2. Hybrid stochastic differential equations

Let us provide a precise description of a hybrid SDE and its solution $(J,X)=\{(J(t),X(t))\}_{t\ge 0}$ , initially adhering to the traditional framework as seen in sources such as [Reference Yin and Zhu39]. For such a construction we require a complete probability space $(\Omega, \mathbb{P}, \mathcal{F})$ that supports the following independent components:

• • A standard Brownian motion ${B}=\{B(t)\}_{t\ge 0}$ ; this will dictate the continuously varying nature of X.

• • A Poisson random measure $\mathfrak{p}(\mathrm{d}t, \mathrm{d}z)$ on $\mathbb{R}_+\times\mathbb{R}_+$ of intensity $\mathrm{d}t\times\mathrm{d}z$ ; this will be used to describe the jump dynamics of J.

Then, (J, X) is the solution to the SDE

(2.1) \begin{align} \mathrm{d}X(t) = \mu(J(t), X(t))\,\mathrm{d}t + \sigma(J(t), X(t))\,\mathrm{d}B(t), \qquad X(0) = x_0\in\mathbb{R}, \\[-35pt] \nonumber \end{align}

(2.2) \begin{align} \mathrm{d}J(t) = \int_{\mathbb{R}_+}h(J(t{-}),X(t{-}), z)\,\mathfrak{p}(\mathrm{d}t, \mathrm{d}z), \qquad\quad J(0) = i_0\in\mathcal{E}, \end{align}

where X is an a.s. continuous real process, J is a càdlàg jump process with finite state space $\mathcal{E}=\{1,\dots,p\}$ , $\mu\colon\mathcal{E}\times \mathbb{R} \mapsto \mathbb{R}$ , $\sigma\colon\mathcal{E}\times \mathbb{R} \mapsto \mathbb{R}$ , and $h\colon\mathcal{E}\times \mathbb{R} \times \mathbb{R} \mapsto \mathbb{Z}$ . Note that the value of h in $\mathbb{Z}$ dictates how many steps J moves (backward or forward) in $\mathcal{E}$ at each time epoch of $\mathfrak{p}$ . Furthermore, we require that (J, X) be adapted to the $\mathbb{P}$ -completed filtration $\mathcal{F}_t$ generated by $\{B(s), \mathfrak{p}(s, \cdot);\ s \le t\}$ . A pair (J, X) that satisfies the aforementioned characteristics is referred to as a strong solution of (2.1)–(2.2). For pedagogical purposes, using a uniformization perspective [Reference van Dijk, van Brummelen and Boucherie35], in this section we will demonstrate the existence of such a solution under simple regularity assumptions in due course.

Here, we will focus on hybrid SDEs with an environmental process J that switches at a state-dependent rate. Specifically, we aim to let J evolve according to

(2.3) \begin{equation} \mathbb{P}(J(t+\mathrm{d}t) = j\mid X(t), J(t)=i) = \delta_{ij} + \Lambda_{ij}(X(t))\,\mathrm{d}t\end{equation}

for a certain family of intensity matrices $\{\boldsymbol{\Lambda}(x)\}_{x\in \mathbb{R}}$ that are assumed to be càdlàg, and thus measurable. Though (2.2) and (2.3) may initially seem like unrelated definitions for J, we only need to relate them by aligning the jump structure of h in (2.2) with the intensities in (2.3). Our approach to achieving this alignment differs slightly from the method presented in [Reference Skorokhod33]. However, this new construction enables us to translate (2.2) into a uniformization framework, thereby allowing us to utilize the existing toolbox for such methods. In our case, we need the following boundedness condition.

Assumption 2.1. The family $\{\boldsymbol{\Lambda}(x)\}_{x\in\mathbb{R}}$ is uniformly bounded, i.e.

\begin{align*} \gamma\,:\!=\,\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}} |\Lambda_{ii}(x)| < \infty.\end{align*}

For our construction, we define h by

(2.4) \begin{equation} h(i,x,z)=\sum_{j=1}^p (j-i)1_{\{z\in \Delta_{ij}(x)\}}.\end{equation}

For each fixed i and $x\in\mathbb{R}$ , $\{\Delta_{ij}(x)\colon j\in\mathcal{E}\}$ is a measurable partition of the interval $[0,\gamma)$ , where the Lebesgue measure of $\Delta_{ij}(x)$ is given by $\mbox{Leb} (\Delta_{ij}(x))=\gamma\delta_{ij} + \Lambda_{ij}(x)$ . Note that we ought to ensure that h as specified in (2.4) is a measurable function of its three entries, so that we need to choose the families $\{\Delta_{ij}(x)\colon j\in\mathcal{E}\}$ carefully. One straightforward way to guarantee this measurability is by defining $\Delta_{ij}(x)=[\ell_{ij}(x), r_{ij}(x))$ , where $\ell_{i1}(x)= 0$ , $r_{i,j}(x)= \ell_{ij}(x)+ (\gamma\delta_{ij} + \Lambda_{ij}(x))$ , $\ell_{i,{j+1}}(x)=r_{ij}(x)$ , and $r_{ip}(x)=\gamma$ . Unless otherwise stated, this is the standard choice used throughout the text.

With the above choice, all the points of $\mathfrak{p}$ that land outside of $\mathbb{R}_+\times[0,\gamma)$ do not contribute to jumps in (2.2). In other words, we can rewrite (2.2) as

(2.5) \begin{equation} \mathrm{d}J(t) = \int_{[0,\gamma)}h(J(t{-}), X(t{-}), z)\,\mathfrak{p}^{*}(\mathrm{d}t,\mathrm{d}z), \quad J(0)=i_0\in\mathcal{E},\end{equation}

where $\mathfrak{p}^*$ is the Poisson random measure $\mathfrak{p}$ restricted to $\mathbb{R}_+\times[0,\gamma)$ . The advantage of considering $\mathfrak{p}^*$ instead of $\mathfrak{p}$ is that the placement of the atoms of $\mathfrak{p}^*$ may be regarded as a Poisson process $\{\theta_k\}_{k\ge 1}$ of intensity $\gamma$ over $\mathbb{R}_+$ (the t-axis), with each arrival $\theta_k$ carrying an independent mark $U_k$ that is uniformly distributed over the interval $[0,\gamma)$ (the z-axis).

With the aforementioned choice of h and the characterization of $\mathfrak{p}^*$ in terms of $\{(\theta_k, U_k)\}_{k\ge 1}$ , we are now ready to explicitly construct the strong solution to (2.1) and (2.2). To facilitate our construction, from now on we assume the following.

Under Assumption 2.2, it is standard that a strong solution to the (ordinary) SDE

(2.6) \begin{equation} X^{[i,x,v]}(t) = x + \int_0^t\mu\big(i,X^{[i,x,v]}(r)\big)\,\mathrm{d}r + \int_0^t\sigma\big(i,X^{[i,x,v]}(r)\big)\,\mathrm{d}B^{[v]}(r) \end{equation}

exists for all $i\in\mathcal{E}$ , $x\in\mathbb{R}$ , and $v,t\ge 0$ , where $B^{[v]}(t)\,:\!=\, B(v+t)-B(v)$ is a time-shifted version of B. In other words, ${X}^{[i,x,v]}$ corresponds to the solution of the SDE driven by ${B}^{[v]}$ , with coefficients $\mu(i,\cdot)$ and $\sigma(i,\cdot)$ , and starting point x.

The existence of a pair (J, X) that solves (2.1) and (2.2) (alternatively, (2.1) and (2.5)) is directly derived from [Reference Ikeda and Watanabe22, Chapter IV.9]. Since its construction is crucial to our convergence developments, we will now offer a concise review for the sake of completeness. Define $X(0)=x_0$ and $J(0)=i_0$ , and for $t\in (0,\theta_1)$ , let $J(t)=i_0$ and $X(t)=X^{[i_0,x_0, 0]}(t)$ : we have defined the processes X and J in $[0,\theta_1)$ . Since we want X to be continuous, we ought to define $X(\theta_1)=X(\theta_1{-})$ . According to (2.5), J jumps at time $\theta_1$ to the state $i_0 + h(X(\theta_1), i_0, U_1)$ , which is equal to state k if and only if

(2.7) \begin{equation} U_1\in \Delta_{i_0,k}(X(\theta_1)).\end{equation}

In simple terms, we are using $U_1$ and $X(\theta_1)$ in such a way that J jumps to state k at time $\theta_1$ with a mass given by $\mbox{Leb}(\Delta_{i_0,k}(X(\theta_1)))/\gamma$ , corresponding to the $(i_0,k)$ th entry of the probability matrix $\boldsymbol{I}+\boldsymbol{\Lambda}(X(\theta_1))/\gamma$ , the uniformized version of $\boldsymbol{\Lambda}(X(\theta_1))$ . After having defined (J, X) in $[0,\theta_1]$ , we construct X in subsequent intervals $(\theta_\ell,\theta_{\ell + 1}]$ by concatenating strong solutions of the type (2.6) with appropriate chosen values for (i, x, v), as well as using a decision rule similar to (2.7) to establish which states J visits. More specifically, in a recursive manner, for $\ell=1,2,3,\dots$ and $t\in(0,\theta_{\ell+1}-\theta_\ell)$ let

(2.8) \begin{equation} \begin{aligned} X(\theta_{\ell} + t) & = X^{[i_\ell, x_\ell,\theta_\ell]}(t), \\ X(\theta_{\ell+1}) & = X(\theta_{\ell+1}{-}), \\ J(\theta_{\ell} + t) & = J(\theta_{\ell}), \\ J(\theta_{\ell+1}) & = k \quad\mbox{if and only if}\ U_{\ell+1}\in \Delta_{i_\ell,k}(X(\theta_{\ell+1})). \end{aligned}\end{equation}

A few aspects that are straightforward to verify about this construction:

• • X and J are $\mathcal{F}_t$ -adapted;

• • X is continuous at $\theta_1, \theta_2,\dots$ , and thus has a.s. continuous paths;

• • X uniquely solves the hybrid SDE (2.1) on every interval $(\theta_\ell,\theta_{\ell+1})$ , $\ell=0,1,2,\dots$ ;

• • J is càdlàg and uniquely solves (2.5).

Thus, (J, X) is indeed the strong solution to (2.2).

We still need to verify that (2.3) holds, which we prove in a slightly more general scenario next. Recall that (2.3) only makes sense if $\boldsymbol{\Lambda}(x)$ is càdlàg with respect to x. A generalization of this property for discontinuous $\boldsymbol{\Lambda}( \! \cdot \! )$ is given by

(2.9) \begin{align} \mathbb{P}\big(&J(t+v) = J(t) \mbox{ for all }v\in[0,s] \nonumber \\ & \mid \mathcal{F}_t\otimes\mathcal{F}^{{B}}_{t+s}, X(t)=x, J(t)=i\big) = \exp\bigg(\int_0^s\Lambda_{ii}\big(X^{[i,x,t]}(v)\big)\,\mathrm{d}v\bigg), \\[-37pt] \nonumber \end{align}

(2.10) \begin{align} \mathbb{P}(&J(\theta_\ell) = j \mid J(\theta_\ell{-})=i, J(\theta_\ell{-})\neq J(\theta_\ell), X(\theta_\ell)) = \frac{\Lambda_{ij}(X(\theta_\ell))}{|\Lambda_{ii}(X(\theta_\ell))|}, \quad i\neq j, \end{align}

where $\mathcal{F}^{{B}}_t$ is the $\mathbb{P}$ -completion of the $\sigma$ -algebra generated by $\{B(s)\colon 0\le s\le t\}$ , and $\otimes$ denotes the product $\sigma$ -algebra. In essence, (2.9) and (2.10) correspond to how inhomogeneous Markov jump processes are classically constructed through their integrated jump intensities/hazard rates (see, e.g., [Reference Trivedi and Bobbio34, Chapter 13]). That (2.9) and (2.10) imply (2.3) for continuous $\boldsymbol{\Lambda}( \! \cdot \! )$ is readily obtained by noting that $\exp\big(\int_0^{\mathrm{d}t}\Lambda_{ii}\big(X^{[i,x,t]}(v)\big)\,\mathrm{d}v\big) =1 + \Lambda_{ii}(x)\,\mathrm{d}t$ , where we employed the continuity of $X^{[i,x,t]}$ too. For the sake of completeness, we now show that (2.9) and (2.10) indeed hold.

