2.1. Nearly density-dependent families

Let $\mathcal{I} \subseteq \mathbb{R}_+$ be a closed interval. In this article, we consider two cases: $\mathcal{I}$ is either [0, a], where $a > 0$ , or $[0,\infty)$ . Consider a positive parameter $r > 0$ , which can be thought of, for example, as representing volume, area, or total population. Define

\begin{align*} \mathcal{I}^r &\;:\!=\; r\mathcal{I} \cap \mathbb{Z} = \{k \in \mathbb{Z} \;|\; k= rx, \; \text{ for some } x \in \mathcal{I} \}, \\[5pt] \overline{\mathcal{I}}^r &\;:\!=\; \mathcal{I} \cap \left\{ \frac{k}{r} \;|\; k \in \mathbb{Z} \right\}. \end{align*}

Notice that $\overline{\mathcal{I}}^r = \frac{1}{r}\mathcal{I}^r$ . Let $\mathcal{R} = \{r_n\}_{n \geq 1}$ be a positive increasing sequence such that $r_n \to \infty$ as $n \to \infty$ .

Definition 2.1. (NDDF.) A family $\{X^r\}_{r \in \mathcal{R}}$ of continuous-time Markov chains with state spaces $\{\mathcal{I}^r\}_{r \in \mathcal{R}}$ will be called a nearly density-dependent family (NDDF) if there exist a function $\beta\;:\; \mathcal{I} \times (\mathbb{Z}\setminus\{0\}) \longrightarrow [0,\infty)$ which does not depend on r and a family of functions $\{\epsilon^r \;:\; \mathcal{I} \times (\mathbb{Z}\setminus\{0\}) \longrightarrow \mathbb{R} \}_{r \in \mathcal{R}}$ such that (i)–(iv) below hold. For convenience, we define

(2.1) \begin{equation} \lambda^r(x,\ell) \;:\!=\; \beta(x,\ell) + \epsilon^r\left(x,\ell\right), \qquad x \in \mathcal{I},\; \ell \in \mathbb{Z}\setminus\{0\}, \: r \in \mathcal{R}. \end{equation}

1. (i) For each $\ell \in \mathbb{Z}\setminus\{0\}$ , $\beta(\cdot,\ell)$ is continuous.

2. (ii) For each compact subset $\mathcal{K} \subseteq \mathcal{I}$ and $\ell \in \mathbb{Z}\setminus\{0\}$ ,

   (2.2) \begin{equation} \sup_{x \in \mathcal{K}} |\epsilon^r(x,\ell)| \leq \frac{M_{\mathcal{K},\ell}}{r} \qquad \text{for every } r \in \mathcal{R}, \end{equation}
   where $M_{\mathcal{K},\ell} > 0$ is a constant not depending on r.

3. (iii) For each $r \in \mathcal{R}$ , $k \in \mathcal{I}^r$ , and $\ell \in \mathbb{Z} \setminus \{0\}$ , if $k +\ell \notin \mathcal{I}^r$ , then $\lambda^r(\frac{k}{r},\ell) = 0$ .

4. (iv) For each $r \in \mathcal{R}$ , $X^r$ has an infinitesimal generator $Q^r=(q^{r}_{k,j})_{k,j \in {\mathcal{I}}^r}$ whose off-diagonal entries are given by

   (2.3) \begin{equation} q^r_{k,k+\ell} = r\lambda^r\left(\frac{k}{r},\ell\right), \qquad k \in \mathcal{I}^r, \, \ell \in \mathbb{Z}\setminus\{0\}, \: k + \ell \in \mathcal{I}^{r}. \end{equation}

As a special case, if the family $\{\epsilon^r \;:\; \mathcal{I} \times(\mathbb{Z}\setminus\{0\}) \longrightarrow \mathbb{R} \}_{r \in \mathcal{R}}$ is such that $\epsilon^r(\cdot,\cdot)=0$ for every $r \in \mathcal{R}$ , then $\{X^r\}_{r \in \mathcal{R}}$ will be called a density-dependent family (DDF).

Definition 2.1 is a blend of definitions in Kurtz [Reference Kurtz14], Ethier and Kurtz [Reference Ethier and Kurtz9, Chapter 11, Equation (1.13)], and Angius et al. [Reference Angius4, Definition 2]. Their definitions are valid in a finite-dimensional space. Here, we focus on the one-dimensional case, but this definition can be generalized to higher dimensions. Ethier and Kurtz [Reference Ethier and Kurtz9] introduced the notion of near density dependence, although the term was coined afterwards by Angius et al. [Reference Angius4], motivated by CRNs. Our notion of a DDF (a special case of Definition 2.1) is consistent with the one introduced by Kurtz [Reference Kurtz14]. It is implicit in Definition 2.1 that the Markov chains do not explode in finite time. We will assume this throughout the paper.

