2. Construction of $\{(\mathcal{R}^\boldsymbol{n}, \mathcal{J}^\boldsymbol{n})\}_{\boldsymbol{n} \geq 0}$

First, we construct a probability space suitable for proving Theorems 1.1 and 1.2. Fix $\boldsymbol{p}$ , Q, $\{\mu_i\}_{i \in\mathcal{S}}$ , and $\{\sigma_i >0\}_{i\in\mathcal{S}}$ of Theorem 1.1. Let $\lambda_0=2\max_{i\in\mathcal{S}}|Q_{ii}|$ , and consider a sequence $\{\lambda_n\}_{n\ge 1}$ such that $\lambda_n\ge \lambda_{n-1}$ for $n\ge 1$ and $\lim_{n\rightarrow\infty}\lambda_n = \infty$ . Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space that supports:

• a standard Brownian motion $\mathcal{B}=\{B(t)\}_{t\ge 0}$ ,

• a Poisson process $\mathcal{M}^0=\{M^0(t)\}_{t\;\ge\;0}$ of rate $\lambda_0/2$ ,

• a sequence of Poisson processes $\{\mathcal{\widetilde{M}}^n\}_{n\;\ge\;1}$ , where $\mathcal{\widetilde{M}}^n$ has rate $(\lambda_{n}-\lambda_{n-1})/2$ ,

• a discrete-time Markov chain $\mathcal{X}^0=\{X^0(k)\}_{k\;\ge\;0}$ with state space $\mathcal{S}$ , initial distribution $\boldsymbol{p}$ , and transition probability matrix $P_0\;:\!=\;I + (\lambda_0/2)^{-1}Q$ ,

with $\mathcal{B}$ , $\mathcal{M}^0$ , $\{\mathcal{\widetilde{M}}^n\}_{n\ge 1}$ , and $\mathcal{X}^0$ being independent of each other. All the elements stated in Theorem 1.1 (and the rest of the paper) will be constructed in $(\Omega, \mathcal{G}, \mathbb{P})$ . To construct $(\mathcal{R},\mathcal{J})$ on $(\Omega, \mathcal{G}, \mathbb{P})$ , let

(2.1) \begin{equation} J(t)=X^0(M^0(t)), \qquad t\ge 0.\end{equation}

The uniformisation method implies that $\mathcal{J}=\{J(t)\}_{t\ge 0}$ is a Markov jump process with initial distribution $\boldsymbol{p}$ and intensity matrix Q. Let $\mathcal{R}=\{R(t)\}_{t\ge 0}$ be defined as in (1.3), so that $(\mathcal{R},\mathcal{J})$ corresponds to a Markov-modulated Brownian motion.

Next, for each $n \geq 0$ , we construct the process $(\mathcal{R}^n,\mathcal{J}^n)$ as follows. Define the arrival process $\mathcal{M}^{n} = \{M^n(t)\}_{t \geq 0}$ to be the superposition of $\big\{\mathcal{M}^0, \mathcal{\widetilde{M}}^{1}, \mathcal{\widetilde{M}}^{2},\dots, \mathcal{\widetilde{M}}^{n}\big\}$ . Then, $\mathcal{M}^n$ is itself a Poisson process of intensity $\lambda_0/2 + \sum_{\ell=1}^n(\lambda_\ell - \lambda_{\ell-1})/2 = \lambda_{n}/2$ , and its arrival epochs form a subset of the arrival epochs of $\mathcal{M}^{n + m}$ for any $m\ge 0$ . In other words, $\{\mathcal{M}^n\}_{n\;\ge\;0}$ is a sequence of Poisson processes with nested time epochs whose new arrivals, as n increases, are created independently of the existing ones. We emphasise that choosing to have Poissonian observations with rates $\lambda_{n}/2$ allows a direct comparison of our construction with the models of [Reference Gorostiza and Griego9,Reference Ramaswami and Latouche18] in the special case of flip-flop approximations to a standard Brownian motion.

Intuitively, our aim is to construct $(\mathcal{R}^n,\mathcal{J}^n)$ in such a way that $\mathcal{R}^n$ visits the levels of $\mathcal{R}$ inspected at the arrival epochs of the Poisson process $\mathcal{M}^n$ . To that end, we employ the well-known Wiener–Hopf factorisation for a Brownian motion with drift; see [Reference Bladt and Nielsen5, Corollary 2.4.10] for a proof.

Theorem 2.1. (Wiener-Hopf factorisation for Brownian motion.) Let $\{W_t\}_{t\;\ge\;0}$ be a Brownian motion with variance $\sigma^2>0$ , drift $\mu$ , and initial point $W_0=0$ . Let S be a stopping time, and for fixed $\beta>0$ let $T \sim \exp(\beta)$ , independent of $\{W_t\}_{t\;\ge\;0}$ . Then, $W_S - \min_{0\;\le\;t\le\;T} W_{S + t}$ and $W_{S+T} - \min_{0\;\le\;t\;\le\;T} W_{S + t}$ are independent and exponentially distributed with respective rates

\begin{equation*} \omega = \sqrt{\frac{\mu^2}{\sigma^4} + \frac{2\beta}{\sigma^2}} + \frac{\mu}{\sigma^2} , \qquad \eta = \sqrt{\frac{\mu^2}{\sigma^4} + \frac{2\beta}{\sigma^2}} - \frac{\mu}{\sigma^2} . \end{equation*}

Theorem 2.1 implies that, restricted to an exponentially distributed time interval, we can track both the value of the minimum over this period and that value at the right endpoint of a Brownian motion with drift. Moreover, such minimum and right endpoint values can be described using exponential random variables. Our aim is to exploit Theorem 2.1 to construct the process $\mathcal{R}^n$ in such a way that it sequentially visits the points $R_s$ for all observation epochs $s\in\mathcal{M}^n$ , as well as the minima that $\mathcal{R}$ reaches between those observation epochs. This means that heuristically $\mathcal{R}^n$ resembles $\mathcal{R}$ . Furthermore, we will show that, as n tends to infinity, since the observation epochs $\mathcal{M}^n$ get denser, $\mathcal{R}^n$ converges in a pathwise sense to $\mathcal{R}$ . We explicitly construct $\mathcal{R}^n$ next.

Let $\{T_k^n\}_{k\;\ge\;1}$ be the interarrival times of the process $\mathcal{M}^n$ , and define $\theta_0^n \;:\!=\; 0$ , $\theta_k^n\;:\!=\; \sum_{j=1}^k T^n_j$ ( $k \geq 0$ ); thus, $\{\theta_k^n\}_{n\;\geq\;0}$ are the arrival epochs of $\mathcal{M}^n$ . See Figure 1 for an illustration.

Figure 1:

Blue dots correspond to arrivals $\{\theta^{0}_k\}_{k \geq 0}$ of $\mathcal{M}^0$ , red diamonds the arrivals $\{\theta^n_k\}_{k \geq 0}$ of $\mathcal{M}^n$ , and black squares the jump epochs of $\mathcal{J}$ . As $\mathcal{J}$ is given by (2.1), its jump epochs form a subset of the arrival times of $\mathcal{M}^0$ .

As $\{\theta_k^n\}_{n \geq 0}$ contain all the arrival epochs of $\mathcal{M}^0$ , (2.1) implies that $\{J(t)\}_{t\ge 0}$ remains constant on each interval $[\theta_k^n, \theta_{k\;+\;1}^n)$ , $k\ge 0$ . Consequently, given $J({\theta_k^n})=i$ on $[\theta_k^n, \theta_{k\;+\;1}^n)$ , $\{R(t)\}_{t\;\ge\;0}$ behaves like a Brownian motion with drift $\mu_i$ and variance $\sigma^2_i$ . Thus, by sequentially using the Wiener–Hopf factorisation between arrival epochs of $\mathcal{M}^n$ , we can keep track of $\{ R(\theta_k^n)\}_{k\ge 0}$ and of $\big\{\!\min_{\theta_k^n\le t\le \theta_{k+1}^n} R(t)\big\}_{k\ge 0}=\big\{\!\min_{0\le t\le T_{k+1}^n} R(\theta_k^n + t)\big\}_{k\ge 0}$ in a simple manner, which we explain in detail next.