Proof. Integrating with respect to the number and position of Poisson arrivals of $\mathfrak{p}^*$ in $[t,t+s]$ , say $\{\theta'_\ell\}_\ell$ ,

\begin{align*} & \mathbb{P}\big(J(t+v) = J(t)\mbox{ for all }v\in[0,s] \mid \mathcal{F}_t\otimes\mathcal{F}^{{B}}_{t+s},X(t)=x,J(t)=i\big) \\ & = \sum_{k=0}^\infty\frac{(\gamma s)^k}{k!}\mathrm{e}^{-\gamma s} \\ & \qquad\quad \times \int_0^s\cdots\int_0^s\frac{\prod_{j=1}^k\mathbb{P}\big(J(\theta'_j)=J(\theta'_j{-}) \mid \theta'_j=t+v_j,X^{[i,x,t]}(v_j),J(\theta'_j)=i\big)} {s^k}\,\mathrm{d}v_1\cdots\mathrm{d}v_k \\ & = \sum_{k=0}^\infty\frac{(\gamma s)^k}{k!}\mathrm{e}^{-\gamma s}\int_0^s\cdots\int_0^s \frac{\prod_{j=1}^k\big(1+\Lambda_{ii}(X^{[i,x,t]}(v_j))/\gamma\big)}{s^k}\,\mathrm{d}v_1\cdots\mathrm{d}v_k \\ & = \sum_{k=0}^\infty\frac{(\gamma s)^k}{k!}\mathrm{e}^{-\gamma s} \Bigg(1+\frac{\int_0^s\Lambda_{ii}\big(X^{[i,x,t]}(v)\big)\,\mathrm{d}v}{\gamma s}\Bigg)^k = \exp\bigg(\int_0^s\Lambda_{ii}\big(X^{[i,x,t]}(v)\big)\,\mathrm{d}v\bigg), \end{align*}

where in the first equality we used that on the event $\{J(t+v) = J(t) \mbox{for all }v\in[0,s],$ $J(t)=i, X(t)=t\}$ , $X(t+v)=X^{[i,x,t]}(v)$ for all $v\in[0,s]$ by construction. Finally, for $i\neq j\in\mathcal{E}$ ,

\begin{multline*} \mathbb{P}\big(J(\theta_\ell)=j\mid J(\theta_\ell{-})=i,J(\theta_\ell{-})\neq J(\theta_\ell),X(\theta_\ell)\big) \\ = \frac{\mathbb{P}\big(J(\theta_\ell)=j\mid J(\theta_\ell{-})=i,X(\theta_\ell)\big)} {\mathbb{P}\big(J(\theta_\ell{-})\neq J(\theta_\ell)\mid J(\theta_\ell{-})=i,X(\theta_\ell)\big)} = \frac{\Lambda_{ij}(X(\theta_\ell))}{|\Lambda_{ii}(X(\theta_\ell))|}, \end{multline*}

completing the proof.

We note that the construction of strong solutions for hybrid SDEs can be carried out in more general scenarios. For instance, Brownian-driven multidimensional hybrid SDEs with unbounded jump intensities are considered in [Reference Nguyen and Yin29].

3. Space-grid approximation of hybrid SDEs

Although solutions to state-dependent hybrid SDEs are easy to simulate following the steps in (2.8), studying their distributional properties is a challenging task (see [Reference Yin and Zhu39] and references therein). Here we advocate studying hybrid SDEs using an approximation method via discretizing the space grid. This approximation differs from the well-known methods based on Wong–Zakai [Reference Nguyen and Peralta30] and Euler–Maruyama [Reference Yin and Zhu39, (5.4)] methods, and thus requires an analysis of its own, which is carried out next.

For each $i\in\mathcal{E}$ , let $\widehat{\mu}(i,\cdot)$ and $\widehat{\sigma}(i,\cdot)$ be piecewise constant càdlàg approximations of ${\mu}(i,\cdot)$ and ${\sigma}(i,\cdot)$ on a space grid $\{\zeta_m\}$ of $\mathbb{R}$ , with $|\widehat{\mu}(i,\cdot)|\le |\mu (i,\cdot)|$ and $|\widehat{\sigma}(i,\cdot)|\le |\sigma(i,\cdot)|$ . Furthermore, let $\widehat{{\boldsymbol{\Lambda}}}( \! \cdot \! )$ be a piecewise constant càdlàg approximation of $\boldsymbol{\Lambda}( \! \cdot \! )$ over the space grid $\{\zeta_m\}$ , with $\widehat{{\boldsymbol{\Lambda}}}(x)$ being an intensity matrix such that $\sup\nolimits_{i}|\widehat{{\Lambda}}_{ii}(x)|\le\gamma$ for all $x\in\mathbb{R}$ . Additionally, for each $i\in\mathcal{E}$ and $x\in\mathbb{R}$ , let $\{\widehat{\Delta}_{ij}(x)\}_j$ be a partition of $[0,\gamma)$ with $\mbox{Leb}(\widehat{\Delta}_{ij}(x))= \gamma\delta_{ij} + \widehat{\Lambda}_{ij}(x)$ , where each aforementioned set is such that the following function is measurable:

\begin{equation*} \widehat{h}(i,x,z)=\sum_{j=1}^p (j-i)1_{\{z\in \widehat{\Delta}_{ij}(x)\}}.\end{equation*}

For such (fixed) functions and partitions, we assume the following.

• • $\{\mathcal{G}_t\}_{t\ge 0}$ is admissible with respect to B and $\mathfrak{p}^*$ , i.e. $B(t+s)-B(t)$ and $\mathfrak{p}^*((t,t+s],\cdot)$ are independent of $\mathcal{G}_t$ for all $s,t\ge 0$ ;

• • $(\widehat{J}, \widehat{X})$ is the $\mathcal{G}_t$ -adapted solution of the hybrid SDE

  (3.1) \begin{equation} \begin{alignedat}{2} \mathrm{d}\widehat{X}(t) & = \widehat{\mu}(\widehat{J}(t),\widehat{X}(t))\,\mathrm{d}t + \widehat{\sigma}(\widehat{J}(t),\widehat{X}(t))\,\mathrm{d}B(t), \quad & \widehat{X}(0) & = x_0\in\mathbb{R}, \\ \mathrm{d}\widehat{J}(t) & = \int_{[0,\gamma)}\widehat{h}(\widehat{J}(t{-}),\widehat{X}(t{-}),z)\, \mathfrak{p}^*(\mathrm{d}t,\mathrm{d}z), \quad & \widehat{J}(0) & = i_0\in\mathcal{E}. \end{alignedat} \end{equation}

In Assumption 3.1, we essentially seek a weak solution for the discretized hybrid SDE (3.1). Namely, in the presence of discontinuous coefficients we need to assume the existence of some probability space (endowed with a filtration that may be larger than the one generated by $\{B(s), \mathfrak{p}(s, \cdot); s \le t\}$ ) such that $(\widehat{J}, \widehat{X})$ solves (3.1). This requirement is necessary because SDEs with discontinuous coefficients generally do not have strong solutions. For example, in [Reference Krylov and Röckner25], the existence of a strong solution for SDEs with discontinuous locally integrable drift is established only under a unit diffusion coefficient, while in [Reference Leobacher and Szölgyenyi27], the same conclusion is reached for piecewise Lipschitz-continuous drift coefficients and Lipschitz-continuous diffusion coefficients. Despite this limitation, we discuss two approximating schemes below for which Assumption 3.1 is guaranteed to hold.

Example 3.1. Suppose that, for all $i\in\mathcal{E}$ , ${\sigma}(i,\cdot)$ and $\widehat{\sigma}(i,\cdot)$ are equal to some $\sigma_i\ge 0$ . By [Reference Krylov and Röckner25], for each $i\in\mathcal{E}$ , $x\in\mathbb{R}$ , and $v\ge 0$ , there exists a strong solution $\widehat{X}^{[i,x,v]}$ to the SDE

(3.2) \begin{equation} \widehat{X}^{[i,x,v]}(t) = x + \int_0^t\widehat{\mu}\big(i,\widehat{X}^{[i,x,v]}(t)\big)\,\mathrm{d}r + \int_0^t\widehat{\sigma}\big(i,\widehat{X}^{[i,x,v]}(t)\big)\,\mathrm{d}B^{[v]}(r). \end{equation}

Under these conditions, using the same concatenating procedure as in Section 2 with $\mu$ and $\sigma$ replaced by $\widehat{\mu}$ and $\widehat{\sigma}$ , we can readily construct the strong solution to (3.1), ensuring that Assumption 3.1 holds.

Example 3.2. Suppose that, for all $i\in\mathcal{E}$ , ${\sigma}(i,\cdot)=\sigma( \! \cdot \! )$ , where $\sigma$ is a positive function bounded away from 0 and $\infty$ . Then the solution to the hybrid SDE (2.1) and (2.2) can be injectively mapped into a unit diffusion hybrid SDE, denoted by $(J,{X}^{\dagger})$ , where $X^{\dagger}(t)\,:\!=\,g(X(t))$ and

\begin{align*}g(x)=\int_{x_0}^x\frac{1}{\sigma(y)}\,\mathrm{d}y. \end{align*}

This mapping, which is a homeomorphism in space, corresponds to the well-known Lamperti transform (see [Reference Lamperti26] and [Reference Nguyen and Peralta30] for the hybrid framework generalization). Consequently, we can consider an approximation $(\widehat{J}^\dagger,\widehat{X}^\dagger)$ of $(J,{X}^{\dagger})$ that falls under the category of Example 3.1 and is thus guaranteed to have a strong solution. It is worth noting that while the original hybrid SDE is not the one being approximated, its first-passage properties can be readily studied through those of $(\widehat{J}^\dagger,\widehat{X}^\dagger)$ , given the homeomorphic nature of the Lamperti transform.

Due to the discretization characteristics of $\widehat{\mu}$ and $\widehat{\sigma}$ , the process $\widehat{X}$ falls within the class of multi-regime Markov-modulated Brownian motions studied in [Reference Horváth and Telek21] (see [Reference Mandjes, Mitra and Scheinhardt28] for an earlier reference in the case $\widehat{\sigma}\equiv 0$ ). Indeed, the piecewise-constant nature of $\widehat{\mu}$ , $\widehat{\sigma}$ , and $\widehat{\boldsymbol{\Lambda}}$ implies that, when restricted to a space interval $(\zeta_m,\zeta_{m+1})$ and to the time intervals for which $\widehat{J}$ is equal to i, $\widehat{X}$ behaves like a Brownian motion with drift $\widehat{\mu}(i,\zeta_m)$ and noise coefficient $\widehat{\sigma}(i,\zeta_m)$ . Moreover, since the functions $\widehat{\mu}$ , $\widehat{\sigma}$ , and $\widehat{\boldsymbol{\Lambda}}$ are space-grid approximations of $\mu$ , $\sigma$ , and $\boldsymbol{\Lambda}$ , we can expect that $\widehat{X}$ approximates the original solution X. A main purpose of this paper is to formalize this statement.

