow, Lando and Turnbull18]. Furthermore, we denote $ \mathbf{Q}^{\unicode{x1D7D9}}$ for the ${\unicode{x1D7D9}}$ -generator of $ \mathbf{P}$ . In this section we discuss the specifics of both approaches and compare the resulting rate matrices $ \mathbf{Q}^{\text{JLT}}$ and $ \mathbf{Q}^{\unicode{x1D7D9}}$ as approximate generators of $ \mathbf{P}$ . To effectively represent the distinction between the two approaches, we use the recursively defined jump times $ \tau_0\,:\!=\, 0$ and $ \tau_k=\inf\{t\geq \tau_{k-1} \colon X_t\neq X_{\tau_{k-1}}\}$ ( $ k=1,2,\ldots$ ) of the associated CTHMC $ (X_t)_{t\geq 0}$ .

In their interpretation of the one-jump scenario, Jarrow et al. [Reference Jarrow, Lando and Turnbull18] deduce the rate matrix $ \mathbf{Q}^{\text{JLT}}=(q_{ij}^{\text{JLT}})$ directly from $ \mathbf{P}=(p_{ij})$ through

(4.1) \begin{align} q^{\text{JLT}}_{ii}=\ln{p_{ii}} ,\quad q^{\text{JLT}}_{ij} = \begin{cases} \dfrac{p_{ij} \ln{p_{ii}}}{p_{ii} - 1} & \text{if}\ 0 \lt p_{ii} \lt 1, \\[7pt]0 & \text{if}\ p_{ii}=1,\end{cases} \quad i \neq j,\end{align}

assuming $ 0 \lt p_{ii}$ for all i. From a probabilistic perspective considering jump times, the elements $ q_{ij}^{\text{JLT}}$ are defined as the solutions to the equations

(4.2) \begin{align} \mathbb{P}(\tau_1 \gt 1 \mid X_0=i)=p_{ii},\quad \mathbb{P}(\tau_1\leq 1, X_{\tau_1}=j \mid X_0=i)= p_{ij},\quad i \neq j.\end{align}

Essentially, their one-jump approach designates $ p_{ij}$ to represent the probability of jumping from state i to state j at or before time $ t=1$ .

In this paper we derive the ${\unicode{x1D7D9}}$ -generator via an alternative approach, as expressed by

(4.3) \begin{align} \mathbb{P}(X_1=j \mid X_0=i,\tau_2 \gt 1)=p_{ij}\quad \text{for all \textit{i} and \textit{j},}\end{align}

which implies that $ p_{ij}$ represents the likelihood of going from state i at $ t=0$ to state j at $ t=1$ via at most one jump before or at $ t=1$ . Hence the difference between the methods (4.2) and (4.3) depends on how the empirical transition probability matrix $ \mathbf{P}$ is viewed in the scenario of a single jump. Both approaches have the merit of bypassing the identification and regularization stages (see [Reference Davies10], [Reference Israel, Rosenthal and Wei17], and [Reference Kreinin and Sidelnikova25]) associated with the embedding problem.

In regularization problems where the transition matrix $ \mathbf{P}$ is not embeddable, the goal is to find an approximate Markov generator $ \mathbf{Q}$ for $ \mathbf{P}$ by minimizing the discrepancy between $ \exp{\mathbf{Q}}$ and $ \mathbf{P}$ . The magnitude of this difference $ \|\mathbf{P} - \exp{\mathbf{Q}}\|$ depends on the norm used. Davies [Reference Davies10] explored the use of the infinity norm in addressing the regularization problem. For an $ n\times n$ matrix $ \mathbf{M}=(m_{ij})$ , the infinity norm is given by $ \|\mathbf{M}\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |m_{ij}|$ . This norm proves useful in comparing transition matrices, since both $ \mathbf{P}$ and $ \exp{\mathbf{Q}}$ consist of rows that are stochastic vectors. The Manhattan or Taxicab distance between the corresponding rows, the ith rows in particular, i.e. $ \sum_{j=1}^{n} |\mathbf{P}_{ij} - (\exp{\mathbf{Q}})_{ij}|$ , is a simple indicator of the discrepancy between the transition probability distributions from state i. Consequently, the maximum absolute row sum of $ \mathbf{P} - \exp{\mathbf{Q}}$ offers a distinct measure to evaluate the difference between $ \mathbf{P}$ and $ \exp{\mathbf{Q}}$ . The infinity norm will be used consistently in the remainder of this paper.