In Section 4 we provide examples of NDDFs, including CRNs with only one species (Examples 4.2 and 4.3) and CRNs reduced by mass-conservation conditions (Examples 4.1 and 4.5). Some of these examples will be DDFs, such as the SIS and SI models for epidemics (Examples 4.4 and 4.6 respectively).

Throughout this paper, for any NDDF $\{X^r\}_{r \in \mathcal{R}}$ , we make the following additional assumption.

Assumption 2.1. There exists a finite set $\boldsymbol{L} = \{\ell_1,\ldots,\ell_n\} \subseteq \mathbb{Z}\setminus\{0\}$ such that for each $r \in \mathcal{R}$ and for each $j=1,\ldots,n$ , $\lambda^r(\cdot,\ell_j)$ is not identically zero and

(2.4) \begin{equation} \lambda^r(\cdot,\ell) =0 \quad \textit{for each } \ell \notin \boldsymbol{L}. \end{equation}

We regard $\textbf{L}$ as the set of possible jumps. Assumption 2.1 asserts that the whole family $\{X^r\}_{r \in \mathcal{R}}$ shares a common finite set of possible jumps. Under this condition, because of (2.2), $\beta(\cdot,\ell) =0$ for a given $\ell \notin \textbf{L}$ and, for a given compact set $\mathcal{K}$ , the constants $M_{\mathcal{K},\ell}$ in (2.2) can be chosen independent of $\ell$ . We suppress the $\ell$ and denote this constant by $M_{\mathcal{K}}$ .

For NDDFs coming from CRNs, Assumption 2.1 is usually satisfied. In this context, each $\ell \in \textbf{L}$ comes from a reaction, and usually only finitely many reactions are possible.

By the reasoning of Theorem 6.4.1 of Ethier and Kurtz [Reference Ethier and Kurtz9], we can construct a realization of an NDDF $\{X^r\}_{r \in \mathcal{R}}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with independent unit-rate Poisson processes $N_1, \ldots,N_n$ such that the following holds:

(2.5) \begin{equation} X^r(t) = X^r(0) + \sum_{j=1}^{n} \ell_jN_j\left(\int_{0}^{t}r\lambda^r\left(\frac{X^r(s)}{r},\ell_j\right)ds \right) \end{equation}

for every $r \in \mathcal{R}$ and for every $t \geq 0$ , where $X^r(0) \in \mathcal{I}^r$ . We normalize $X^r$ by considering the concentration process $\overline{X}^r \;:\!=\; \frac{1}{r}X^r$ , which takes values in $\overline{\mathcal{I}}^r \subseteq \mathcal{I}$ .

We next introduce the CLA of $\overline{X}^r$ . This is a slight generalization to NDDFs of the CLA introduced by Leite and Williams [Reference Leite and Williams17] for CRNs.

2.2. Constrained Langevin approximation

Consider an NDDF $\{X^r\}_{r \in \mathcal{R}}$ with its corresponding function $\beta$ . Consider the continuous coefficients $\mu, \sigma \;:\; \mathcal{I} \longrightarrow \mathbb{R}$ defined for $x \in \mathcal{I}$ by

(2.6) \begin{equation} \mu(x) \;:\!=\; \sum_{j=1}^n \ell_j\beta(x,\ell_j), \qquad \sigma(x) \;:\!=\; \left(\sum_{j=1}^{n} \ell_j^2\beta(x,\ell_j)\right)^{1/2}. \end{equation}

Consider the function $\gamma \;:\; \mathcal{I} \longrightarrow \mathbb{R}$ defined by

(2.7) \begin{equation} \gamma= \begin{cases} \mathbb{1}_{\{0\}} - \mathbb{1}_{\{a\}} & \text{ if } \mathcal{I} =[0,a] \; \text{ for } a >0,\\[5pt] \mathbb{1}_{\{0\}} & \text{ if } \mathcal{I} =[0, \infty). \end{cases} \end{equation}

It can be verified, under the conditions of Definition 2.1, that $\mu(x)\gamma(x) \geq 0$ for each $x \in \partial \mathcal{I}$ , where $\partial \mathcal{I} = \{0,a\}$ if $\mathcal{I}=[0,a]$ and $\partial \mathcal{I} = \{0\}$ if $\mathcal{I} =[0,\infty)$ . Consider a family $\{\nu^r\}_{r \in \mathcal{R}}$ of Borel probability measures on $\mathcal{I}$ .