To prove pathwise convergence of $(\widehat{{J}},\widehat{{X}})$ to (J, X) as one of its parameters goes to $\infty$ , we proceed in two steps. First, we investigate the convergence of $\widehat{{X}}$ to X on the event that $\widehat{{J}}$ coincides with J in some time interval $[0,\xi)$ . Then, we let $\xi$ go to infinity and obtain the rate at which the probabilities of $\widehat{{J}}$ being different from J on $[0,\xi)$ converge to 0.

3.1. Measuring $|{X}-\widehat{{X}}|$

Here we focus on measuring $|{X}-\widehat{{X}}|$ over a compact time interval where J and $\widehat{J}$ are identical, where both (J, X) and $(\widehat{J},\widehat{X})$ are defined in the probability space $(\Omega,\mathbb{P},\mathcal{F}, \{\mathcal{G}_t\}_{t\ge 0})$ considered in Assumption 3.1. Generally, measuring the distance between two weak solutions is not feasible, as they may not be defined on the same probability space. However, since (J, X) is guaranteed to possess a strong solution in $(\Omega,\mathbb{P},\mathcal{F}, \{\mathcal{G}_t\}_{t\ge 0})$ , it is reasonable to measure $|{X}-\widehat{{X}}|$ in that space. We emphasize that, due to Assumption 3.1, the probability space $(\Omega,\mathbb{P},\mathcal{F}, \{\mathcal{G}_t\}_{t\ge 0})$ is not necessarily fixed. In particular, two different approximations (along with (J, X)) may exist in distinct probability spaces.

We begin our analysis by introducing a fundamental condition that will be beneficial for our subsequent developments. We offer a simpler proof compared to that in [Reference Yin and Zhu39, Proposition∼2.3], aiming to demonstrate that the proof is valid even with discontinuous coefficients.

Lemma 3.1. Let (J,X) and $(\widehat{{J}},\widehat{{X}})$ satisfy Assumptions 2.1 and 3.1. Then

$$ \mathbb{E}\big(\!\sup\nolimits_{s\le t}|X(s)|^2\big)<\infty, \qquad \mathbb{E}\big(\!\sup\nolimits_{s\le t}|\widehat{{X}}(s)|^2\big)<\infty. $$

Proof. Let

(3.3) \begin{equation} \tau_m=\inf\{s\ge 0\colon X(s)\vee \widehat{X}(s)> m\}. \end{equation}

Then $\int_0^{t\wedge\tau_m}|\sigma(J(s),X(s))|^2\,\mathrm{d}s<\infty$ , so that $V(t)\,:\!=\,\int_0^{t\wedge\tau_m}\sigma(J(s),X(s))\,\mathrm{d}B(s)$ is a local martingale. Employing Doob’s inequaliy and Itô isometry on V(t), along with Tonelli’s theorem, we get

\begin{align*} \mathbb{E}\bigg(\!\!\sup\nolimits_{r\le t\wedge\tau_m}\bigg|\int_0^{r\wedge\tau_m}\sigma(J(s),X(s))\,\mathrm{d}B(s)\bigg|^2\bigg) & \le 4\mathbb{E}\bigg(\int_0^{t\wedge\tau_m}|\sigma(J(s),X(s))|^2\,\mathrm{d}s\bigg) \\ & \le 4\int_0^{t}\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m}|\sigma(J(r),X(r))|^2\big)\,\mathrm{d}s. \end{align*}

Now, the Lipschitz condition of $\sigma(i,\cdot)$ implies a linear growth that takes the form

\begin{align*} \sup\nolimits_{r\le s\wedge\tau_m}|\sigma(J(r),X(r))| \le \sup\nolimits_{i\in\mathcal{S}}|\sigma(i,0)| + K\sup\nolimits_{r\le s\wedge\tau_m}|X(r)|, \end{align*}

which, by the inequality $(a+b)^2\le 2a^2+2b^2$ , means

(3.4) \begin{multline} \mathbb{E}\bigg(\!\!\sup\nolimits_{r\le t\wedge\tau_m}\bigg|\int_0^{r\wedge\tau_m}\sigma(J(s),X(s))\,\mathrm{d}B(s)\bigg|^2\bigg)\\ \le 8t\sup\nolimits_{i\in\mathcal{S}}|\sigma(i,0)| + 8K\int_0^{t}\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m}|X(r)|^2\big)\,\mathrm{d}s. \end{multline}

Employing the Cauchy–Schwartz inequality and analogous steps leads to the following inequalities:

(3.5) \begin{align} \mathbb{E}\bigg(\!\!\sup\nolimits_{r\le t\wedge\tau_m}\bigg|\int_0^{r\wedge\tau_m}\mu(J(s),X(s))\,\mathrm{d}s\bigg|^2\bigg) & \le t\int_0^t\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m}|\mu(J(r),X(r))|^2\big)\,\mathrm{d}s \nonumber \\ & \le 2t^2\sup\nolimits_{i\in\mathcal{S}}|\mu(i,0)| \nonumber \\ & \quad + 2tK\int_0^{t}\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m}|X(r)|^2\big)\,\mathrm{d}s. \end{align}

Using the inequality $(a+b+c)^2\le 3a^2 + 3b^2 + 3c^2$ in (2.1), along with (3.4) and (3.5), we have

\begin{align*} \mathbb{E}\big(\sup\nolimits_{r\le t\wedge\tau_m}|X(r)|^2\big) & \le 3|x_0|^2 + 3\mathbb{E}\bigg(\!\!\sup\nolimits_{r\le t\wedge\tau_m}\bigg|\int_0^{r\wedge\tau_m}\mu(J(s),X(s))\,\mathrm{d}s\bigg|^2\bigg) \\ & \quad + 3\mathbb{E}\bigg(\!\!\sup\nolimits_{r\le t\wedge\tau_m} \bigg|\int_0^{r\wedge\tau_m}\sigma(J(s),X(s))\,\mathrm{d}B(s)\bigg|^2\bigg) \\ & \le K_1 + K_2\int_0^t\mathbb{E}\big(\sup\nolimits_{r\le t\wedge\tau_m}|X(r)|^2\big)\,\mathrm{d}s \end{align*}

for some constants $K_1,K_2>0$ that are not dependent on m. Gronwall’s lemma then implies that $\mathbb{E}\big(\sup\nolimits_{r\le t\wedge\tau_m}|X(r)|^2\big) \le K_1\mathrm{e}^{K_2 t}$ ; taking $m\rightarrow\infty$ yields $\mathbb{E}\big(\sup\nolimits_{s\le t}|X(s)|^2\big)\le K_1\mathrm{e}^{K_2 t}<\infty$ . The steps to prove

\begin{align*} \mathbb{E}\big(\sup\nolimits_{s\le t}|X(s)|^2\big)\le K_1\mathrm{e}^{K_2 t} < \infty \end{align*}

can be repeated verbatim by replacing $\mu$ , $\sigma$ , J, and X with $\widehat{\mu}$ , $\widehat{\sigma}$ , $\widehat{J}$ , and $\widehat{X}$ , employing along the way the inequalities

\begin{align*} \sup\nolimits_{r\le s\wedge\tau_m}|\widehat{\sigma}(\widehat{J}(r), \widehat{X}(r))| & \le \sup\nolimits_{r\le s\wedge\tau_m}|{\sigma}(\widehat{J}(r),\widehat{X}(r))| \\ & \le \sup\nolimits_{i\in\mathcal{S}}|\sigma(i,0)| + K\sup\nolimits_{r\le s\wedge\tau_m}|\widehat{X}(r)|, \\ \sup\nolimits_{r\le s\wedge\tau_m}|\widehat{\mu}(\widehat{J}(r),\widehat{X}(r))| & \le \sup\nolimits_{r\le s\wedge\tau_m}|{\mu}(\widehat{J}(r),\widehat{X}(r))| \\ & \le \sup\nolimits_{i\in\mathcal{S}}|\mu(i,0)| + K\sup\nolimits_{r\le s\wedge\tau_m}|\widehat{X}(r)|. \end{align*}

Now, we quantify the distance between X and $\widehat{X}$ up to the decoupling time $\iota$ defined by $\iota\,:\!=\,\inf\{s\ge 0\colon J(s)\neq \widehat{J}(s)\}$ .

Theorem 3.1. For $t\ge 0$ ,

(3.6) \begin{multline} \mathbb{E}\big(\sup\nolimits_{s\le t\wedge\iota}|X(s)-\widehat{X}(s)|^2\big) \\ \le C(t)\big(\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2\big), \end{multline}

where $C(t)=16(t^2\vee 1)\mathrm{e}^{4K^2(t+4)^2}$ .

Proof. Let $\tau_m$ be as in (3.3), and for $t\ge 0$ define $Z_m(t) = \mathbb{E}\big(\sup\nolimits_{s\le t\wedge\tau_m\wedge\iota}|X(s)-\widehat{X}(s)|^2\big)$ . By Lemma 3.1, $Z_m(t)$ is finite for all $t\ge 0$ . Using the inequality $(a+b)^2\le 2a^2+2b^2$ , together with the fact that $\widehat{J}$ agrees with J on the time interval $[0,\iota)$ , we get

(3.7) \begin{align} Z_m(t) & \le 2\mathbb{E}\bigg(\!\!\sup\nolimits_{s\le t\wedge\tau_m\wedge\iota} \bigg|\int_0^s\mu(J(r),X(r)) - \widehat{\mu}(J(r),\widehat{X}(r))\,\mathrm{d}r\bigg|^2\bigg) \nonumber \\ & \quad + 2\mathbb{E}\bigg(\!\!\sup\nolimits_{s\le t\wedge\tau_m\wedge\iota}\bigg|\int_0^s\sigma(J(r),X(r)) - \widehat{\sigma}(J(r),\widehat{X}(r))\,\mathrm{d}B(r)\bigg|^2\bigg). \end{align}

Employing similar techniques to those in the proof of Lemma 3.1, we get

(3.8) \begin{align} & \mathbb{E}\bigg(\!\!\sup\nolimits_{s\le t\wedge\tau_m\wedge\iota}\bigg|\int_0^s\mu(J(r),X(r)) - \widehat{\mu}(J(r),\widehat{X}(r))\,\mathrm{d}r\bigg|^2\bigg) \nonumber \\ & \hspace{2cm} \le t\int_0^t\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m\wedge\iota} |\mu(J(r),X(r))-\widehat{\mu}(J(r),\widehat{X}(r))|^2\big)\,\mathrm{d}s, \\[-30pt] \nonumber \end{align}

(3.9) \begin{align} & \mathbb{E}\bigg(\!\!\sup\nolimits_{s\le t\wedge\tau_m\wedge\iota}\bigg|\int_0^s\sigma(J(r),X(r)) - \widehat{\sigma}(J(r),\widehat{X}(r))\,\mathrm{d}r\bigg|^2\bigg) \nonumber \\ & \hspace{2cm} \le 4\int_0^t\mathbb{E}\big(\sup\nolimits_{r\le s\wedge\tau_m\wedge\iota} |\sigma(J(r),X(r))-\widehat{\sigma}(J(r),\widehat{X}(r))|^2\big)\,\mathrm{d}s. \end{align}

Furthermore, the Lipschitz continuity of $\mu$ implies that

(3.10) \begin{align} & |\mu(J(r),X(r)) - \widehat{\mu}(J(r),\widehat{X}(r))|^2 \nonumber \\ & \qquad \le 2|\mu(J(r),X(r)) - \mu(J(r),\widehat{X}(r))|^2 + 2|\mu(J(r),\widehat{X}(r)) - \widehat{\mu}(J(r),\widehat{X}(r))|^2 \nonumber \\ & \qquad \le 2K^2|X(r) - \widehat{X}(r)|^2 + 2\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2; \end{align}

similarly,

(3.11) \begin{equation} |\sigma(J(r),X(r)) - \widehat{\sigma}(J(r),\widehat{X}(r))|^2 \le 2K^2|X(r) - \widehat{X}(r)|^2 + 2\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2. \end{equation}

By virtue of (3.7)–(3.11), we get

\begin{align*} Z_m(t) & \le 4t^2\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + 16t\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2 \\ & \quad + 4K^2(t+4)\int_0^t Z_m(r)\,\mathrm{d}r \\ & \le 16(t^2\vee 1)\big(\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2\big) \\ & \quad + 4K^2(t+4)\int_0^t Z_m(r)\,\mathrm{d}r. \end{align*}

Finally, Gronwall’s lemma implies that

\begin{align*} Z_m(t) & \le 16(t^2\vee 1)\mathrm{e}^{4K^2(t+4)t}\big(\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}} |\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2\big) \\ & \le C(t)\big(\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2\big), \end{align*}

so that (3.6) holds by taking $m\rightarrow\infty$ .