For each $k\ge 0$ , define the random variables $X^n(k) \;:\!=\; J(\theta_k^n)$ , $L^{n}_{k+1} \;:\!=\; R(\theta_{k}^n) - \min_{0\le t\le T_{k+1}^n} R(\theta_k^n + t)$ , and $H^{n}_{k+1} \;:\!=\; R(\theta_{k+1}^n) - \min_{0\le t\le T_{k+1}^n} R(\theta_k^n + t)$ . By Theorem A.1 in the Appendix, $\mathcal{X}^n=\{X^n(k)\}_{k\ge 0}$ is a discrete-time Markov chain with transition probability matrix $P_n\;:\!=\; I + (\lambda_n/2)^{-1}Q$ . The strong Markov property of $\{(R(t), J(t))\}_{t\ge 0}$ and Theorem 2.1 imply that, conditioned on $\mathcal{X}^n$ , $\{L^{n}_{k+1}\}_{k\ge 0}$ is a collection of independent random variables. More specifically, given $X^n_k=i$ , $L^{n}_{k+1}$ is exponentially distributed with rate

\begin{equation*} \omega_i^n \;:\!=\; \sqrt{\frac{\mu^2_i}{\sigma^4_i} + \frac{\lambda_n}{\sigma^2_i}} + \frac{\mu_i}{\sigma^2_i}.\end{equation*}

Similarly, $\{H^{n}_{k+1}\}_{k\ge 0}$ is a collection of conditionally independent random variables for which, given $X^n_k=i$ , $H^{n}_{k+1}$ is exponentially distributed with rate

\begin{align*} \eta_i^n \;:\!=\; \sqrt{\frac{\mu^2_i}{\sigma^4_i} + \frac{\lambda_n}{\sigma^2_i}} - \frac{\mu_i}{\sigma^2_i}.\end{align*}

Moreover, $\{L^{n}_{k+1}\}_{k\ge 0}$ is conditionally independent of $\{H^{n}_{k+1}\}_{k\ge 0}$ . Note that $\{L^{n}_{k+1}\}_{k\ge 0}$ and $\{H^{n}_{k+1}\}_{k\ge 0}$ completely describe $\{ R(\theta_k^n)\}_{k\ge 0}$ and $\big\{\!\min_{\theta_k^n\le t\le \theta_{k+1}^n} R(t)\big\}_{k\ge 0}$ , in the sense that, for all $k\ge 0$ ,

(2.2) \begin{equation} R(\theta_k^n) = \sum_{j=1}^k \left( -L^{n}_{j} +H^{n}_{j}\right), \qquad \min_{\theta_k^n\le t\le \theta_{k+1}^n} R(t) = \sum_{j=1}^k \left(-L^{n}_{j} +H^{n}_{j}\right) - L^{n}_{k+1}. \end{equation}

For all $k\ge 1$ , if $X^n(k-1) = i$ , define $\widehat{L}^{n}_{k} \;:\!=\; \lambda_n^{-1}\omega_i^n L^{n}_{k}$ and $\widehat{H}^{n}_{k} \;:\!=\; \lambda_n^{-1}\eta_i^n H^{n}_{k}$ . Then, the collections $\big\{\widehat{L}^{n}_{k}\big\}_{k\ge 1}$ and $\big\{\widehat{H}^{n}_{k}\big\}_{k\ge 1}$ are independent and identically distributed random variables exponentially distributed with parameter $\lambda_n$ . Let $\chi_n^0 \;:\!=\; 0$ , and define, for all $k\ge 1$ , $\chi_k^n\;:\!=\;\sum_{j=1}^k \left(\widehat{L}^{n}_{j} + \widehat{H}^{n}_{j}\right)$ . Let $\mathcal{J}^n = \{J^n(t)\}_{t\ge 0}$ be the process with state space $\{+,-\}\times\mathcal{S}$ defined by

\begin{align*} J^n(t) =\left\{\begin{array}{cl}(-,i)\;\;\;\;\;\; & \mbox{if } t\in[\chi^n_k,\chi^n_k + \widehat{L}^{n}_{k+1})\mbox{ for some $k\ge 0$ and }X_k^n = i,\\[5pt] (+,i)\;\;\;\;\;\; & \mbox{if } t\in[\chi^n_k + \widehat{L}^{n}_{k+1}, \chi^n_{k+1})\mbox{ for some $k\ge 0$ and }X_k^n = i. \end{array}\right.\end{align*}

Figure 2 shows a sample path of $\mathcal{J}^n$ with $\mathcal{S} = \{1, 2, 3\}$ .

Figure 2:

A sample path of $\{J^n(t)\}$ with $\mathcal{S} = \{1,2, 3\}$ .

The process $\mathcal{J}^n$ jumps alternately between $\{-\}\times\mathcal{S}$ and $\{+\}\times\mathcal{S}$ with intensity given by $\lambda_n$ ; furthermore, changes in its second coordinate, which occur according to $P_n$ , are only possible at jump instants from $\{+\}\times\mathcal{S}$ to $\{-\}\times\mathcal{S}$ . Thus, $\mathcal{J}^n$ is a Markov jump process with state space $\{+,-\}\times\mathcal{S}$ (ordered lexicographically), initial distribution $({\bf{{0}}},{\bf{{p}}})$ (where ${\bf{{0}}}$ denotes a row vector of zeros of size $|\mathcal{S}|$ ), and intensity matrix given by

\begin{equation*} \left[\begin{array}{c@{\quad}c} -\lambda_n I & \lambda_n P_n\\\lambda_n I & -\lambda_n I\end{array}\right] = \left[\begin{array}{c@{\quad}c} -\lambda_n I & 2 Q + \lambda_n I\\\lambda_n I& -\lambda_n I \end{array}\right].\end{equation*}

Note that the sequence of states in $\mathcal{S}$ visited by $\pi_2(\mathcal{J}^n)$ coincides with that of $\mathcal{J}$ , or, more precisely,

(2.3) \begin{equation} \pi_2({J}^n(\chi^n_k))=J(\theta^n_k)\qquad\mbox{for all } k\ge 0.\end{equation}

Also notice that the jumps of $\pi_2(\mathcal{J}^n)$ can occur only at $\{\chi^n_k\}_{k\ge 0}$ while the jumps of $\mathcal{J}$ can occur only at $\{\theta^n_k\}_{k\ge 0}$ . In general, $\{\chi^n_k\}_{k\ge 0}\neq\{\theta^n_k\}_{k\ge 0}$ ; nevertheless,

\begin{align*}\mathbb{E}\left[\theta^n_k\right]& = \mathbb{E}\left[\sum_{j=1}^k T^n_j\right] = k \mathbb{E}\left[T^n_1\right] = \frac{2k}{\lambda_n},\\\mathbb{E}\left[\chi^n_k\right]&=\mathbb{E}\left[\sum_{j=1}^k \left(\widehat{L}^{n}_{j} + \widehat{H}^{n}_{j}\right)\right] = k \mathbb{E}\left[\widehat{L}^{n}_{1} + \widehat{H}^{n}_{1}\right] = \frac{2k}{\lambda_n}.\end{align*}

In words, the average jump times of $\pi_2(\mathcal{J}^n)$ coincide with the average jump times of $\mathcal{J}$ , so that the process $\pi_2(\mathcal{J}^n)$ is indeed similar to $\mathcal{J}$ . A more precise and stronger version of this statement is proven in Section 3.