The CLA $Z^r$ behaves like the Langevin approximation to $\overline{X}^r$ in the interior of $\mathcal{I}$ . By (2.8), the continuous process $L^r$ can only increase when $Z^r$ is at the boundary of $\mathcal{I}$ , and this is used to keep the process $Z^r$ in the interval $\mathcal{I}$ .

3.1. Bounded-interval case

We make the following assumption in this section.

Assumption 3.1. The interval $\mathcal{I}$ is of the form $\mathcal{I} = [0,a]$ where $a > 0$ . Furthermore, there exists a constant $A > 0$ such that

(3.1) \begin{equation} |\beta(x,\ell_j)-\beta(y,\ell_j)| \leq A|x-y| \end{equation}

for every $x,y \in \mathcal{I}$ and $1 \leq j \leq n$ .

Then, for each $r \in \mathcal{R}$ , $\mathcal{I}^r$ is a finite set, and this implies that the Markov chain $X^r$ automatically does not explode in finite time. Additionally, by (2.2), we can pick a constant $M_{\mathcal{I}} > 0$ , not depending on r or $\ell$ , such that

(3.2) \begin{equation} \sup_{x \in \mathcal{I}} |\epsilon^r(x,\ell_j)| \leq \frac{M_{\mathcal{I}}}{r} \end{equation}

for every $r \in \mathcal{R}$ and $1 \leq j \leq n$ . Under Assumption 3.1, we can prove that the CLA is well defined.

This result is a consequence of Theorem 5.2 and Proposition 5.1. In Theorem 5.2 and Corollary 5.1 we show that strong existence and pathwise uniqueness in fact hold for Equation (2.9). We focus on existence and uniqueness in law here because those are the notions needed for the next theorem. Note that by (3.1), the coefficient $\sigma$ is Hölder continuous and not necessarily Lipschitz continuous.

Now we give our main result.

Loosely speaking, this result proves that the pathwise error of the CLA on a bounded interval is $O\big(\frac{\log r}{r}\big)$ on compact time intervals. The proof of Theorem 3.2 can be found in Section 6.2. This constructs an appropriate coupling of the processes $Z^r$ and $\overline{X}^r$ .

We can derive from Theorem 3.2 the following result concerning the Wasserstein distance $\mathcal{W}_p$ between the laws of $\overline{X}^r$ and $Z^r$ . For $r \in \mathcal{R}$ and $T > 0$ , denote by $P^r_T$ the probability measure on $(\mathcal{D}[0,T],\mathcal{M}_T)$ that is the law of $\{\overline{X}^r(t), \; 0 \leq t \leq T\}$ under $\overline{X}^r(0)=x^r_0$ , $\mathbb{P}$ -a.s. Also, denote by $\tilde{P}^r_T$ the probability measure on $(\mathcal{D}[0,T], \mathcal{M}_T)$ that is the law of $\{Z^r(t), \; 0 \leq t \leq T\}$ under the initial condition $Z^r(0)=x^r_0$ , $\mathbb{P}^r$ -a.s.

Corollary 3.1. Suppose Assumption 3.1 holds. Then, for every $T,p \geq 1$ , there exists a constant $C=C(T,p,\mathcal{I},\boldsymbol{L},M_{\mathcal{I}},\beta)> 0$ , depending only on T, p, $\mathcal{I}$ , $\boldsymbol{L}$ , $M_{\mathcal{I}}$ , and $\beta$ , such that

(3.5) \begin{equation} \mathcal{W}_p(P^r_T,\tilde{P}^r_T) \leq C\frac{\log r}{r} \end{equation}

for every $r \geq 8$ in $\mathcal{R}$ .

The proof of this result is given in Section 6.3.

3.2. Half-line case

We make the following assumption in this section.

Under (3.6), both $\mu$ and $\sigma^2$ are locally Lipschitz continuous on $[0,\infty)$ . Also, (3.7) implies that $\beta(x,\ell_j) \leq C\ell^{-2}_j(1+x^2)$ for $x \geq 0$ and $1 \leq j \leq n$ . In the context of CRNs, this allows for the possibility of some binary reactions (see Example 4.3). Observe that the conditions (3.6) and (3.7) alone may not prevent the explosion of the members of the NDDF. Consequently, we include (iii) in our assumptions.

As in the previous case, we can prove that the CLA $Z^r$ is properly defined.