As usual, mean-square distance provides bounds on pointwise distance, which in our case takes the following form.

Corollary 3.1. Suppose that there exists some $\alpha>0$ such that

\begin{equation*} \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2 \le \alpha^2. \end{equation*}

Then, for $\delta>0$ and for all sufficiently large $n\ge 1$ ,

\begin{align*} \mathbb{P}\big(\sup\nolimits_{s\le t\wedge\iota}|X(s)-\widehat{X}(s)| \ge \alpha\sqrt{n^{\delta}C(t)}\big) \le n^{-\delta}. \end{align*}

Proof. Markov’s inequality and Theorem 3.1 imply that, for any $\beta_{n,t,\alpha}>0$ ,

\begin{align*} & \mathbb{P}\big(\!\sup\nolimits_{s\le t\wedge\iota}|X(s)-\widehat{X}(s)| \ge \beta_{n,t,\alpha}\big) \\ & \qquad \le \frac{\mathbb{E}\big(\!\sup\nolimits_{s\le t\wedge\iota}|X(s)-\widehat{X}(s)|^2\big)}{\beta_{n,t,\alpha}^2} \\ & \qquad \le \frac{C(t)\big(\!\sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\mu(i,x)-\widehat{\mu}(i,x)|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}|\sigma(i,x)-\widehat{\sigma}(i,x)|^2\big)}{\beta_{n,t,\alpha}^2} \le \frac{C(t)\alpha^2}{\beta_{n,t,\alpha}^2}. \end{align*}

The result follows by choosing $\beta_{n,t,\alpha}=\alpha \sqrt{n^{\delta}C(t)}$ .

Corollary 3.1 implies that we can choose appropriate values of t and $\alpha$ that are dependent on n, say $t_n$ and $\alpha_n$ , such that, for $\delta>0$ , $t_n\rightarrow\infty$ and $\alpha_n\sqrt{n^\delta C(t_n)}\rightarrow 0$ as $n\rightarrow\infty$ , and then conclude that $|X-\widehat{{X}}|$ converges in probability to 0 uniformly over increasing compact intervals (up to time $\iota$ ). Here we choose $t_n=\sqrt{\log\log n}$ and $\alpha_n= n^{-\delta_*-\delta/2}$ for some $\delta_* > 0$ . In such a case, for all $n\ge 1$ with $\sqrt{\log n\log n}> 4$ and $\log n > \log\log n$ ,

\begin{equation*} \alpha_n\sqrt{n^\delta C(t_n)} = 16(t_n^2\vee 1)\mathrm{e}^{4K^2(t_n+4)^2}n^{-\delta_*} \le 16(\!\log n)\mathrm{e}^{8K^2\log\log n}n^{-\delta_*} = 16(\!\log n)^{\beta_*}n^{-\delta_*} , \end{equation*}

where $\beta_* = 1+8K^2$ . In light of this, we assume the following.

Assumption 3.2. Consider the functions $\widehat{\mu}$ and $\widehat{\sigma}$ associated with the process $(\widehat{J},\widehat{X})$ , all of which now exhibit an explicit dependence on $n\ge 1$ via their superscript. Then, there exist $\delta_*,\delta>0$ such that

\begin{align*} \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}\big|\mu(i,x)-\widehat{\mu}^{(n)}(i,x)\big|^2 + \sup\nolimits_{i\in\mathcal{E},\,x\in\mathbb{R}}\big|\sigma(i,x)-\widehat{\sigma}^{(n)}(i,x)\big|^2 \le n^{-\delta_*-\delta/2}. \end{align*}

To summarize, under Assumptions 2.1, 3.1, and 3.2, for all sufficiently large $n\ge 1$ ,

(3.12) \begin{equation} \mathbb{P}\big(\!\sup\nolimits_{s\le t_n\wedge\iota}\big|X(s)-\widehat{X}^{(n)}(s)\big| \ge 16(\!\log n)^{\beta_*}n^{-\delta_*}\big) \le n^{-\delta}, \quad\mbox{where}\ \beta_* = 1+8K^2.\end{equation}

From (3.12), we can determine various types of convergence in different situations. If the existence of a solution under Assumption 3.1 can be established in a strong sense with a fixed probability space for all approximations, then (3.12) demonstrates the convergence in probability of $\widehat{X}^{(n)}$ to X up to time $\iota$ . Furthermore, if $\delta>1$ , then, using Borel–Cantelli arguments, the convergence holds in a strong sense. In the general case where the solution considered in Assumption 3 is weak, (3.12) establishes the weak convergence of $\widehat{X}^{(n)}$ to X up to time $\iota$ . Refer to [Reference Csörgö and Horváth12, Reference Csörgo and Révész13] for instances where measuring pointwise distances in distinct probability spaces leads to weakly convergent approximations; an additional example using the analogous formulation of triangular arrays can be found in [Reference De Acosta15].

Now that we have assessed the convergence of $\widehat{X}^{(n)}$ to X within the interval $[0,\iota)$ , in the following section we establish some probabilistic properties of the decoupling time $\iota$ .

3.2. Measuring the decoupling event

In order to guarantee that the decoupling time $\iota$ occurs at a large epoch, we need to ensure that the jumps of $\widehat{J}$ remain ‘similar’ to those of J. The way to guarantee this is through the sets $\{\widehat{\Delta}_{ij}(x)\}_{i,j\mathcal{E}}$ associated to the discretized hybrid SDE (3.1). The idea is to make each set $\widehat{\Delta}_{ij}(x)$ as close as possible to ${\Delta}_{ij}(x)$ , with the difference that the $\mbox{Leb}(\widehat{\Delta}_{ij}(x))= \gamma\delta_{ij} + \widehat{\Lambda}_{ij}(x)$ while $\mbox{Leb}({\Delta}_{ij}(x))= \gamma\delta_{ij} + {\Lambda}_{ij}(x)$ .

Recall that, in Section 2, for a fixed $x\in\mathbb{R}$ and $i\in\mathcal{E}$ , we let $\Delta_{ij}(x)=[\ell_{ij}(x), r_{ij}(x))$ , where $\ell_{i1}(x)= 0$ , $r_{ij}(x)= \ell_{ij}(x)+ (\gamma\delta_{ij} + \Lambda_{ij}(x))$ , $\ell_{i,{j+1}}(x)=r_{ij}(x)$ , and $r_{ip}(x)=\gamma$ . Next, we introduce $\Gamma_{ij}(x)=[\ell_{ij}(x), r_{ij}(x)\wedge \widehat{r}_{ij}(x))$ , where $\widehat{r}_{ij}(x)= \ell_{ij}(x)+ (\gamma\delta_{ij} + \widehat{\Lambda}_{ij}(x))$ . Note that

\begin{align*}\mbox{Leb}\big(\Gamma_{ij}(x)\big)=(\gamma\delta_{ij} + {\Lambda}_{ij}(x))\wedge (\gamma\delta_{ij} + \widehat{\Lambda}_{ij}(x))\le { \mbox{Leb}\big(\widehat{\Delta}_{ij}(x)\big)}; \end{align*}

in simple words, since $\widehat{\Lambda}_{ij}(x)$ is thought of as being close to ${\Lambda}_{ij}(x)$ , the length of $\Gamma_{ij}(x)$ approximates the required Lebesgue measure of $\widehat{\Delta}_{ij}(x)$ from below. For the last step, for a fixed $i\in\mathcal{E}$ and for all $x\in\mathbb{R}$ , we let $\{\Gamma^*_{ij}(x)\}_j$ be a partition of the set $[0,\gamma)\setminus(\cup_j \Gamma_{ij}(x))$ that keeps the mapping $(x,z)\mapsto 1_{\{z\in\Gamma_{ij}(x)\}}$ measurable, with $\mbox{Leb}\big(\Gamma^*_{ij}(x)\big)={ \mbox{Leb}\big(\widehat{\Delta}_{ij}(x)\big)}-\mbox{Leb}\big(\Gamma_{ij}(x)\big)$ . (Employing the measurability properties of $\Lambda_{ij}( \! \cdot \! )$ and $\widehat{\Lambda}_{ij}( \! \cdot \! )$ , constructing a partition $\{\Gamma^*_{ij}(x)\}_j$ with the desired characteristics is always achievable. Presenting an explicit construction is tedious and, more importantly, it is not relevant to subsequent developments; for this reason, we refrain from detailing it.) We then define $\widehat{\Delta}_{ij}(x)\,:\!=\,\Gamma_{ij}(x)\cup\Gamma^*_{ij}(x)$ , from which we can readily verify that $\widehat{\Delta}_{ij}(x)$ indeed has a Lebesgue measure equal to $\gamma\delta_{ij} + \widehat{\Lambda}_{ij}(x)$ , $\{\widehat{\Delta}_{ij}(x)\}_j$ is a partition of $[0,\gamma)$ , and the mapping $(x,z)\mapsto 1_{\{z\in\widehat{\Delta}_{ij}(x)\}}$ is measurable. This implies the characteristics needed for $\widehat{\Delta}_{ij}(x)$ as discussed in Section 3. We now relate the sets $\Gamma_{ij}(x)$ to the decoupling time $\iota$ .

Lemma 3.2. Define $\kappa = \inf\big\{m\ge 1\colon U_m\notin\bigcup_{j=1}^p\Gamma_{J(\theta_{m-1}),j}(X(\theta_m)) \cap \Gamma_{\widehat{J}(\theta_{m-1}),j}(\widehat{X}(\theta_m))\big\}$ . Then $\theta_\kappa\le \iota$ .