An interesting question arises as to whether systematically $ \exp{\mathbf{Q}}^{\text{JLT}}$ or $ \exp{\mathbf{Q}}^{\unicode{x1D7D9}}$ is the best approximation to $ \mathbf{P}$ , or whether this answer depends on the transition matrix $ \mathbf{P}$ . In the following sections, we analyse the discrepancies $ \|\mathbf{P} - \exp{\mathbf{Q}}^{\text{JLT}}\|_{\infty}$ and $ \|\mathbf{P} - \exp{\mathbf{Q}}^{\unicode{x1D7D9}}\|_{\infty}$ for specific categories of the transition matrix $ \mathbf{P}$ .

4.1. Credit rating transition matrix

Consider the empirical transition matrix

\[\renewcommand{\arraystretch}{.8}\mathbf{P}=\left[ \begin{array}{r@{\quad}r@{\quad}r@{\quad}r@{\quad}r@{\quad}r@{\quad}r@{\quad}r} 0.8910 & 0.0963 & 0.0078 & 0.0019 & 0.0030 & 0.0000 & 0.0000 & 0.0000 \\[3pt] 0.0086 & 0.9010 & 0.0747 & 0.0099 & 0.0029 & 0.0029 & 0.0000 & 0.0000 \\[3pt] 0.0009 & 0.0291 & 0.8896 & 0.0649 & 0.0101 & 0.0045 & 0.0000 & 0.0009 \\[3pt] 0.0006 & 0.0043 & 0.0656 & 0.8428 & 0.0644 & 0.0160 & 0.0018 & 0.0045 \\[3pt] 0.0004 & 0.0022 & 0.0079 & 0.0719 & 0.7765 & 0.1043 & 0.0127 & 0.0241 \\[3pt] 0.0000 & 0.0019 & 0.0031 & 0.0066 & 0.0517 & 0.8247 & 0.0435 & 0.0685 \\[3pt] 0.0000 & 0.0000 & 0.0116 & 0.0116 & 0.0203 & 0.0754 & 0.6492 & 0.2319 \\[3pt] 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 \\[3pt] \end{array} \right]\! ,\]

which is derived from Table 3 in Jarrow et al. [Reference Jarrow, Lando and Turnbull18, p. 506]. We have made minor adjustments to five entries on the main diagonal to ensure that all row sums are equal to one, as some rows in Table 3 did not add up to one because of rounding. For this matrix, the vector function $ \mathbf{T} \colon \mathcal{X}\to\mathcal{X}$ , defined by (3.2), is a contraction mapping with Lipschitz constant $ K \lt 1$ ( $ K\approx 0.7833$ ); see Proposition 3.5. Using fixed-point iteration and (3.6), we find that the unique ${\unicode{x1D7D9}}$ -generator, truncated to four decimal places, is given by

\[\renewcommand{\arraystretch}{.8}\mathbf{Q}^{\unicode{x1D7D9}}=\left[ \begin{array}{r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\quad}r} -0.1221 & 0.1075 & 0.0088 & 0.0022 & 0.0036 & 0.0000 & 0.0000 & 0.0000 \\[3pt] 0.0096 & -0.1114 & 0.0836 & 0.0114 & 0.0035 & 0.0034 & 0.0000 & 0.0000 \\[3pt] 0.0010 & 0.0325 & -0.1271 & 0.0752 & 0.0122 & 0.0053 & 0.0000 & 0.0009 \\[3pt] 0.0007 & 0.0049 & 0.0755 & -0.1874 & 0.0798 & 0.0192 & 0.0024 & 0.0049 \\[3pt] 0.0005 & 0.0026 & 0.0094 & 0.0886 & -0.2759 & 0.1301 & 0.0178 & 0.0270 \\[3pt] 0.0000 & 0.0022 & 0.0036 & 0.0079 & 0.0647 & -0.2121 & 0.0592 & 0.0746 \\[3pt] 0.0000 & 0.0000 & 0.0152 & 0.0157 & 0.0287 & 0.1031 & -0.4460 & 0.2834 \\[3pt] 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\[3pt] \end{array} \right]\! .\]