In order to construct $\mathcal{R}^n$ , define $r^n \colon \{+,-\}\times\mathcal{S} \mapsto \mathbb{R}$ by $r^n(+,i) \;:\!=\; \lambda_n/\eta_i^n$ , $r^n(-,i) \;:\!=\; -\lambda_n/\omega_i^n$ . Let $R^n(t) \;:\!=\; \int_0^t r^n(J^n(s)) \, \textrm{d} s$ , $t\ge 0$ . The pair $(\mathcal{R}^n, \mathcal{J}^n)$ is indeed a stochastic fluid process. Moreover, from the construction of $\mathcal{J}^n$ and (2.2), it follows that, for all $k \ge 0$ ,

(2.4) \begin{align} R^n\left(\chi_k^n\right) & = \sum_{j=1}^k\left(-L^{n}_{j} + H^{n}_{j}\right) = R(\theta_k^n), \end{align}

(2.5) \begin{align} R^n\left(\chi_k^n + \widehat{L}^{n}_{k+1}\right) = \sum_{j=1}^k \left(- L^{n}_{j} + H^{n}_{j}\right) - L^{n}_{k+1}=\min_{\theta_k^n\le t\le \theta_{k+1}^n} R(t). \end{align}

This implies that the values at the inflection points of the level process $\mathcal{R}^n$ coincide with the values of $\{ R(\theta_k^n)\}_{k\ge 0}$ and $\big\{\!\min_{\theta_k^n\le t\le \theta_{k+1}^n} R(t)\big\}_{k\ge 0}$ . In conclusion, the values of $\mathcal{R}$ at the arrival epochs of $\mathcal{M}^n$ , and the minimum level attained between them, are embedded in $\mathcal{R}^n$ . Figure 3 illustrates the construction of the stochastic fluid process $(\mathcal{R}^n, \mathcal{J}^n)$ corresponding to the Markov-modulated Brownian motion $(\mathcal{R}, \mathcal{J})$ .

Figure 3:

(Left) A sample path of an MMBM $(\mathcal{R},\mathcal{J})$ with $\mathcal{J}$ being on $\mathcal{S} = \{1, 2\}$ : arrivals corresponding to $\mathcal{M}^n$ occur at $\{\theta_i^n\}_{i \geq 0}$ , and the minima of $\mathcal{R}$ attained between these arrivals are highlighted with green crosses. When $J(t) = 1$ (red), $\mu_1 = 5$ and $\sigma_1^2 = 4$ . When $J(t) = 2$ (blue), $\mu_2 = -2$ and $\sigma_2^2 = 1$ . (Right) An associated sample path of a stochastic fluid process $(\mathcal{R}^n,\mathcal{J}^n)$ : jumps from $\{+\}\times\mathcal{S}$ to $\{-\}\times\mathcal{S}$ occur at $\{\chi_i^n\}_{i \geq 1}$ . The values of $R^n(\chi_i^n)$ match with those of $R(\theta_i^n)$ for $i = 1, \ldots, 4$ .

3. Proof of Theorem 1.2

In order to prove Theorem 1.2 we measure the absolute difference between $\mathcal{R}$ and $\mathcal{R}^n$ . First, we make use of the triangle inequality to cover the sample space by a number of possibly overlapping events. Then, using the subadditivity property of the probability measure, we bound the difference between $\mathcal{R}$ and $\mathcal{R}^n$ on each of these events. Below we define such events and obtain bounds for their probability.

As $\lambda_n \rightarrow \infty$ , the grids $\{\theta^n_k\}_{k\ge 0}$ and $\{\chi^n_k\}_{k\ge 0}$ become finer. Intuitively, this and (2.4) indicate that $\mathcal{R}^n$ approximates $\mathcal{R}$ as $n\rightarrow\infty$ , which is stated more precisely in Theorems 1.1 and 1.2. We devote this section to rigorously proving Theorem 1.2, from which Theorem 1.1 follows as a corollary.

Let $\lambda_n=2n^2$ ; this makes our results and rates comparable to those of [Reference Gorostiza and Griego9] and related papers. Fix $T \in [0,1)$ , $q>0$ , and without loss of generality consider $n\ge 2$ throughout.

Proof of Theorem 1.2(i). In order to prove (1.4), notice that

(3.1) \begin{align} \mathbb{P}\left(\sup_{0 \le s\le T} |R(s) - R^n(s)| > \alpha \varepsilon_n\right)\le \mathbb{P}(A^n) + \mathbb{P}(\chi^n_{n^2} < T),\end{align}

where

\begin{align*} A^n \;:\!=\; \left\{\sup_{0 \le s\le \chi^{n}_{n^2}} |R(s) - R^n(s)| > \alpha \varepsilon_n\right\} = \left\{\max_{1 \leq k \leq n^2} \sup_{\chi^{n}_{k-1} \le s \le \chi^{n}_{k}} |R(s) - R^n(s)| > \alpha \varepsilon_n\right\},\end{align*}

with $\alpha$ a constant to be determined later. We now show that each of the quantities $\mathbb{P}(A^n)$ and $\mathbb{P}(\chi_{n^2}^n < T)$ are $o(n^{-q})$ .

The triangle inequality implies that $A^n \subseteq B_1^n\cup B_2^n\cup B_3^n\cup B_4^n$ , where

\begin{align*}B_1^n & \;:\!=\; \left\{\max_{1 \le k\le n^2}\sup_{\chi_{{k - 1}}^n \le s\le \chi_{{k}}^n } |R(s) - R(\chi_k^n)| > \alpha \varepsilon_n/4\right\},\\[5pt] B_2^n & \;:\!=\; \left\{\max_{1 \le k\le n^2} |R(\chi_k^n) - R(k/n^2)| > \alpha \varepsilon_n/4\right\},\\[5pt] B_3^n & \;:\!=\; \left\{\max_{1 \le k\le n^2} |R(k/n^2) - R^n(\chi_k^n)| > \alpha \varepsilon_n/4\right\},\\[5pt] B_4^n & \;:\!=\; \left\{\max_{1 \le k\le n^2}\sup_{\chi_{{k - 1}}^n \le s\le \chi_{{k}}^n} |R^n(\chi_k^n) - R^n(s)| > \alpha \varepsilon_n/4\right\}.\end{align*}

By (2.4), $B_3^n$ can be rewritten as $B_3^n = \left\{\max_{1 \le k\le n^2} |R(\theta_k^n) - R(k/n^2)| > \alpha \varepsilon_n/4\right\}$ , so that the events $B_1^n$ , $B_2^n$ , and $B_3^n$ concern only the process $\mathcal{R}$ , not $\mathcal{R}^n$ .