Proof. It is enough to prove that $\{\kappa > m\}\subseteq \{\iota > \theta_m\}$ for all $m\ge 1$ . Assuming that this holds for some fixed $m\ge 1$ , We proceed by induction. Then,

\begin{align*} \{\kappa > m+1\} & = \{\kappa > m\} \cap \bigg\{U_{m+1} \in \bigcup_{j=1}^p\Gamma_{J(\theta_{m}),j}(X(\theta_{m+1})) \cap \Gamma_{\widehat{J}(\theta_{m}),j}(\widehat{X}(\theta_{m+1}))\bigg\} \\ & \subseteq \{\iota > \theta_m\} \cap \bigg\{U_{m+1} \in \bigcup_{j=1}^p \Gamma_{J(\theta_{m}),j}(X(\theta_{m+1})) \cap \Gamma_{\widehat{J}(\theta_{m}),j}(\widehat{X}(\theta_{m+1}))\bigg\} \nonumber\\[-2pt] & = \bigcup_{j=1}^p\{\iota > \theta_m\} \cap \big\{U_{m+1}\in\Gamma_{J(\theta_{m}),j}(X(\theta_{m+1})) \cap \Gamma_{\widehat{J}(\theta_{m}),j}(\widehat{X}(\theta_{m+1}))\big\} \\ & \subseteq \bigcup_{j=1}^p\{\iota > \theta_m\} \cap \big\{U_{m+1}\in\Delta_{J(\theta_{m}),j}(X(\theta_{m+1})) \cap \widehat{\Delta}_{\widehat{J}(\theta_{m}),j}(\widehat{X}(\theta_{m+1}))\big\} \\ & = \bigcup_{j=1}^p\{\iota > \theta_m\} \cap \{J(\theta_{m+1})=\widehat{J}(\theta_{m+1})=j\} = \{\iota > \theta_{m+1}\}, \end{align*}

where in the fourth line we employed that $\Gamma_{ij}(x)$ is contained in $\Delta_{ij}(x)$ and $\widehat{\Delta}_{ij}(x)$ for all $i,j\in\mathcal{E}$ , $x\in\mathbb{R}$ . This completes the induction and the proof.

As a preliminary step, let us provide bounds for the probability of a possible decoupling at a given epoch $\theta_{m}$ for fixed values of $X(\theta_{m})$ , $\widehat{X}^{(n)}(\theta_{m})$ , and $J(\theta_{m-1})=\widehat{J}^{(n)}(\theta_{m-1})$ .

Lemma 3.3. For a square matrix $\boldsymbol{A}=\{a_{ij}\}$ , let $\Vert\cdot\Vert$ denote the norm defined by $\Vert \boldsymbol{A}\Vert=\sup\nolimits_{i,j}\{|a_{ij}|\}$ . Then, for $i\in\mathcal{E}$ and $x,y\in\mathbb{R}$ ,

\begin{align*} \mathbb{P}\Bigg(U_m \notin \bigcup_{j=1}^p \Gamma_{ij}(x) \cap \Gamma_{ij}(y)\Bigg) & \le \frac{p}{\gamma}\big(p\Vert\boldsymbol{\Lambda}(x) - \boldsymbol{\Lambda}(y)\Vert + 2p\sup\nolimits_{z\in\mathbb{R}}\big\Vert\widehat{\boldsymbol{\Lambda}}^{(n)}(z) - \boldsymbol{\Lambda}(z)\big\Vert\big) \\ & \le \frac{2p^2}{\gamma}\big(\Vert\boldsymbol{\Lambda}(x) - \boldsymbol{\Lambda}(y)\Vert + \sup\nolimits_{z\in\mathbb{R}}\big\Vert\widehat{\boldsymbol{\Lambda}}^{(n)}(z) - \boldsymbol{\Lambda}(z)\big\Vert\big). \end{align*}

Proof. Note that $\Gamma_{ij}(x) \cap \Gamma_{ij}(y)=[\ell_{ij}(x,y), r_{ij}(x,y))$ , where $\ell_{ij}(x,y)=\ell_{ij}(x)\vee \ell_{ij}(y)$ and $r_{ij}(x,y)= (r_{ij}(x)\wedge\widehat{r}_{ij}(x))\wedge (r_{ij}(y)\wedge\widehat{r}_{ij}(y))$ , with the understanding that [a, b) is an empty interval if $a\ge b$ . Moreover, since $\ell_{i1}(x,y)=0$ and, for all $1\le j< p$ we have $\ell_{ij}(x,y)\le \ell_{i,{j+1}}(x,y)$ and $r_{ij}(x,y)\le r_{i,{j+1}}(x,y)$ , the set where $U_k$ may fall, leading to a potential decoupling, can be written as

\begin{equation*} [0,\gamma) \setminus \Bigg(\bigcup_{j=1}^p\Gamma_{ij}(x) \cap \Gamma_{ij}(y)\Bigg) = [0,\gamma) \setminus \Bigg(\bigcup_{j=1}^p[\ell_{ij}(x,y),r_{ij}(x,y))\Bigg) = \bigcup_{j=1}^{p}[r_{ij}(x,y), \ell_{i,j+1}(x,y)), \end{equation*}

where $\ell_{i,p+1}(x,y)\,:\!=\,\gamma$ . Thus, the probability of decoupling is bounded as follows:

\begin{align*} & \mathbb{P}\Bigg(U_k \notin \bigcup_{j=1}^p\Gamma_{ij}(x) \cap \Gamma_{ij}(y)\Bigg) \\ & \qquad = \frac{1}{\gamma}\operatorname{Leb}\Bigg([0,\gamma) \setminus \bigcup_{j=1}^p\Gamma_{ij}(x) \cap \Gamma_{ij}(y)\Bigg) \\ & \qquad = \frac{1}{\gamma}\operatorname{Leb}\Bigg(\bigcup_{j=1}^{p}(r_{ij}(x,y),\ell_{i,{j+1}}(x,y)]\Bigg) \\ & \qquad \le \frac{1}{\gamma}\sum_{j=1}^{p}\operatorname{Leb}((r_{ij}(x,y),\ell_{i,{j+1}}(x,y)]) \nonumber\\[-2pt] & \qquad \le \frac{1}{\gamma}\sum_{j=1}^{p}|\ell_{i,{j+1}}(x,y)-r_{ij}(x,y)| \\ & \qquad = \frac{1}{\gamma}\sum_{j=1}^{p}|(\ell_{i,{j+1}}(x)\vee\ell_{i,{j+1}}(y)) - ((r_{ij}(x)\wedge\widehat{r}_{ij}(x))\wedge(r_{ij}(y)\wedge\widehat{r}_{ij}(y)))| \\ & \qquad = \frac{1}{\gamma}\sum_{j=1}^{p}|(r_{ij}(x) \vee r_{ij}(y)) - ((r_{ij}(x)\wedge\widehat{r}_{ij}(x))\wedge(r_{ij}(y)\wedge\widehat{r}_{ij}(y)))| \\ & \qquad \le \frac{1}{\gamma}\sum_{j=1}^{p}(|r_{ij}(x)-r_{ij}(y)| + |r_{ij}(y)-\widehat{r}_{ij}(y)| + |r_{ij}(x)-\widehat{r}_{ij}(x)|). \end{align*}

Note that from the recursive definition of $r_{ij}(z) = \sum_{k=1}^j(\gamma\delta_{ik} + \Lambda_{ik}(z))$ , it holds that, for all $x,y,z\in\mathbb{R}$ ,

\begin{align*} |r_{ij}(x)-r_{ij}(y)| \le \sum_{k=1}^j|\Lambda_{ik}(x)-\Lambda_{ik}(y)| \le j\Vert\boldsymbol{\Lambda}(x)-\boldsymbol{\Lambda}(y)\Vert \le p\Vert\boldsymbol{\Lambda}(x)-\boldsymbol{\Lambda}(y)\Vert, \end{align*}

and since $r_{ij}(z) - \widehat{r}_{ij}(z) = (\ell_{ij}(z) + \gamma\delta_{ij} + \Lambda_{ij}(z)) - (\ell_{ij}(z) + \gamma\delta_{ij} + \widehat{\Lambda}_{ij}(z)) = \Lambda_{ij}(z)-\widehat{\Lambda}_{ij}(z)$ ,

\begin{align*} |r_{ij}(z)-\widehat{r}_{ij}(z)| = |\Lambda_{ij}(z)-\widehat{\Lambda}_{ij}(z)| \le \Vert\boldsymbol{\Lambda}(z)-\widehat{\boldsymbol{\Lambda}}(z)\Vert \le p\Vert\boldsymbol{\Lambda}(z)-\widehat{\boldsymbol{\Lambda}}(z)\Vert. \end{align*}

Substituting these bounds into the sum yields the result.

Next, we study the arrivals $\{\theta_k\}_{k\ge 1}$ that occur in the interval $[0,t_n]$ . To do so, let us introduce $\{N(t)\}_{t\ge 0}$ , the Poisson counting process of intensity $\gamma$ associated to $\{\theta_k\}_{k\ge 1}$ . Each arrival represents an opportunity for J and $\widehat{J}$ to get decoupled, so that providing a tight tail estimate for $N(t_n)$ is relevant to our analysis.

Lemma 3.4. $\mathbb{P}(N(t_n) > \lfloor\log\log n\rfloor)=o(1)$ as $n\rightarrow\infty$ .

Proof. Using [Reference Glynn19, Proposition 1], for sufficiently large n,

\begin{equation*} \mathbb{P}(N(t_n) > \lfloor\log\log n\rfloor) \le \bigg(1-\frac{\gamma t_n}{\lfloor\log\log n\rfloor + 1}\bigg)^{-1} \bigg(\frac{(\gamma t_n)^{\lfloor\log\log n\rfloor}}{\lfloor\log\log n\rfloor!}\mathrm{e}^{-\gamma t_n}\bigg); \end{equation*}

the first element in the product on the right-hand side clearly converges to 1, while the second element is bounded by

\begin{align*} & \frac{(\gamma t_n)^{\lfloor\log\log n\rfloor}}{\lfloor\log\log n\rfloor!} \\ & \quad \le \big((\gamma t_n)^{\lfloor\log\log n\rfloor}\big) \bigg(\sqrt{2\pi\lfloor\log\log n\rfloor} \bigg(\frac{\lfloor\log\log n\rfloor}{\mathrm{e}}\bigg)^{\lfloor\log\log n\rfloor} \exp\bigg\{\frac{1}{12\lfloor\log\log n\rfloor + 1}\bigg\}\bigg)^{-1} \\ & \quad = O\bigg(\bigg(\frac{\mathrm{e}\gamma t_n}{\lfloor\log\log n\rfloor}\bigg)^{\lfloor\log\log n\rfloor}\bigg) = o(1). \end{align*}

From Lemma 3.3, it is clear that we need some regularity assumptions on $\boldsymbol{\Lambda}$ and $\widehat{\boldsymbol{\Lambda}}$ in order to bound the probability of a decoupling occuring. Similarly to Assumption 3.2, let us now make explicit the dependence of $\widehat{\boldsymbol{\Lambda}}$ , $\widehat{J}$ , $\widehat{X}$ , $\iota$ , and $\kappa$ on n via a superscript.

Assumption 3.3. The intensity matrix $\boldsymbol{\Lambda}( \! \cdot \! )$ is $\log$ -Hölder continuous, in the sense that there exists a constant $G>0$ such that, for any $x,y\in\mathbb{R}$ ,

\begin{align*} \Vert\boldsymbol{\Lambda}(x) - \boldsymbol{\Lambda}(y)\Vert \le \frac{G}{-\log|x-y|}. \end{align*}

Furthermore, $\widehat{\boldsymbol{\Lambda}}^{(n)}$ is right-continuous with left limits and converges uniformly to $\boldsymbol{\Lambda}$ at a $\log$ rate; more specifically,

\begin{align*} \sup\nolimits_{z\in\mathbb{R}}\big\Vert\widehat{\boldsymbol{\Lambda}}^{(n)}(z) - \boldsymbol{\Lambda}(z)\big\Vert \le \frac{G}{\log n}. \end{align*}

Remark 3.2. Under Assumption 3.3, $\boldsymbol{\Lambda}$ does not admit any discontinuities. However, recall that $\log$ -Hölder continuity is considerably less restrictive than Hölder or Lipschitz continuity. This means that we can mimic discontinuous behaviour for $\boldsymbol{\Lambda}$ at a point $z_0$ by considering instead some steep function that behaves like $(\log(z-z_0))^{-1}$ for sufficiently close $z> z_0$ .