In contrast, the rate matrix $ \mathbf{Q}^{\text{JLT}}$ , published in Jarrow et al. [Reference Jarrow, Lando and Turnbull18] and defined by equation (4.1), is

\[\renewcommand{\arraystretch}{.8}\mathbf{Q}^{\text{JLT}}=\left[\begin{array}{r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\,\,\,}r@{\quad}r} -0.1154 & 0.1020 & 0.0083 & 0.0020 & 0.0032 & 0.0000 & 0.0000 & 0.0000 \\[3pt] 0.0091 & -0.1043 & 0.0787 & 0.0104 & 0.0031 & 0.0031 & 0.0000 & 0.0000 \\[3pt] 0.0010 & 0.0308 & -0.1170 & 0.0688 & 0.0107 & 0.0048 & 0.0000 & 0.0010 \\[3pt] 0.0007 & 0.0047 & 0.0714 & -0.1710 & 0.0701 & 0.0174 & 0.0020 & 0.0049 \\[3pt] 0.0005 & 0.0025 & 0.0089 & 0.0814 & -0.2530 & 0.1180 & 0.0144 & 0.0273 \\[3pt] 0.0000 & 0.0021 & 0.0034 & 0.0073 & 0.0568 & -0.1927 & 0.0478 & 0.0753 \\[3pt] 0.0000 & 0.0000 & 0.0143 & 0.0143 & 0.0250 & 0.0929 & -0.4320 & 0.2856 \\[3pt] 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\[3pt] \end{array} \right]\! .\]

We find $ \|\mathbf{P}-\exp{\mathbf{Q}}^{\unicode{x1D7D9}}\|_{\infty}\approx 0.0247$ and $ \|\mathbf{P}-\exp{\mathbf{Q}}^{\text{JLT}}\|_{\infty}\approx 0.0277$ (accurate to four decimal places). Hence it appears that $ \mathbf{Q}^{\unicode{x1D7D9}}$ is a better approximate generator of $ \mathbf{P}$ than $ \mathbf{Q}^{\text{JLT}}$ , when using the infinity norm.

Observe that the magnitudes of the diagonal entries in $ \mathbf{Q}^{\text{JLT}}$ are smaller compared to those in the ${\unicode{x1D7D9}}$ -generator. This observation persists across all matrices $ \mathbf{P}$ with non-zero diagonal entries, as demonstrated by the following proposition.

Proposition 4.1. Suppose $ \mathbf{P}=(p_{ij})$ is an $ n\times n$ stochastic matrix that satisfies $ p_{ii} \gt 0$ for all i. Then $ q_{ii}^{\unicode{x1D7D9}}\leq q_{ii}^{\mathrm{JLT}}$ for all i.

Proof. By Theorem 3.2, $ q_{ii}^{\unicode{x1D7D9}}=1-\ln{\theta_i}$ for all i, where $ \theta=(\theta_1,\ldots,\theta_n)$ is the unique fixed point of the vector function $ \mathbf{T}=(T_1,\ldots,T_n)$ , defined by (3.2). For $ i\in\mathcal{S}$ , we have by (3.3) that

\[ \theta_i\ln{\theta_i}=\dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}}p_{ij}\rho(\theta_i,\theta_j). \]

Using the definition of the function $ \rho$ in (3.1), the above equation can be rewritten as

(4.4) \begin{align} \biggl(\dfrac{\theta_i}{\mathrm{e}}-1\biggr)\rho(\theta_i,\mathrm{e})=\theta_i\ln{\theta_i}-\theta_i= \dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}\setminus\{i\}}p_{ij}\rho(\theta_i,\theta_j). \end{align}

Now, since $ 1-\ln{\theta_j}=q_{jj}^{\unicode{x1D7D9}}\leq 0$ , we have $ \theta_j\geq\mathrm{e}$ for all j. By Lemma A.2(iii), $ \rho(\theta_i,\theta_j)\geq \rho(\theta_i,\mathrm{e})$ for all j. Hence, from (4.4), it follows that

\[ \biggl(\dfrac{\theta_i}{\mathrm{e}}-1\biggr)\rho(\theta_i,\mathrm{e})\geq \dfrac{1}{p_{ii}}\sum_{j\in\mathcal{S}\setminus\{i\}}p_{ij}\rho(\theta_i,\mathrm{e})=\biggl(\dfrac{1}{p_{ii}}-1\biggr)\rho(\theta_i,\mathrm{e}), \]

which simplifies to $ \theta_i\geq \mathrm{e}/p_{ii}$ . Taking logarithms from both sides of this inequality and using $ q_{ii}^{\unicode{x1D7D9}}=1-\ln{\theta_i}$ , we arrive at $ q_{ii}^{\unicode{x1D7D9}}\leq\ln{p_{ii}}=q_{ii}^{\mathrm{JLT}}$ .