Let $\{\delta_n\}_{n\ge 0}$ be any positive and decreasing sequence. Making a further partition, we obtain

(3.2) \begin{align} A^n \subseteq \left(\left(B_1^n\cup B_2^n\cup B_3^n\right) \cap \left( C^n_\chi \cup C^n_\theta\right)^\textrm{c} \right) \cup \left( C^n_\chi \cup C^n_\theta\right) \cup B_4^n,\end{align}

where $C^n_\chi \;:\!=\; \left\{\max_{1 \le k\le n^2} |\chi_k^n - k/n^2|>\delta_n\right\}$ , $C^n_\theta \;:\!=\; \left\{\max_{1 \le k\le n^2} |\theta_k^n - k/n^2|>\delta_n\right\}$ . On $(C^n_\chi\cup C^n_\theta)^\textrm{c}$ , for all $k = 1, \ldots, n^2$ we have $\chi_k^n, \theta_k^n, \chi_{k + 1}^n \in [k/n^2 - \delta_n, (k + 1)/n^2 + \delta_n]$ . Let $x_+\;:\!=\;\max\{x,0\}$ for $x\in\mathbb{R}$ . If there exists $a, b \in [k/n^2 - \delta_n, (k + 1)/n^2 + \delta_n]$ such that $|R(a_+) - R(b_+)| > \alpha \varepsilon_n/4$ , then by the triangle inequality either $|R([k/n^2 - \delta_n]_+) - R(a_+)| > \alpha \varepsilon_n / 8$ or $|R([k/n^2 - \delta_n]_+) - R(b_+)| > \alpha \varepsilon_n / 8$ . Therefore,

(3.3) \begin{align} \left(B_1^n\cup B_2^n\cup B_3^n\right) \cap \left(C^n_\chi \cup C^n_\theta\right)^\textrm{c} \subseteq D^n, \end{align}

where

\begin{align*} D^n &\;:\!=\; \left\{\max_{1 \le k\le n^2}\sup_{a \in [k/n^2 - \delta_n, (k + 1)/n^2 + \delta_n]} \left|R\left((k/n^2 - \delta_n)_+\right) - R\left(a_+\right)\right| > \alpha \varepsilon_n/8\right\}\\[5pt] &\;= \left\{\max_{1 \le k\le n^2}\sup_{s \in [0, n^{-2} + 2\delta_n]} \left|R\left((k/n^2 - \delta_n + s)_+\right) - R\left((k/n^2 - \delta_n)_+\right)\right| > \alpha \varepsilon_n/8\right\}. \end{align*}

Thus, by (3.2) and (3.3),

(3.4) \begin{align} \mathbb{P}(A^n) \le \mathbb{P}(D^n) + \mathbb{P}(C^n_\chi ) + \mathbb{P}(C^n_\theta) + \mathbb{P}(B^n_4).\end{align}

In the following, we show that with an appropiate choice of $\alpha$ and $\{\delta_n\}_{n\ge 1}$ , each summand on the right-hand side of (3.4) is an $o(n^{-q})$ function. For the remainder of the section, $K_j$ ( $j \in \mathbb{N}$ ) denote generic constants that are used to simplify bounds and are not dependent on q or n.

Bounding $\mathbb{P}(B^n_4)$ . Let $\omega_{\min}^n \;:\!=\; \min\limits_{i\in\mathcal{S}}\omega_i^n$ , $\eta_{\min}^n \;:\!=\; \min\limits_{i\in\mathcal{S}}\eta_i^n$ , $\kappa_n \;:\!=\;\omega_{\min}^n\wedge\eta_{\min}^n$ . Then,

\begin{align}\mathbb{P}(B_4^n)&\le \sum_{1\le k\le n^2}\mathbb{P}\left(\sup_{\chi_{{k - 1}}^n \le s\le \chi_{{k}}^n} \left| R^n(\chi_k^n) - R^n(s)\right| > \alpha \varepsilon_n/4\right) \nonumber \\[5pt] &\le \sum_{1\le k\le n^2}\mathbb{P}\left(H^{n}_k>\alpha \varepsilon_n/4\right) + \mathbb{P}\left(L^{n}_k>\alpha \varepsilon_n/4\right) \nonumber\\[5pt] &\le n^2\left(\textrm{e}^{-\omega^n_{\min}\alpha \varepsilon_n/4} + \textrm{e}^{-\eta^n_{\min}\alpha \varepsilon_n/4}\right)\nonumber\\[5pt] &\le n^2 \left( 2\textrm{e}^{-\kappa_n(\alpha \varepsilon_n/4)}\right). \nonumber\end{align}

By definition,

\begin{align*} \kappa_n \;:\!=\; \min_{i\in\mathcal{S}}\left\{\left(\sqrt{\frac{\mu^2_i}{\sigma^4_i} + \frac{2n^2}{\sigma^2_i}} + \frac{\mu_i}{\sigma^2_i}\right)\wedge\left( \sqrt{\frac{\mu^2_i}{\sigma^4_i} + \frac{2n^2}{\sigma^2_i}} - \frac{\mu_i}{\sigma^2_i}\right)\right\}= O(n),\end{align*}

where the notation O(g(n)), for $g \colon \mathbb{N}\mapsto \mathbb{R}_+$ , denotes a function $f \colon \mathbb{N}\mapsto \mathbb{R}$ such that $\limsup_{n \rightarrow \infty} |f(n)|/g(n) \le M$ for some $M\in\mathbb{R}$ . Then, there exists $n_1$ such that $\kappa_n >n^{1/2}$ for all $n\ge n_1$ , and so, for $n\ge n_1$ ,

\begin{align*} \mathbb{P}(B_4^n)&\le n^2 \left(2\textrm{e}^{-\kappa_n(\alpha/4)n^{-1/2}\log(n)}\right) \le n^2 \left(2\textrm{e}^{-(\alpha/4)\log(n)}\right) =K_1n^{2-\alpha/4} . \end{align*}

Choose $\alpha$ to be larger than $\alpha_1\;:\!=\;8q+8$ . Then, $\mathbb{P}(B_4^n)$ is an $O(n^{-2q})$ function, and thus it is an $o(n^{-q})$ function.

Bounding $\mathbb{P}(C^n_{\chi})$ and $\mathbb{P}(C^n_{\theta})$ . Let $\{p_n\}_{n\ge 0}$ be a sequence taking values in $\mathbb{N}$ . By Doob’s $L_p$ -maximal inequality,

(3.5) \begin{align} \mathbb{P}(C^n_\chi )\le\frac{\mathbb{E}[(\chi_{n^2}^n - 1)^{2p_n}]}{(\delta_n)^{2p_n}}. \end{align}

Since $\chi_{n^2}^n$ is a convolution of $2n^2$ exponential random variables of rate $2n^2$ , $\chi_{n^2}^n\sim\mbox{Erlang}(2n^2, 2n^2)$ , so that (3.5) and Lemma A.1 (in the Appendix) imply that

(3.6) \begin{align}\mathbb{P}(C^n_\chi )&\le(\delta_n)^{-2p_n} \frac{(2p_n)!\sqrt{2n^2}}{(2n^2)^{2p_n}} \frac{\sqrt{2n^2}^{2p_n+1}-1}{\sqrt{2n^2}-1}\nonumber\\[5pt] & \le K_2 \frac{(\delta_n)^{-2p_n}(2p_n)! }{n^{2p_n-1}}\nonumber\\[5pt] & \le K_2 n\left(\frac{2p_n }{\delta_n n}\right)^{2p_n}.\end{align}

Similarly, since $\theta_{n^2}^n\sim \mbox{Erlang}(n^2, n^2)$ , we have, for $n \geq 2$ ,

(3.7) \begin{align}\mathbb{P}(C^n_\theta)\le K_3 n\left(\frac{2p_n }{\delta_n n}\right)^{2p_n}.\end{align}

Set $\delta_n \;:\!=\; 2p_nn^{(q+1/2)/p_n - 1}$ , $n\ge 1$ . With this choice of $\{\delta_n\}$ , both (3.6) and the right-hand side of (3.7) are proportional to $n^{-2q}$ , so that $\mathbb{P}(C^n_\chi )$ and $\mathbb{P}(C^n_\theta)$ are $o(n^{-q})$ functions.