Lemma 3.5. Fix some small $\varepsilon_1>0$ . Then, as $n\rightarrow\infty$ ,

\begin{equation*} \mathbb{P}\big(\kappa^{(n)} \le N(t_n),\,N(t_n) \le \lfloor\log\log n\rfloor,\, \sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| \le n^{-\delta_*+\varepsilon_1}\big) = o(1). \end{equation*}

Proof. For brevity, in this proof let us write $\widehat{\boldsymbol{\Lambda}}$ , $\widehat{J}$ , $\widehat{X}$ , $\iota$ , and $\kappa$ instead of $\widehat{\boldsymbol{\Lambda}}^{(n)}$ , $\widehat{J}^{(n)}$ , $\widehat{X}^{(n)}$ , $\iota^{(n)}$ , and $\kappa^{(n)}$ , respectively. Note that

(3.13) \begin{align} & \big\{\kappa \le N(t_n),\,N(t_n) \le \lfloor\log\log n\rfloor,\, \sup\nolimits_{s\le t_n\wedge\iota}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}\big\} \nonumber \\ & \qquad = \bigcup_{\ell=1}^{\lfloor\log\log n\rfloor}\bigcup_{k=1}^\ell \big\{\kappa=k,\,N(t_n) = \ell,\,\sup\nolimits_{s\le t_n\wedge\iota}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}\big\} \nonumber \\ & \qquad \subseteq \bigcup_{\ell=1}^{\lfloor\log\log n\rfloor}\bigcup_{k=1}^\ell \big\{\kappa=k,\,N(t_n) = \ell,\,\sup\nolimits_{s\le\theta_k}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}\big\}. \end{align}

Letting $F = \big\{\kappa>k-1,\,{N(t_n) = \ell,\,\sup\nolimits_{s\le\theta_k}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}}\big\}$ and

\begin{align*} \mathcal{H} = \sigma\big(N(t_n), \{J(s)\}_{s\le\theta_{k-1}}, \{\widehat{J}(s)\}_{s\le\theta_{k-1}}, \{X(s)\}_{s\le\theta_k}, \{\widehat{X}(s)\}_{s\le\theta_k}\big), \end{align*}

we get

\begin{align*} &\mathbb{P}\big(\kappa=k,\,N(t_n) = \ell,\,\sup\nolimits_{s\le\theta_k}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1} \mid \mathcal{H}\big) \\ & \qquad = \mathbb{P}(\iota=\theta_k \mid \mathcal{H}) \times 1_{F} = \mathbb{P}\Bigg(U_k\notin\bigcup_{j=1}^p\Gamma_{J(\theta_{k-1}),j}(X(\theta_k)) \cap \Gamma_{\widehat{J}(\theta_{k-1}),j}(\widehat{X}(\theta_k))\Bigg) \times 1_{F} \\ & \qquad \le \frac{2p^2}{\gamma}\big(\Vert\boldsymbol{\Lambda}(X(\theta_k)) - \boldsymbol{\Lambda}(\widehat{X}(\theta_k))\Vert + \sup\nolimits_{z\in\mathbb{R}}\Vert\widehat{\boldsymbol{\Lambda}}(z) - \boldsymbol{\Lambda}(z)\Vert\big) \times 1_{F} \\ & \qquad \le \frac{2p^2}{\gamma}\bigg(\frac{G}{-\log|X(\theta_k)-\widehat{X}(\theta_k)|} + \frac{G}{\log n}\bigg) \times 1_{F} \le \frac{2p^2G}{\gamma}\bigg[\frac{1}{\delta_*-\varepsilon_1} + 1\bigg](\log n)^{-1}. \end{align*}

From (3.13), we get

\begin{align*} & \mathbb{P}\big(\kappa \le N(t_n),\,N(t_n)\le\lfloor\log\log n\rfloor,\, \sup\nolimits_{s\le\theta_k\wedge\iota}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}\big) \\ & \qquad \le \sum_{\ell=1}^{\lfloor\log\log n\rfloor}\sum_{k=1}^\ell \mathbb{P}\big(\kappa=k,\,N(t_n) = \ell,\,\sup\nolimits_{s\le\theta_k}|X(s)-\widehat{X}(s)| \le n^{-\delta_*+\varepsilon_1}\big) \\ & \qquad \le \sum_{\ell=1}^{\lfloor\log\log n\rfloor}\sum_{k=1}^\ell \frac{G}{\gamma}\bigg[\frac{1}{\delta_*-\varepsilon_1} + 1\bigg](\log n)^{-1} \le \frac{G}{\gamma}\bigg[\frac{1}{\delta_*-\varepsilon_1} + 1\bigg]\frac{\lfloor\log\log n\rfloor^2}{\log n} = o(1), \end{align*}

which concludes the proof.

Now we are ready to prove the main result of the paper.

Theorem 3.2. Let Assumptions 2.1, 2.2, 3.1, 3.2, and 3.3 hold. Then, for any fixed $\varepsilon_1>0$ ,

(3.14) \begin{equation} \mathbb{P}\big(\big\{\sup\nolimits_{s\le t_n}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\} \cup \{\iota^{(n)} \le t_n\}\big) = o(1) \quad\mbox{as}\ n\rightarrow\infty. \end{equation}

Proof. By simple set manipulations,

\begin{align*} & \big\{\!\sup\nolimits_{s\le t_n}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\} \cup \{\iota^{(n)}\le t_n\} \\ & \qquad = \big\{\iota^{(n)}> t_n,\,\sup\nolimits_{s\le t_n}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\} \cup \{\iota^{(n)} \le t_n\} \\ & \qquad = \big\{\iota^{(n)}> t_n,\,\sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\} \cup \{\iota^{(n)} \le t_n\} \\ & \qquad \subseteq \big\{\sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\} \cup \{\iota^{(n)} \le t_n\}. \end{align*}

Moreover,

\begin{align*} \{\iota^{(n)}\le t_n\} & \subseteq \{\kappa^{(n)}\le N(t_n)\} \\ & \subseteq \big\{\kappa^{(n)} \le N(t_n),\,N(t_n)\le\lfloor\log\log n\rfloor,\, \sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| \le n^{-\delta_*+\varepsilon_1}\big\} \\ & \qquad \cup \{N(t_n) > \lfloor\log\log n\rfloor\} \cup \big\{\sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big\}. \end{align*}

Thus, the left-hand side of (3.14) is bounded by

\begin{multline*} \mathbb{P}\big(\kappa^{(n)} \le N(t_n),\,N(t_n) \le \lfloor\log\log n\rfloor,\, \sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| \le n^{-\delta_*+\varepsilon_1}\big) \\ + \mathbb{P}(N(t_n) > \lfloor\log\log n\rfloor) + \mathbb{P}\big(\!\sup\nolimits_{s\le t_n\wedge\iota^{(n)}}\big|X(s)-\widehat{X}^{(n)}(s)\big| > n^{-\delta_*+\varepsilon_1}\big); \end{multline*}

since each one of the aforementioned summands is shown to be o(1) as $n\rightarrow \infty$ in Lemma 3.5, Lemma 3.4, and (3.12), respectively, the result follows.

Theorem 3.2 essentially indicates that under Assumptions 2.1, 2.2, 3.1, 3.2, and 3.3, there exists a sequence of multi-regime Markov-modulated Brownian motions $\widehat{X}^{(n)}$ that weakly converge to the solution X of a hybrid stochastic differential equation, as well as their respective underlying processes $\widehat{J}^{(n)}$ and J. This convergence occurs uniformly over increasing compact intervals. If, additionally, the solution for each approximation in Assumption 3.1 is strong, the probability space $(\Omega,\mathbb{P},\mathcal{F}, \{\mathcal{G}_t\}_{t\ge 0})$ may remain fixed for all $n\ge 1$ , allowing the result to be upgraded to convergence in probability. In any case, this implies that the first-passage probabilities of X into the set $(\!-\infty,a)\cup(a,+\infty)$ can be approximated by the corresponding first-passage probabilities of $\widehat{X}^{(n)}$ . In the following section we leverage this to offer efficient approximations for certain first-passage probabilities for hybrid stochastic differential equations.

4. First-passage probabilities and expected occupation times for solutions of hybrid SDEs

Define the first-passage times $\tau_0^- = \inf\{t>0\colon X(t) \le 0\}$ , $\tau_a^+ = \inf\{t>0\colon X(t) \ge a\}$ , where $a> 0$ . For $q\ge 0$ , we are interested in providing approximations for the Laplace transform of the first-passage times,

\begin{align*} m^-_{ij}(u,q,a) & = \mathbb{E}\big(\mathrm{e}^{-q(\tau_0^-\wedge\tau_a^+)} 1_{\{J(\tau_0^-\wedge\tau_a^+)=j,X(\tau_0^-\wedge\tau_a^+)=0\}} \mid J(0)=i, X(0)=u\big), \\ m^+_{ij}(u,q,a) & = \mathbb{E}\big(\mathrm{e}^{-q(\tau_0^-\wedge\tau_a^+)} 1_{\{J(\tau_0^-\wedge\tau_a^+)=j,X(\tau_0^-\wedge\tau_a^+)=a\}} \mid J(0)=i, X(0)=u\big),\end{align*}

for $0< u < a\le \infty$ and $i,j\in\mathcal{E}$ . In essence, the functions $m^+_{ij}$ and $m^-_{ij}$ characterize the distributional law of the exit times of X from the band (0, a). Employing the law of total probability, it is straightforward to reinterpret $m^-_{ij}$ and $m^-_{ij}$ as

\begin{align*} m^-_{ij}(u,q,a) & = \mathbb{P}\big(\tau_0^-\wedge\tau_a^+< e_q, J(\tau_0^-\wedge\tau_a^+)=j, X(\tau_0^-\wedge\tau_a^+)=0 \mid J(0)=i, X(0)=u\big), \\ m^+_{ij}(u,q,a) & = \mathbb{P}\big(\tau_0^-\wedge\tau_a^+< e_q, J(\tau_0^-\wedge\tau_a^+)=j, X(\tau_0^-\wedge\tau_a^+) = a \mid J(0)=i, X(0)=u\big),\end{align*}

where $e_q$ is an $\mbox{Exp}(q)$ random variable independent of everything else; from now on, we adopt this reinterpretation.