4.2. Transition matrices with identical diagonal entries

When the stochastic matrix $ \mathbf{P}$ has identical diagonal entries $ p_{ii}=p \gt 0$ , for all i, Proposition 3.6 indicates that the ${\unicode{x1D7D9}}$ -generator can be explicitly represented as

\[\mathbf{Q}^{\unicode{x1D7D9}}=\frac{1}{p}(\mathbf{P}-\mathbf{I}),\]

with $ \mathbf{I}$ denoting the $ n\times n$ identity matrix. Furthermore, by (4.1), we have

\[ \mathbf{Q}^{\text{JLT}} = \frac{\ln{p}}{p-1}(\mathbf{P}-\mathbf{I}).\]

Both $ \mathbf{Q}^{\unicode{x1D7D9}}$ and $ \mathbf{Q}^{\text{JLT}}$ share a similar structure; indeed, they can be expressed as

\[\mathbf{Q}^{\unicode{x1D7D9}}=\mathbf{Q}\biggl(\frac{1}{p}\biggr)\quad\text{and}\quad\mathbf{Q}^{\text{JLT}}=\mathbf{Q}\biggl(\frac{\ln{p}}{p-1}\biggr),\]

where

(4.5) \begin{align}\mathbf{Q}(k)=k(\mathbf{P}-\mathbf{I}), \quad k\geq 0.\end{align}

In this section, we explore which matrix, either $ \mathbf{Q}^{\unicode{x1D7D9}}$ or $ \mathbf{Q}^{\text{JLT}}$ , is the better approximate generator of $ \mathbf{P}$ according to the infinity norm.

For the $ 2 \times 2$ case,

\begin{equation*}\mathbf{P}= \left[\begin{array} {c@{\quad}c} p & 1-p \\[3pt] 1-p & p \end{array} \right]\!, \quad 0 \lt p \lt 1,\end{equation*}

it is known from [Reference Kingman24] that $ \mathbf{P}$ is embeddable if and only if its trace exceeds 1, i.e. $ p \gt \tfrac{1}{2}$ . The unique Markov generator of $ \mathbf{P}$ is then given by

\[\mathbf{Q}\biggl(\frac{\ln{(2p-1)}}{2(p-1)}\biggr);\]

see [Reference Singer and Spilerman28, p. 392]. If $ \mathbf{P}$ is not embeddable ( $ p \leq \tfrac{1}{2}$ ), it turns out that $ \|\mathbf{P}-\exp{\mathbf{Q}^{\unicode{x1D7D9}}}\|_{\infty} \lt \|\mathbf{P}-\exp{\mathbf{Q}^{\text{JLT}}}\|_{\infty}$ , an inequality that even holds for all $ p\in(0,1)$ . This is proved in Lemma A.8 in the Appendix.

For the $ n \times n$ case where $ n\geq 3$ , the answer to our question seems no longer clear-cut. For instance, if

(4.6) \begin{align} \mathbf{P} = \left[\begin{array} {c@{\quad}c@{\quad}c} p & 1-p & 0 \\[3pt] \tfrac{1}{2}(1-p) & p & \tfrac{1}{2}(1-p) \\[3pt] 0 & 1-p & p \\[3pt] \end{array} \right]\!,\quad 0 \lt p \lt 1,\end{align}

we have $ \|\mathbf{P}-\exp{\mathbf{Q}^{\unicode{x1D7D9}}}\|_{\infty} \lt \|\mathbf{P}-\exp{\mathbf{Q}^{\text{JLT}}}\|_{\infty}$ ; see Lemma A.9. However, if

(4.7) \begin{align} \mathbf{P} = \left[\begin{array} {c@{\quad}c@{\quad}c} p & 0 & 1-p \\[3pt] 0 & p & 1-p \\[3pt] 0 & 1-p & p \\[3pt]\end{array} \right]\!, \quad 0 \lt p \lt 1,\end{align}

we have the reversed inequality, i.e. $ \|\mathbf{P}-\exp{\mathbf{Q}^{\unicode{x1D7D9}}}\|_{\infty} \gt \|\mathbf{P}-\exp{\mathbf{Q}^{\text{JLT}}}\|_{\infty}$ ; see Lemma A.10. We note that the transition matrices of (4.6) and (4.7) are not embeddable, since for some $ i\neq j$ we have $ \mathbf{P}_{ij}=0$ but $ (\mathbf{P}^2)_{ij} \gt 0$ ; see [Reference Chung8, Theorem 5, p. 126], highlighting the importance of investigating $ \mathbf{Q}^{\unicode{x1D7D9}}$ and $ \mathbf{Q}^{\text{JLT}}$ as approximate Markov generators.