Bounding $\mathbb{P}(D^n)$ . Set $p_0\;:\!=\;0$ and $p_n \;:\!=\; \lfloor\log(n)\rfloor$ for $n\ge 1$ ; we can verify that with this choice of $\{p_n\}_{n\ge 0}$ , the sequence $\{\delta_n\}_{n\ge 0}$ is an $O(n^{-1}\log(n))$ function. Define $\mu_{\max}\;:\!=\;\max_{i\in\mathcal{S}}|\mu_i|$ , $\sigma_{\max}\;:\!=\;\max_{i\in\mathcal{S}}\sigma_i$ , and ${\delta}'_{\!\!n} \;:\!=\; 2\delta_n + n^{-2}$ , and let $n_2\ge n_1$ be such that $\alpha_1\varepsilon_n/8-\mu_{\max}{\delta}'_{\!\!n}>0$ for all $n\ge n_2$ . Then, for any $\alpha\ge \alpha_1$ , we have $\alpha\varepsilon_n/8-\mu_{\max}{\delta}'_{\!\!n}>0$ for all $n\ge n_2$ . Thus,

\begin{align*} \mathbb{P}(D^n) & \leq {\sum_{k = 1}^{n^2} \mathbb{P} \left(\sup_{0 \leq s \leq {\delta}'_{\!\!n}} \left| R\left((k/n^2 - \delta_n)_+\right) - R\left((k/n^2 - \delta_n + s)_+\right) \right| > \alpha \varepsilon_n/8\right)}\nonumber \\[5pt] & \leq { \sum_{k = 1}^{n^2} \sum_{i \in \mathcal{S}} \mathbb{P}\left( \sup_{0 \leq s \leq {\delta}'_{\!\!n}} \left| R\left((k/n^2 - \delta_n)_+\right) - R\left((k/n^2 - \delta_n + s)_+\right) \right| > \alpha \varepsilon_n/8 \; \Big | \; G_{i, k, n} \right)}, \nonumber \\ & {\rm{where}} \, G_{i,k,n} \;:\!=\; \{J(k/n^2 - \delta_n) = i\}. {\rm{By \, the \,strong \,Markov \,property, \,we \,can \,rewrite \,the \,above}} \nonumber \\[5pt] & {\rm{right-hand \,side \,to \,obtain}} \nonumber \\[5pt] & \mathbb{P}(D^n) = {n^2\sum_{i \in \mathcal{S}} \mathbb{P}\left(\sup_{0 \leq s \leq {\delta}'_{\!\!n}} \left| R(s) \right| > \alpha \varepsilon_n/8 \; \Big | \; J(0) = i \right)} \nonumber \\[10pt] &\le n^2m \left[\frac{4}{\sqrt{2\pi}\,}\frac{\sqrt{\sigma_{\max} {\delta}'_{\!\!n}}}{\alpha \varepsilon_n / 8 - \mu_{\max} {\delta}'_{\!\!n}}\exp\left(-\frac{(\alpha\varepsilon_n/8-\mu_{\max}{\delta}'_{\!\!n})^2}{2\sigma_{\max}{\delta}'_{\!\!n}}\right)\right]\nonumber \\[10pt] &\le K_4n^2\exp\left(-\frac{(\alpha\varepsilon_n/8-\mu_{\max}{\delta}'_{\!\!n})^2}{2\sigma_{\max}{\delta}'_{\!\!n}}\right),\qquad n\ge n_2, \end{align*}

where the first inequality follows from Lemma A.2 (in the Appendix).

Let $n_3\ge n_2$ be such that $\varepsilon_n\le 1$ and ${\delta}'_{\!\!n}\le 3\delta_n$ for all $n\ge n_3$ . Then

(3.8) \begin{align}\mathbb{P}(D^n)&\le K_4n^2 \exp\left(-\frac{(\alpha\varepsilon_n/8)^2 + (\mu_{\max}{\delta}'_{\!\!n})^2 - 2(\alpha\varepsilon_n/8)(\mu_{\max}{\delta}'_{\!\!n}) }{2\sigma_{\max}{\delta}'_{\!\!n}}\right) \nonumber\\[5pt] & \le K_4 n^2 \exp\left(-\frac{(\alpha\varepsilon_n/8)^2 - 2(\alpha\varepsilon_n/8)(\mu_{\max}{\delta}'_{\!\!n}) }{2\sigma_{\max}{\delta}'_{\!\!n}}\right) \nonumber\\[5pt] & \le K_4 n^2 \exp\left(-\frac{(\alpha\varepsilon_n/8)^2}{2\sigma_{\max}{\delta}'_{\!\!n}} + \frac{\alpha\mu_{\max}}{8\sigma_{\max}}\right)\nonumber\\[5pt] & \le K_5 n^2 \exp\left(-\frac{(\alpha\varepsilon_n/8)^2}{6\sigma_{\max}\delta_n}\right)\nonumber\\[5pt] &= K_5 n^2 \exp\left(-\frac{\alpha^2n^{-1}(\log(n))^2}{K_6\lfloor\log(n)\rfloor n^{(q+1/2)/\lfloor\log(n)\rfloor -1}}\right)\nonumber \\[5pt] & \le K_5 n^2 \exp\left(-\frac{\alpha^2\log(n)}{K_6 n^{(q+1/2)/\lfloor\log(n)\rfloor}}\right),\qquad n\ge n_3.\end{align}

Let $\gamma(q)=\sup_{n\ge n_3} K_6 n^{(q+1/2)/\lfloor\log(n)\rfloor}$ , which is finite since $n^{(q+1/2)/\lfloor\log(n)\rfloor}$ converges to $\textrm{e}^{q + 1/2}$ . If $\alpha > \alpha_2\;:\!=\;\sqrt{(2q+2)\gamma(q)}$ , by (3.8) we have

\begin{align*}\mathbb{P}(D^n) & \le K_5 n^2 \exp\left(-\frac{(2q + 2)\gamma(q)\log(n)}{ K_6 n^{(q+1/2)/\lfloor\log(n)\rfloor}} \right) \\& \le K_5 n^2 \exp\left(-(2q + 2)\log(n)\right) \\& = K_5 n^{-2q}, \qquad n\ge n_3,\end{align*}

which implies $\mathbb{P}(D^n)$ is an $o(n^{-q})$ function. Thus, all four terms on the left-hand side of (3.4) are $o(n^{-q})$ functions, and so is $\mathbb{P}(A^n)$ .

Finally, let $n_4\ge n_3$ be such that $\delta_n<1-T$ for all $n\ge n_4$ . Then, $\mathbb{P}\left(\chi^n_{n^2} < T\right)\le \mathbb{P}\left(C^n_{\chi}\right)$ for all $n\ge n_4$ , meaning that (3.1) is an $o(n^{-q})$ function. The proof of (1.4) is now complete.

Proof of Theorem 1.2(ii). Now, let $\{\rho_{\ell}\}_{\ell\ge 1}$ be a sequence with $\rho_{\ell}\downarrow 0$ , and define $E^\ell \;:\!=\; \bigcap_{j=1}^\infty \bigcup_{n=j}^\infty \left\{\pi_2(J^n(s))\neq J(s) \mbox{ for some }s\in(T-\rho_\ell, T + \rho_\ell)\right\}$ . Proving (1.5) is equivalent to showing that $\mathbb{P}\big(\cap_{\ell=1}^\infty E^\ell\big) = 0$ , which in turn is equivalent to proving that $\lim_{\ell\rightarrow\infty}\mathbb{P}(E^\ell)=0$ .