In the existing literature, explicit solutions to $m^+_{ij}$ and $m^-_{ij}$ only exist in very simple scenarios: either $q=0$ and $\mathcal{E}=\{1\}$ (see, e.g., [Reference Itô and McKean23, Chapter 4]), or $\mu_i$ and $\sigma_i$ are constant for all $i\in\mathcal{E}$ (see, e.g., [Reference Ivanovs24]). Moreover, we are also interested in the present value of the expected ocupation times in state $j\in\mathcal{E}$ while below level $b\in (0,a)$ , defined as

\begin{align*}O_{ij}(u,q,a,b) = \mathbb{E}\bigg(\int_{0}^{\tau_0^-\wedge\tau_a^+}\mathrm{e}^{-qs} 1_{\{J(s)=j,\,0 < X(s)\le b\}}\,\mathrm{d}s \mid J(0)=i, X(0)=u\bigg). \end{align*}

Similarly to $m^-_{ij}$ and $m^+_{ij}$ , here we reinterpret $O_{ij}$ as

\begin{align*} O_{ij}(u,q,a,b) = \mathbb{E}\bigg(\int_{0}^{\tau_0^-\wedge\tau_a^+\wedge e_q}1_{\{J(s)=j,\,0 < X(s)\le b\}}\,\mathrm{d}s \mid J(0)=i, X(0)=u\bigg). \end{align*}

In this section we propose to exploit the approximations laid out in Theorem 3.2, along with the existing theory for multi-regime Markov-modulated Brownian motions, to provide an efficient approximation scheme based on computing

\begin{align*} \widehat{m}^-_{ij}(u,q,a) & = \mathbb{P}\big(\widehat{\tau}^-_0\wedge\widehat{\tau}^+_a < e_q, \widehat{J}(\widehat{\tau}^-_0\wedge\widehat{\tau}_a^+)=j, \widehat{X}(\widehat{\tau}^-_0\wedge\widehat{\tau}_a^+)=0 \mid \widehat{J}(0)=i, \widehat{X}(0)=u\big), \\ \widehat{m}^+_{ij}(u,q,a) & = \mathbb{P}\big(\widehat{\tau}^-_0\wedge\widehat{\tau}^+_a < e_q, \widehat{J}(\widehat{\tau}^-_0\wedge\widehat{\tau}_a^+)=j, \widehat{X}(\widehat{\tau}^-_0\wedge\widehat{\tau}_a^+)=a \mid \widehat{J}(0)=i, \widehat{X}(0)=u\big), \\ \widehat{O}_{ij}(u,q,a,b) & = \mathbb{E}\bigg(\int_{0}^{\widehat{\tau}_0^-\wedge\widehat{\tau}_a^+\wedge e_q} 1_{\{\widehat{J}(s)=j,\,0 < \widehat{X}(s)\le b\}}\,\mathrm{d}s \mid \widehat{J}(0)=i, \widehat{X}(0)=u\bigg),\end{align*}

where $\widehat{\tau}_0^- = \inf\{t>0\colon\widehat{X}(t) \le 0\}$ and $\widehat{\tau}_a^+ = \inf\{t>0\colon\widehat{X}(t) \ge a\}$ .

In the case that $\widehat{\tau}_0^-$ converges in probability to $\tau_0^-$ and that $\widehat{\tau}_a^+$ converges in probability to $\tau_a^+$ , and using the continuity of the paths of $\widehat{X}$ and X, it follows that $\widehat{m}^-_{ij}$ and $\widehat{m}^+_{ij}$ converge pointwise to $m^-_{ij}$ and $m^+_{ij}$ , respectively. Furthermore, if $q>0$ then, by the dominated convergence theorem, $\widehat{O}_{ij}$ converges to $O_{ij}$ .

While the functions $\widehat{m}^-_{ij}$ and $\widehat{m}^+_{ij}$ are not readily available in the existing literature, we can compute them by embedding independent copies of the excursion $(\widehat{J}_*, \widehat{X}_*)\,:\!=\,\{(\widehat{J}(t),\widehat{X}(t)) \colon t\le \widehat{\tau}_0^-\wedge \widehat{\tau}_a^+\wedge e_q\}$ into a certain recurrent queue on the strip [0, a] (that is, a process reflected on the level boundaries 0 and a). This procedure is similar to those exploited in [Reference Akar, Gursoy, Horvath and Telek1, Reference Van Houdt and Blondia36, Reference Yazici and Akar37] to provide finite-time ruin probabilities for multi-regime Markov-modulated risk processes or its subclasses. Below we spell out the details of our construction.

Let $\big\{\big(\widehat{J}^{\{\ell\}}_*, \widehat{X}^{\{\ell\}}_*\big)\big\}_{\ell\ge 1}$ be a sequence of independent copies of $(\widehat{J}_*, \widehat{X}_*)$ . Here we assume that the total length of each excursion $\widehat{X}^{\{\ell\}}_*$ , say $T^{\{\ell\}}$ , has finite mean; this trivially holds if $q>0$ , or if either $\widehat{\tau}_0^-$ or $\widehat{\tau}_a^+$ have a finite mean. Additionally, let our probability space support three independent sequences of $\mbox{Exp}(1)$ -distributed independent random variables $\big\{\rho_{-M}^{\{\ell\}}\big\}$ , $\big\{\rho_0^{\{\ell\}}\big\}$ , and $\big\{\rho_M^{\{\ell\}}\big\}$ . Based on the construction of multi-regime queues in [Reference Akar, Gursoy, Horvath and Telek1, Reference Horváth and Telek21], here we define Y on a space grid $\{\zeta_m\}_{-M\le m\le M}$ (with $\zeta_{-M}=0$ , $\zeta_{0}=u$ , and $\zeta_M=a$ ) modulated by the jump process L that has a state space $\mathcal{E}\cup\{\partial_0\}$ , both of which evolve as follows:

1. (i) On the time interval $\big[0,\nu_1^{\{1\}}\big)$ with $\nu_1^{\{1\}}=T^{\{1\}}$ , let (L, Y) coincide with $\big(\widehat{J}^{\{\ell\}}_*,\widehat{X}^{\{1\}}_*\big)$ .

2. (ii) At time $\nu_1^{\{1\}}$ , one of three events happens:

   • • If $\widehat{X}^{\{1\}}\big(\nu_1^{\{1\}}\big)=0$ , let $\nu_2^{\{1\}} = \nu_1^{\{1\}} + \rho_{-M}^{\{1\}}$ and $\big(Y(t)=0,L(t)=L\big(\nu_1^{\{1\}}-\big)\big)$ for all $t\in \big[\nu_1^{\{1\}}, \nu_2^{\{1\}}\big)$ .

   • • If $\widehat{X}^{\{1\}}\big(\nu_1^{\{1\}}\big)=a$ , let $\nu_2^{\{1\}} = \nu_1^{\{1\}} + \rho_M^{\{1\}}$ and $\big(Y(t)=a,L(t)=L\big(\nu_1^{\{1\}}-\big)\big)$ for all $t\in \big[\nu_1^{\{1\}}, \nu_2^{\{1\}}\big)$ .

   • • If $\widehat{X}^{\{1\}}\big(\nu_1^{\{1\}}\big)\in(0,a)$ , let $\nu_2^{\{1\}} = \nu_1^{\{1\}}$ .

3. (iii) At time $\nu_2^{\{1\}}$ , one of two events happens:

   • • If $Y\big(\nu_2^{\{1\}}\big)\le u$ , let Y increase at unit rate up to reaching level u, say, at time $\nu_3^{\{1\}}$ . Let $L(t)=\partial_0$ for all $t\in\big[\nu_2^{\{1\}}, \nu_3^{\{1\}}\big)$ .

   • • If $Y\big(\nu_2^{\{1\}}\big)> u$ , let Y decrease at unit rate up to reaching level u, say, at time $\nu_3^{\{1\}}$ . Let $L(t)=\partial_0$ for all $t\in\big[\nu_2^{\{1\}}, \nu_3^{\{1\}}\big)$ .

4. (iv) Let $P^{\{1\}}_*=\nu_3^{\{1\}} + \rho_0{\{1\}}_u$ and $(Y(t)=u,L(t)=\partial_0)$ for all $t\in \big[\nu_3^{\{1\}}, \nu_4^{\{1\}}\big)$ .

5. (v) Repeat steps (i)–(iv) for $\ell=2,3,\dots$ , shifting the time accordingly in order to concatenate the excursions along with the increasing sequences $\big\{\nu_i^{\{\ell\}}\big\}_{\ell}$ , $i\in\{1,2,3,4\}$ . This produces a process (L, Y) that regenerates at the epochs $\big\{\nu_4^{\{\ell\}}\big\}_{\ell\ge 1}$ .

See Figure 1 for a graphical description of this construction.

Figure 1.

Queue model resulting from concatenating $\{\widehat{X}^{\{\ell\}}_*\}_{\ell\ge 1}$ using steps (i)–(iv), which are shown in the colors blue, mustard, green, and red, respectively.

As pointed out earlier, the process (L, Y) falls within the class of multi-regime Markov-modulated Brownian motion queueing models (see [Reference Akar, Gursoy, Horvath and Telek1, Reference Horváth and Telek21] for details), which has a law characterized by the level-dependent matrices $(Q(x), R(x), S(x))_{0\le x\le a}$ with the following interpretation:

• • $Q_{ij}(x)$ is the jump intensity of L from i to j while Y is in level x. We further specify the dependence on the level x by employing intensity matrices $\{Q^{(m)}\}_{-M+1\le m \le M}$ and $\{\tilde{Q}^{(m)}\}_{-M\le m \le M}$ , where

  \begin{align*} Q(x) = \left\{ \begin{array}{c@{\quad}c@{\quad}l} Q^{(m)} & \mbox{for} & \zeta_{m-1} < x <\zeta_m, \\ \tilde{Q}^{(m)} & \mbox{for} & x= \zeta_m. \end{array}\right. \end{align*}

• • $R_{ii}(x)$ is the drift of Y at level x while L is in state i (by convention we let $R_{ij}(x)=0$ for all $i\neq j$ ). We further specify this dependence on the level x by employing diagonal matrices $\{R^{(m)}\}_{-M+1\le m \le M}$ and $\{\tilde{R}^{(m)}\}_{-M\le m \le M}$ , where

  \begin{align*} R(x) = \left\{ \begin{array}{c@{\quad}c@{\quad}l} R^{(m)} & \mbox{for} & \zeta_{m-1} < x <\zeta_m, \\ \tilde{R}^{(m)} & \mbox{for} & x= \zeta_m. \end{array}\right. \end{align*}

• • $S_{ii}(x)$ is the diffusion coefficient of Y at level x while L is in state i (by convention we let $S_{ij}(x)=0$ for all $i\neq j$ ). We further specify the dependence on the level x by employing nonnegative diagonal matrices $\{S^{(m)}\}_{-M+1\le m \le M}$ , where $S(x)=S^{(m)}$ for $\zeta_{m-1} \le x <\zeta_m$ .

Under the previous considerations, the corresponding matrices for our particular model with state space $\mathcal{E}\cup\{\partial_0\}$ take the form

\begin{alignat*}{2} Q^{(m)} & = \begin{pmatrix} \widehat{\Lambda}(\zeta_{m-1})-q\boldsymbol{I} &\quad q\boldsymbol{1} \\\boldsymbol{0} &\quad 0 \end{pmatrix}, & & m \in \{-M+1, -M+2,\dots, M\}, \\\tilde{Q}^{(m)} & = \begin{pmatrix} \widehat{\Lambda}(\zeta_{m})-q\boldsymbol{I} &\quad q\boldsymbol{1} \nonumber \\[-2pt]\boldsymbol{0} &\quad 0 \end{pmatrix}, & & m\in \{-M+1, -M+2,\dots, M-1\}\setminus\{0\}, \\ \tilde{Q}^{(-M)} & = \tilde{Q}^{(M)} = \begin{pmatrix} -\boldsymbol{I} &\quad \boldsymbol{1} \\\boldsymbol{0} &\quad 0 \end{pmatrix}, & & \\\tilde{Q}^{(0)} & = \begin{pmatrix} \widehat{\Lambda}(\zeta_{0})-q\boldsymbol{I} &\quad q\boldsymbol{1} \\\boldsymbol{e}^{\top}_{i} &\quad -1 \end{pmatrix}, & & \\R^{(m)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\mu}_i(\zeta_{m-1})\} &\quad \boldsymbol{0} \\\boldsymbol{0} &\quad 1 \end{pmatrix}, & & m\in \{-M+1, -M+2,\dots, 0\}, \\ R^{(m)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\mu}_i(\zeta_{m-1})\} &\quad \boldsymbol{0} \\\boldsymbol{0} &\quad -1 \end{pmatrix}, \qquad & & m\in \{1, 2,\dots, M\}, \\ \tilde{R}^{(m)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\mu}_i(\zeta_{m})\} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad 1 \end{pmatrix}, & & m\in \{-M+1, -M+2,\dots, -1\}, \\ \tilde{R}^{(m)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\mu}_i(\zeta_{m})\} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad -1 \end{pmatrix}, & & m\in \{1, -M+2,\dots, M-1\}, \\ \tilde{R}^{(-M)} & = \begin{pmatrix} \boldsymbol{0} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad 1 \end{pmatrix}, & & \\ \tilde{R}^{(M)} & = \begin{pmatrix} \boldsymbol{0} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad -1 \end{pmatrix}, & & \\ \tilde{R}^{(0)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\mu}_i(\zeta_{0})\} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad 0 \end{pmatrix}, & & \\ S^{(m)} & = \begin{pmatrix} \mbox{diag}\{\widehat{\sigma}_i(\zeta_{m-1})\} &\quad \boldsymbol{0} \\ \boldsymbol{0} &\quad 0 \end{pmatrix}, & & m\in \{-M+1, -M+2,\dots, M\},\end{alignat*}

where $\boldsymbol{e}^{\top}_{i}$ denotes the ith canonical row vector, and $\mbox{diag}\{a_i\}$ denotes the diagonal matrix with the elements $\{a_1,\dots,a_p\}$ filling the diagonal. In [Reference Horváth and Telek21], the authors provide an efficient algorithm to compute the steady-state distribution of (L, Y). In particular, using their method we are able obtain:

• • the steady-state probability atoms for levels $\zeta_{-M}$ and $\zeta_{M}$ while in the states $\mathcal{E}$ in row vector form, say $\boldsymbol{p}_{-M}$ and $\boldsymbol{p}_{M}$ , where

  \begin{equation*} (\boldsymbol{p}_{-M})_j = \lim_{t\rightarrow \infty}\mathbb{P}(L(t)=j, Y(t)= 0), \qquad (\boldsymbol{p}_{M})_j = \lim_{t\rightarrow \infty}\mathbb{P}(L(t)=j, Y(t)= a); \end{equation*}

• • the steady-state probability atoms for level $\zeta_0$ while in the state $\partial_0$ , say $p_0$ , where

  \begin{equation*} p_0 = \lim_{t\rightarrow \infty}\mathbb{P}(L(t)=\partial_0, Y(t)= u); \end{equation*}

• • the steady-state probability distribution function while in state $j\in\mathcal{E}$ over (0,a), say $F_j$ , where $F_j(b)=\lim_{t\rightarrow \infty}\mathbb{P}(L(t)=j, 0<Y(t)\le b)$ , $b\in (0,a)$ .

Below we link these quantities with the first-passage probabilities $\widehat{m}^+_{ij}$ and $\widehat{m}^-_{ij}$ , as well as with the expected occupation times $\widehat{O}_{ij}$ .

Theorem 4.1. Suppose that $\mathbb{E}(\widehat{\tau}_0^- \wedge \widehat{\tau}_a^+ \wedge e_q \mid \widehat{J}(0)=i, \widehat{X}(0)=u)<\infty$ . Then

(4.1) \begin{equation} \widehat{m}^-_{ij}(u,q,a)=\frac{(\boldsymbol{p}_{-M})_j}{p_0}, \qquad \widehat{m}^+_{ij}(u,q,a)=\frac{(\boldsymbol{p}_{M})_j}{p_0}. \end{equation}

Moreover,

(4.2) \begin{equation} \widehat{O}_{ij}(u,q,a,b)=\frac{F_j(b)}{p_0}, \quad b\in(0,a). \end{equation}

Proof. The process (L, Y) regenerates at the epochs $\{\nu_4^{\{\ell\}}\}_{\ell\ge 1}$ , all of which have a (common) finite first moment. This in turn implies that (L, Y) is positive recurrent, so by [Reference Glynn20, Theorem 1],

\begin{align*} \boldsymbol{p}_M\boldsymbol{1} = \frac{\mathbb{E}\big(\!\int_0^{\nu_4^{\{1\}}}1_{\{Y(s)=u,L(s)\in\mathcal{E}\}}\,\mathrm{d}s\big)} {\mathbb{E}\big(\nu_4^{\{1\}}\big)} & = \frac{\mathbb{P}\big(Y\big(\nu_1^{\{1\}}\big)=u,L\big(\nu_1^{\{1\}}\big)\in\mathcal{E}\big) \mathbb{E}\big(\rho_0^{\{1\}}\big)}{\mathbb{E}\big(\nu_4^{\{1\}}\big)} \\ & = \frac{\widehat{m}^+(u,q,a)}{\mathbb{E}\big(\nu_4^{\{1\}}\big)}, \end{align*}

where we used that $\mathbb{E}\big(\rho_0^{\{1\}}\big)=1$ . Employing similar arguments, we get

\begin{equation*} \boldsymbol{p}_{-M}\boldsymbol{1} = \frac{\widehat{m}^-(u,q,a)}{\mathbb{E}\big(\nu_4^{\{1\}}\big)}, \qquad p_0 = \frac{1}{\mathbb{E}\big(\nu_4^{\{1\}}\big)}, \end{equation*}

so that (4.1) follows. Additionally,

\begin{align*} F_j(b) = \frac{\mathbb{E}\big(\!\int_0^{\nu_4^{\{1\}}}1_{\{L(s)=j,\,0 < Y(s)\le b\}}\,\mathrm{d}s\big)} {\mathbb{E}\big(\nu_4^{\{1\}}\big)} & = \frac{\mathbb{E}\big(\!\int_{0}^{\widehat{\tau}_0^-\wedge\widehat{\tau}_a^+\wedge e_q} 1_{\{\widehat{J}(s)=j,\,0 < \widehat{X}(s)\le b\}}\,\mathrm{d}s\big)}{\mathbb{E}\big(\nu_4^{\{1\}}\big)} \\ & = \frac{\widehat{O}_{j,b}(u,q,a)}{\mathbb{E}\big(\nu_4^{\{1\}}\big)}, \end{align*}

which in turn implies (4.2).

In short, with the help of the algorithms developed in [Reference Horváth and Telek21], we are able to efficiently compute first-passage probabilities and their expected occupation times for $(\widehat{J},\widehat{X})$ using Theorem 4.1, which in turn approximates the first-passage probabilities and their expected occupation times for (J, X) by virtue of Theorem 3.2. Below we explore a couple of synthetic examples.

Example 4.1. Consider a three-state hybrid SDE (X, J) evolving in the space band (0,1) with parameters

\begin{equation*} \mu_1(x)= 0.5, \qquad \mu_2(x)=0.5(1-x), \qquad \mu_3(x)=0.5(1-x)^2, \qquad \sigma_1=\sigma_2=\sigma_3=1. \end{equation*}

In essence, the process X has a force pushing it upwards while J is in any of the three states. The force is weakest while in state 3 and strongest while in 1; note that the difference between regimes is accentuated as X gets closer to level 1. In the context of actuarial science, this may, e.g., describe the case of the surplus process of an insurance company with three possible occurring regimes, all with positive drift, but some being more responsive with premium reductions when the surplus approaches 1. This can be described, for instance, by an intensity matrix function of the form

\begin{align} \boldsymbol{\Lambda}(x)=10\times \begin{pmatrix} -x &\quad x &\quad 0 \\ (1-x) &\quad -1 &\quad x \\ 0 &\quad 1-x &\quad -(1-x) \end{pmatrix}. \end{align}

Employing the MATLAB coded provided at https://www.hit.bme.hu/~ghorvath based on [Reference Horváth and Telek21] together with Theorem 4.1, we are able to compute the first-passage probabilities and expected occupation times of the multi-regime Markov-modulated Brownian motion $(\widehat{J},\widehat{X})$ that approximates (J, X). We perform such computations for $q=0$ , $a=1$ , $i=2$ , and $M=50$ , with the results shown in Figure 2.

Figure 2.

Left: Plots of $\widehat{m}^-_{2j}$ as a function of u. Right: Plots of $\widehat{O}_{2j}$ as a function of b with $u=0.5$ . Each plot contains the cases $j=1,2,3$ .

For small values of u, downcrossings of level 0 are most likely while in state 2; this can be explained by noting that the random oscillations while in state 2 (the initial state of J) can produce a downcrossing before a switching even occurs. For medium values of u, the probabilities of downcrossing level 0 while in states 1 and 2 are comparable, both of which are considerably larger than that corresponding to state 3; this is due to the state-dependent switching behaviour of $\Lambda$ , which favours states 1 and 2 while the level is low, and 2 and 3 while the level is high. For high values of u, the probability of downcrossing 0 is uniformly low for all states. On the other hand, we can observe that the expected occupation time while in state 2 is larger than that in states 1 or 2 uniformly over all u; this is a consequence of state 2 being the initial state of the system.

While Theorem 3.2 guarantees that $\widehat{m}^-_{ij}$ converges to $m^-_{ij}$ as the space grid becomes denser, in Figure 3 we (empirically) show how fast this convergence happens. As we can appreciate from the plot, convergence is quickly achieved, with differences between the cases being essentially negligible for $M\ge 40$ .

Figure 3.

Plots of $\widehat{m}^-_{2j}$ as a function of M for the cases $j=1,2,3$ .

Example 4.2. Consider now the same parameters as in Example 4.1, but take $\mu_3(x)=-0.5x^2$ and $\sigma_3=0$ . While this alternative scenario does not necessarily have an immediate practical interpretation, it does help to investigate some of the consequences of having a state that lacks any random noise behaviour. From Figure 4, we note that the downcrossing probabilities for states 1 and 3 are larger than those in Example 4.1; this is because we now have one of the states pushing the level process downwards. However, note that the probability of downcrossing 0 while in state 3 is null; this is because $\mu_3$ approaches 0 as the level gets closer to 0, meaning that the process simply cannot cross level 0 while in state 3. Moreover, we note that for large values of b, the expected occupation time while in state 3 is larger than in states 1 or 2; this suggests that before exiting the band (0,1), the process switches to and stays in state 3 for a larger amount of time than in Example 4.1.

Figure 4.

Left: Plots of $\widehat{m}^-_{2j}$ as a function of u. Right: Plots of $\widehat{O}_{2j}$ as a function of b with $u=0.5$ . Each plot contains the cases $j=1,2,3$ .

5. Summary and extensions

Under Assumptions 2.1, 2.2, 3.1 3.2, and 3.3, we considered the class of hybrid SDEs and their space-grid approximations, the latter belonging to a family of multi-regime Markov-modulated Brownian motions. We provided a rigorous proof of their pathwise convergence, which holds in a weak sense uniformly over increasing compact intervals. As an application of our convergence result, we considered the challenging problem of identifying first-passage probabilities and expected occupation times for solutions of hybrid SDEs, for which we suggest an approximate and computationally efficient answer that employs developments in the area of multi-regime Markov-modulated queues.

We point out that applications of our convergence result may also be used to approximate other (and possibly more complex) descriptors of hybrid SDEs. For instance, we can easily build hybrid SDEs with phase-type jumps by means of the fluidization method [Reference Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford7]. Other modifications and extensions that are straightforward to implement in our framework are the Omega model [Reference Albrecher, Gerber and Shiu2, Reference Gerber, Shiu and Yang18], the Erlangian approximations for finite-time probabilities of ruin [Reference Asmussen, Avram and Usabel6], and Parisian ruin problems with phase-type clocks [Reference Bladt, Nielsen and Peralta10].

Furthermore, while it is possible to adapt the methods presented here to a higher-dimensional setting by appropriately modifying the norms and proofs, we chose to focus on the one-dimensional case for two main reasons. First, higher-dimensional arguments introduce significant notational complexity, which may obscure the key ideas and results. Second, the existing literature on algorithmic results for multi-regime Markov-modulated Brownian motions is limited to the one-dimensional setting, which means that extending to higher dimensions would have limited immediate applicability. Future work may explore these extensions in contexts where they become practically relevant.