Define $\beta_0\;:\!=\;0$ , $\beta^n_0\;:\!=\; 0$ for $n\ge 0$ . For $k\ge 0$ , let $\beta_{k+1} \;:\!=\;\inf\left\{s>\beta_k\;:\;J(s^-)\neq J(s)\right\}$ , $\beta_{k+1}^n \;:\!=\; \inf \left\{s>\beta_k^n\;:\;\pi_2(J^n(s^-))\neq \pi_2(J^n(s))\right\}$ ( $n\ge 0$ ). For any $a,b\in\mathbb{R}$ , define $M^0[a,b]\;:\!=\; M(b_+)-M(a_+)$ ; recall that $\mathcal{M}^0$ is a Poisson process of rate $\lambda_0/2$ defined in Section 2. Then, $E^\ell \subseteq \left\{M^0[T-2\rho_\ell, T+2\rho_\ell]>0\right\} \cup \left(\left\{M^0[T-2\rho_\ell, T+2\rho_\ell]=0\right\} \cap E^{\ell}\right)$ . Note that $\mathbb{P}(M^0[T-2\rho_\ell, T+2\rho_\ell]>0) \le 1-\textrm{e}^{-(\lambda_0/2)4\rho_\ell} \rightarrow 0$ as $\ell\rightarrow\infty$ . Thus, in order to prove that $\lim_{\ell\rightarrow\infty}\mathbb{P}(E^\ell)= 0$ , it is sufficient to show that

\begin{equation*}\lim_{\ell\rightarrow\infty}\mathbb{P}\left(\{M^0[T-2\rho_\ell, T+2\rho_\ell]=0\}\cap E^\ell\right)=0,\end{equation*}

which we do next. A path inspection reveals that

(3.9) \begin{align}& \{M^0 [T-2\rho_\ell, T+2\rho_\ell]=0\}\cap E^\ell\nonumber\\&\subseteq\left\{\bigcup_{k\ge 0}\left\{ \beta_k<T-2\rho_\ell < T+2\rho_\ell<\beta_{k+1}\right\}\right\}\cap E^\ell \nonumber\\&=\bigcup_{k\ge 0}\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty \left\{\begin{array}{cc}\{\beta_k<T-2\rho_\ell< T+2\rho_\ell<\beta_{k+1}, T-\rho_\ell < \beta^n_k \} \; \cup\\\{\beta_k<T-2\rho_\ell < T+2\rho_\ell<\beta_{k+1}, \beta^n_{k+1}<T+\rho_\ell \}\end{array}\right\}\nonumber\\&\subseteq\bigcup_{k\ge 0}\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty \left(\{|\beta_k - \beta_k^n|>\rho_\ell, \beta_k< T-2\rho_\ell \}\cup\{|\beta_{k+1} - \beta_{k+1}^n|>\rho_\ell, \beta^n_{k+1}< T+\rho_\ell \}\right)\nonumber\\&\subseteq \left(\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty \left\{\max_{k:\theta^n_k< T-2\rho_\ell}|\chi^n_k - \theta^n_k|>\rho_\ell\right\}\right) \; \cup \; \left(\bigcap_{j=1}^\infty \bigcup_{n=j}^\infty \left\{\max_{k:\chi^n_k< T+\rho_\ell}|\chi^n_k - \theta^n_k|>\rho_\ell\right\}\right)\nonumber\\&\subseteq \left(\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty \left\{\max_{1\le k\le n^2}|\chi^n_k - \theta^n_k|>\rho_\ell\right\}\right) \cup \left(\bigcap_{j=1}^\infty \bigcup_{n=j}^\infty\{ \theta^n_{n^2}<T-2\rho_\ell\}\right) \nonumber \\& \quad \cup \left(\bigcap_{j=1}^\infty \bigcup_{n=j}^\infty\{\chi^n_{n^2}<T+\rho_\ell\}\right).\end{align}

Since $\{\delta_n\}_{n\ge 1}$ is a sequence such that $\delta_n\downarrow 0$ , we have, for each $\ell\ge 1$ ,

\begin{align*} \mathbb{P}\left(\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty\Big\{\max_{1\le k\le n^2}|\chi^n_k - \theta^n_k| > \rho_\ell\Big\}\right)\le \mathbb{P}\left(\bigcap_{j=1}^\infty\bigcup_{n=j}^\infty\Big\{\max_{1\le k\le n^2}|\chi^n_k - \theta^n_k| > 2\delta_n\Big\}\right) = 0,\end{align*}

where the last equality follows from the fact that $\mathbb{P}\left(\max_{1\le k\le n^2}|\chi^n_k - \theta^n_k| > 2\delta_n\right)\le \mathbb{P}(C^n_\chi\cup C^n_\theta)=o(n^{-q})$ , and applying Borel–Cantelli (choosing, say, $q = 2$ ). Similar arguments follow for the two other events in (3.9). Thus, $\lim_{\ell\rightarrow\infty}\mathbb{P}(E^\ell)=0$ and so (1.5) follows.

4. An application: First passage probabilities

Theorem 1.1 implies that some first passage properties of $(\mathcal{R},\mathcal{J})$ can be analysed as the limiting first passage properties of $(\mathcal{R}^n,\mathcal{J}^n)$ as $n\rightarrow\infty$ . In particular, for any Borel set $A\subset\mathbb{R}$ define $\tau_A \;:\!=\;\inf\left\{s\ge 0\;:\;R(s)\in A\right\}$ , $\tau^n_A \;:\!=\;\inf\left\{s\ge 0\;:\;R^n(s)\in A\right\}$ ( $n\ge 0$ ). Then, Theorem 1.1 implies that, for any open set A and $j\in\mathcal{S}$ , $\tau_A= \lim_{n\rightarrow\infty}\tau_A^n$ a.s., and on the event $\{\tau_A<\infty\}$ , $\left\{J(\tau_A)=j\right\}=\bigcup_{i=0}^\infty\bigcap_{n=i}^\infty\{\pi_2(J^n(\tau_A))=j\}$ a.s.

In the case that A takes the form $(\!-\!\infty,-x)$ for $x\ge 0$ , we have the following.

Proof. Fix $x\ge 0$ and $n\ge 0$ . Let $N\;:\!=\;\sup\{k\ge 0\;:\;\tau_x\ge \theta^n_k\}$ . This implies that $\tau_x\in[\theta^n_N,\theta^n_{N+1})$ on $\{N<\infty\}$ , and since $\mathcal{J}$ is constant between the epochs $\{\theta^n_k\}_{k\ge 0}$ , then $J(\theta^n_N)=J(\tau_x)$ . Similarly, if we define $N^n\;:\!=\;\sup\{k\ge 0\;:\;\tau_x^n\ge \chi^n_k\}$ , then $\pi_2(J^n(\chi^n_{N^n}))=\pi_2(J^n(\tau_x^n))$ on $\{N^n<\infty\}$ . Equations (2.4) and (2.5) imply that $N_n=N$ , and since $J(\theta^n_k)=\pi_2(J^n(\chi^n_k))$ for all $k\ge 0$ (see (2.3)), $J(\tau_x)=J(\theta^n_N)=\pi_2(J^n(\chi^n_{N^n}))=\pi_2(J^n(\tau_x^n))$ on $\{N<\infty\}$ , and the result follows.

The following result describes one central first passage distributional property of our construction.

Theorem 4.1. For $n\ge 0$ , let $U_n$ denote the infinitesimal generator associated to the process $\{\pi_2(J^n(\tau^n_x))\}_{x\ge 0}$ . Then, $U_n$ is a solution to the quadratic matrix equation

(4.1) \begin{align} X^2 + 2\Delta_{\mu}\Delta_{\sigma}^{-2}X + 2\Delta_{\sigma}^{-2}Q=0 ,\end{align}

where $\Delta_{\mu} = \textrm{diag}\{\mu_i\;:\;i\in\mathcal{S}\}$ , $\Delta_{\sigma}=\textrm{diag}\{\sigma_i\;:\;i\in\mathcal{S}\}$ . Furthermore, $U_n$ corresponds to the infinitesimal generator associated to $\{J(\tau_x)\}_{x\ge 0}$ .

Proof. Let $\Psi_n$ be the $p\times p$ -dimensional matrix defined by

\begin{align*} (\Psi_n)_{ij}=\mathbb{P}\left(\tau^n_0 <\infty, J^n(\tau_0)=(-,j) \mid R^n(0)=0, J^n(0)=(+,i)\right), \qquad i,j\in\mathcal{S}. \end{align*}

Define $ \Delta_{r^n_+} =\mbox{diag}\{r^n(+, i)\;:\;i\in\mathcal{S}\}$ , $\Delta_{r^n_-} =\mbox{diag}\{|r^n(-, i)|\;:\;i\in\mathcal{S}\}$ , and

\begin{align*} \left[\begin{array}{c@{\quad}c} T_{++} & T_{+-} \\ T_{-+} & T_{--}\end{array}\right] \;:\!=\; \left[\begin{array}{c@{\quad}c} -\lambda_n I & 2 Q + \lambda_n I \\\lambda_n I & -\lambda_n I \end{array}\right].\end{align*}

It is known [Reference Bean, O’Reilly and Taylor3] that $\Psi_n$ is the minimal nonnegative solution to the Riccati matrix equation

(4.2) \begin{align} \Delta_{r^n_+}^{-1}T_{++}\Psi_n + \Psi_n\Delta_{r^n_-}^{-1}T_{--} + \Psi_n\Delta_{r^n_-}^{-1}T_{-+}\Psi_n + \Delta_{r^n_+}^{-1}T_{+-} = 0, \end{align}

and that $\mathbb{P}(\tau^n_0 <\infty, J^n(\tau_x)=(-,j) \mid R^n(0)=0, J^n(0)=(-,i))={\bf{{e}}}_i^{\top}e^{U_n x}{\bf{{e}}}_j$ , where

(4.3) \begin{align} U_n & = \Delta_{r^n_-}^{-1} (T_{--} + T_{-+}\Psi_n) =\lambda_n \Delta_{r^n_-}^{-1} (\Psi_n - I),\end{align}

with ${\bf{{e}}}_i$ being the ith unit column vector.

Premultiplying (4.2) by $\Delta_{r^n_-}^{-1}$ and commuting $\Delta_{r^n_-}^{-1}$ with $\Delta_{r^n_+}^{-1}$ give

\begin{align*} -\lambda_n \Delta_{r^n_+}^{-1} \Delta_{r^n_-}^{-1}\Psi_n -\lambda_n \Delta_{r^n_-}^{-1}\Psi_n\Delta_{r^n_-}^{-1} + \lambda_n\Delta_{r^n_-}^{-1}\Psi_n\Delta_{r^n_-}^{-1}\Psi_n + \Delta_{r^n_+}^{-1}\Delta_{r^n_-}^{-1}(2Q + \lambda_nI) = 0,\end{align*}

which leads to $(\Delta_{r^n_-}^{-1}-\Delta_{r^n_+}^{-1})U_n+ \lambda_n^{-1}U_n^2 + 2\Delta_{r^n_+}^{-1}\Delta_{r^n_-}^{-1}Q = 0$ . As $\Delta_{r^n_-}^{-1}-\Delta_{r^n_+}^{-1} = 2\lambda_n^{-1}\Delta_{\mu}\Delta_{\sigma}^{-2}$ and $\Delta_{r^n_+}^{-1}\Delta_{r^n_-}^{-1} = \lambda_n^{-1}\Delta_{\sigma}^{-2}$ , we obtain $U_n^2 + 2\Delta_{\mu}\Delta_{\sigma}^{-2}U_n + 2\Delta_{\sigma}^{-2}Q=0$ . That $U_n$ is also the infinitesimal generator of $\{J(\tau_x)\}_{x\ge 0}$ follows from Proposition 4.1.

A. Appendix

The following are some standalone results used in Sections 2 and 3.

Theorem A.1. Let $\mathcal{A}=\{A(t)\}_{t\ge 0}$ be a Poisson process of parameter $\lambda_a>0$ , and $\mathcal{X}=\{X(n)\}_{n\ge 0}$ an independent discrete-time Markov chain with state space $\mathcal{S}$ and transition probability matrix P. Define the Markov jump process $\mathcal{J}=\{J(t)\}_{t\ge 0}$ as $J(t)= X(A(t))$ , $t\ge 0$ . Let $\mathcal{B}=\{B(t)\}_{t\ge 0}$ be an independent Poisson process of parameter $\lambda_b>0$ . Define $\mathcal{C}$ to be the superposition of the Poisson processes $\mathcal{A}$ and $\mathcal{B}$ , and denote by $\{\tau_k\}_{k\ge 0}$ the arrival times of $\mathcal{C}$ . If we let $Y(n)=J(\tau_n)$ , $n\ge 0$ , then the process $\mathcal{Y}=\{Y(n)\}_{n\ge 0}$ is a Markov chain, independent from $\mathcal{C}$ , with transition probability matrix given by

\begin{equation*} \frac{\lambda_a}{\lambda_a+\lambda_b}P + \frac{\lambda_b}{\lambda_a+\lambda_b}I.\end{equation*}

Proof. First, we show that $\mathcal{Y}$ is a Markov process. Let $i\in\mathcal{S}$ and $k\ge 1$ . Then,

\begin{align*}\mathbb{P}(Y(k) & = \ i \mid Y(0), Y(1), \dots, Y(k-1))\\[5pt] & = \mathbb{P}(J(\tau_{k})=i \mid J(\tau_0), J(\tau_{1}), \dots, J(\tau_{k-1}))\\[5pt] & = \mathbb{P}(J(\tau_{k})=i \mid J(\tau_{k-1}))\qquad\mbox{(strong Markov property of $\mathcal{J}$)}\\[5pt] & = \mathbb{P}(Y(k)=i \mid Y(k-1)),\end{align*}

so that the Markov property holds.

Next, let $\mathcal{C}^*$ be the marked Poisson process with arrivals corresponding to the superposition of $\mathcal{A}$ , arrivals which we mark with an a, and $\mathcal{B}$ , arrivals which we mark with a b. The kth arrival of $\mathcal{C}^*$ occurs at $\tau_k$ carrying a mark $m_k\in\{a,b\}$ , say. Then,

\begin{align*} \mathbb{P}(Y(k) &= j \mid Y(k-1)=i,\mathcal{C}) \\[5pt] & = \mathbb{P}(Y(k)=j, m_k=a \mid Y(k-1)=i,\mathcal{C}) + \mathbb{P}(Y(k)=j, m_k=b \mid Y(k-1)=i,\mathcal{C})\\[5pt] & = \mathbb{P}(Y(k)=j \mid Y(k-1)=i, m_k=a,\mathcal{C})\mathbb{P}(m_k=a \mid Y(k-1)=i,\mathcal{C})\\[5pt] &\quad + \mathbb{P}(Y(k)=j \mid Y(k-1)=i, m_k=b,\mathcal{C})\mathbb{P}(m_k=b \mid Y(k-1)=i,\mathcal{C}).\end{align*}

The event $\{m_k=a\}$ is clearly independent of $\{Y(k-1)=i,\mathcal{C}\}$ : the mark of a given Poisson arrival is independent of the history of arrivals. Thus,

\begin{equation*}\mathbb{P}(m_k=a \mid Y(k-1)=i,\mathcal{C}) = \mathbb{P}(m_k=a) = \frac{\lambda_a}{\lambda_a+\lambda_b}.\end{equation*}

Similarly,

\begin{equation*}\mathbb{P}(m_k=b \mid Y(k-1)=i,\mathcal{C}) = \mathbb{P}(m_k=b) = \frac{\lambda_b}{\lambda_a+\lambda_b}.\end{equation*}

Next, since $\mathcal{J}$ only (possibly) jumps at arrival times marked with a, then $\mathbb{P}(Y(k)=j \mid $ $ Y(k-1)=i, m_k=b,\mathcal{C}) = \delta_{ij}$ , where $\delta_{ij}$ denotes the Kronecker delta. Finally, since $\mathcal{J}$ is piecewise constant between the arrival times $\{\tau_k\}_k$ , $\{Y(k-1)=i\} = \{J(\tau_{k-1})= i\}=\{J(\tau_{k}^-)= i\}$ . This implies that

\begin{align*}\mathbb{P}(Y(k)=j \mid Y(k-1)=i, m_k=a,\mathcal{C}) & = \mathbb{P}(J(\tau_{k})= j \mid J(\tau_{k}^-)= i, m_k=a,\mathcal{C}) = p_{ij},\end{align*}

where time homogeneity is used in the last equality. Consequently,

\begin{align*}\mathbb{P}(Y(k)=j \mid Y(k-1)=i, \mathcal{C}) = p_{ij}\frac{\lambda_a}{\lambda_a+\lambda_b} + \delta_{ij}\frac{\lambda_a}{\lambda_a+\lambda_b},\end{align*}

implying that $\mathcal{Y}$ is independent of $\mathcal{C}$ and completing the proof.

Lemma A.1. For $a\in \mathbb{N}_{+}\backslash\{1\}$ and $b>0$ , let $Y\sim\textrm{Erlang}(a,b)$ . Then

(A.1) \begin{align} \mathbb{E}\left[(Y-\mathbb{E}[Y])^k\right] \le \frac{k!\sqrt{a}}{b^k} \frac{\sqrt{a}^{k+1}-1}{\sqrt{a}-1}, \qquad k\in\mathbb{N}_{+}.\end{align}

Proof. Without loss of generality, suppose that $b=1$ . Equation (A.1) can be rewritten as

(A.2) \begin{align} \mathbb{E}\left[(Y-\mathbb{E}[Y])^k\right] \le {k!}\sum_{j=1}^k\sqrt{a}^j.\end{align}

We use induction to prove that (A.2) holds. First, since $\mathbb{E}[Y-\mathbb{E}[Y]]=0< 1!\sqrt{a}$ , the case $k=1$ holds trivially. Now, suppose (A.1) holds for all $k\in\{1,2,\dots, k_0\}$ for some $k_0\ge 1$ . By [Reference Willink20, third formula on p. 704],

\begin{equation*} \mathbb{E}\left[(Y-\mathbb{E}[Y])^{k_0+1}\right] = k_0!a\sum_{i=0}^{k_0-1} \frac{\mathbb{E}\left[(Y-\mathbb{E}[Y])^{i}\right]}{i!} =k_0!a\left[1 + \sum_{i=2}^{k_0-1}\frac{\mathbb{E}[(Y-\mathbb{E}[Y])^{i}]}{i!}\right]. \end{equation*}

Using the induction hypothesis on the right-hand side of this equation, we get

\begin{align*}\mathbb{E}[(Y-\mathbb{E}[Y])^{k_0+1}] & \le k_0!a\left[1 + \sum_{i=2}^{k_0-1}\sum_{j=1}^i\sqrt{a}^j\right] \le k_0!a\sum_{i=1}^{k_0-1}\sum_{j=1}^i\sqrt{a}^j\\ &= k_0!a\sum_{j=1}^{k_0-1}\sum_{i=j}^{k_0-1}\sqrt{a}^j= k_0!a\sum_{j=1}^{k_0-1}(k_0-j)\sqrt{a}^j\\ & \le (k_0+1)!a\sum_{j=1}^{k_0-1}\sqrt{a}^j\le (k_0+1)!\sum_{j=1}^{k_0+1}\sqrt{a}^j,\end{align*}

which proves (A.2) and thus (A.1).

Lemma A.2. Let $(\mathcal{R}, \mathcal{J}) = \{(R(t), J(t))\}_{t\ge 0}$ be a Markov-modulated Brownian motion defined as in (1.3). Then, for any $i\in\mathcal{S}$ , $t > 0$ , and ${a > \mu_{\max} t}$ ,

\begin{equation*}\mathbb{P}\left(\sup_{0 \le s\le t}|R(s)|> a \; \Big| \; J(0) = i \right)\le \frac{4}{{\sqrt{2 \pi}\,}}{\frac{\sqrt{\sigma_{\max}t}}{a - \mu_{\max}t}}\exp\left(-\frac{(a-\mu_{\max}t)^2}{2\sigma_{\max}t}\right),\end{equation*}

where $\mu_{\max}\;:\!=\;\max_{i\in\mathcal{S}}|\mu_i|$ and $\sigma_{\max}\;:\!=\;\max_{i\in\mathcal{S}}\sigma_i.$

Proof. Let $\{W(t)\}_{t\ge 0}$ be a standard Brownian motion, independent of $(\mathcal{R},\mathcal{J})$ . A standard bound for the Brownian motion based on the reflection principle gives, for $b > 0$ ,

\begin{align*} \mathbb{P}\left(\sup_{0 \le s\le t}|W(s)|> b\right) &\le 2\mathbb{P}\left(\sup_{0 \le s\le t}W(s)> b\right)\le 4 \int_b^{\infty} \frac{1}{\sqrt{2\pi t}\,}\textrm{e}^{-x^2/ 2t} \, \textrm{d} x \\ & \leq 4 \int_{b}^{\infty} \frac{x / b}{\sqrt{2 \pi t}\,} \textrm{e}^{-x^2/ 2t} \, \textrm{d} x \le \frac{4}{\sqrt{2 \pi}\,} \left(\frac{\sqrt{t}}{b}\right)\textrm{e}^{-b^2/2t}.\end{align*}

Note that $\mathcal{R}$ is identically distributed to $\{W(I^\sigma_t) + I^\mu_t\}_{t\ge 0}$ , where $ I^\sigma_t \;:\!=\; \int_0^t\sigma_{J(s)} \, \textrm{d} s$ and $I^{\mu}_t \;:\!=\;\int_0^t\mu_{J(s)} \, \textrm{d} s$ . This implies that

\begin{align*}\mathbb{P}&\left(\sup_{0 \le s\le t}|R(s)|> a \;\Big| \; J(0)=i\right) = \mathbb{P}\left(\sup_{0 \le s\le t}|W\left(I^\sigma_s\right) + I^\mu_s|> a\;\Big| \; J(0)=i\right)\\ & \le \mathbb{P}\left(\sup_{0 \le s\le t}|W(I^\sigma_s)| > a - \sup_{0 \le s\le t}|{I^\mu_s}|\;\Big| \; J(0)=i\right)\\ & = \mathbb{E}\left(\mathbb{P}\left(\sup_{0 \le s\le t}\left|W(I^\sigma_s)\right| > a - \sup_{0 \le s\le t}|{I^\mu_s}| \; \Big| \; J(0)=i, \{I^\sigma_s\}_{0\le s\le t}, \{I^\mu_s\}_{0\le s\le t} \right)\;\Big|\;J(0)=i\right) \\ & \le \mathbb{E}\left(\frac{4}{{\sqrt{2\pi}}\,}{\left(\frac{\sqrt{\sup_{0 \le s\le t}|{I^\sigma_s}|}}{a-\sup_{0 \le s\le t}|{I^\mu_s}|}\right)} \exp\left(-\frac{(a-\sup_{0 \le s\le t}|{I^\mu_s}|)^2}{2\sup_{0 \le s\le t}|{I^\sigma_s}|}\right)\;\Big| \; J(0)=i\right)\\ & \le \mathbb{E}\left(\frac{4}{{\sqrt{2 \pi}}\,}{\frac{\sqrt{\sigma_{\max}t}}{a - \mu_{\max}t}}\exp\left(-\frac{(a-\mu_{\max}t)^2}{2\sigma_{\max}t}\right)\;\Big| \; J(0)=i\right) \\ & = \frac{4}{{\sqrt{2 \pi}}\,}{\frac{\sqrt{\sigma_{\max}t}}{a - \mu_{\max}t}}\exp\left(-\frac{(a-\mu_{\max}t)^2}{2\sigma_{\max}t}\right),\end{align*}

which completes the proof.