Similarly to Theorem 3.1, this result is a consequence of Corollary 5.1, Theorem 5.2, and Proposition 5.1. Strong existence and pathwise uniqueness hold for Equation (2.9). The following is our main result for the half-line case.

The proof of this result can be found in Section 6.4. In this case, the coefficients $\beta(\cdot, \ell_j)$ could be unbounded. This is the main reason for truncation by $\tau^r_{\mathcal{K}}$ in Theorem 3.4. As in the bounded-interval case, an analogue of Remark 3.1 holds here.

5.1. SDER

As before, let $\mathcal{I} \subseteq \mathbb{R}_+$ be a closed interval of the form [0, a], where $a > 0$ , or the interval $[0,\infty)$ . Let $d \geq 1$ be an integer and let $b \;:\; \mathcal{I} \longrightarrow \mathbb{R}$ and $\rho \;:\; \mathcal{I} \longrightarrow \mathbb{R}^d$ be Borel-measurable functions. Additionally, consider a real number $\alpha > 0$ and the function $\gamma \;:\; \mathcal{I} \longrightarrow \mathbb{R}$ given by (2.7).

Definition 5.1. Given $(b,\rho,\alpha,\gamma)$ , we say that $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z,W,L)$ is a solution of the SDER

(5.1) \begin{equation} Z(t)=Z(0)+\int_{0}^{t} b(Z(s))ds + \int_{0}^{t}\rho(Z(s))dW(s) + \alpha\int_{0}^{t}\gamma(Z(s))dL(s), \end{equation}

where

1. (i) $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and $\{\mathcal{F}_t\}$ is a filtration on $\mathcal{F}$ such that the filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\},\mathbb{P})$ satisfies the usual conditions,

2. (ii) $Z =\{Z(t), \; 0 \leq t < \infty\}$ is a continuous, $\{\mathcal{F}_t\}$ -adapted process such that $\mathbb{P}$ -a.s. $Z(t) \in \mathcal{I}$ for all $t \geq 0$ ,

3. (iii) $W = \{W(t)=(W_1(t),\ldots,W_d(t)), \; 0 \leq t < \infty\}$ is a d-dimensional standard Brownian motion and an $\{\mathcal{F}_t\}$ -martingale under $\mathbb{P}$ ,

4. (iv) $L= \{L(t), \; 0 \leq t < \infty\}$ is a continuous, $\{\mathcal{F}_t\}$ -adapted, one-dimensional, $\mathbb{P}$ -a.s. nondecreasing process, with $\mathbb{P}[L(0)=0]=1$ and such that $\mathbb{P}$ -a.s.

   (5.2) \begin{equation} L(t) = \int_{0}^{t} \mathbb{1}_{\{Z(s) \in \partial \mathcal{I} \}}dL(s), \qquad \text{for every } t \geq 0, \end{equation}

5. (v) for every $t \geq 0$ , $\mathbb{P}$ -a.s.

   (5.3) \begin{equation} \int_{0}^{t} |b(Z(s))|ds < \infty, \quad \text{and} \quad \int_{0}^{t} |\rho(Z(s))|^2ds < \infty, \end{equation}
   and

6. (vi) the triple (Z, W, L) $\mathbb{P}$ -a.s. satisfies

   (5.4) \begin{equation} Z(t)=Z(0)+\int_{0}^{t}b(Z(s))ds + \sum_{i=1}^{d}\int_{0}^{t}\rho_i(Z(s))dW_i(s) + \alpha \int_{0}^{t}\gamma(Z(s))dL(s) \end{equation}
   for every $t \geq 0$ .

For a Borel probability measure $\nu$ on $\mathcal{I}$ , we will say that a solution to (5.1) has initial distribution $\nu$ provided Z(0) is distributed as $\nu$ , under $\mathbb{P}$ .

The notion in Definition 5.1 is commonly referred to as a weak solution. For the sake of simplicity, we will just call it a solution in this paper. Additionally, we will say that Z is a solution of (5.1), meaning that $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z,W,L)$ is a solution of (5.1).

For a better understanding of the CLA, we explore further properties of SDERs. We start by describing two notions of uniqueness for Equation (5.1).

Definition 5.2. We say that uniqueness in law or weak uniqueness holds for Equation (5.1) if for every pair of solutions

\begin{equation*} (\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z,W,L) \; \text{ and } \; (\tilde{\Omega},\tilde{\mathcal{F}},\{\tilde{\mathcal{F}}_t\},\tilde{\mathbb{P}},\tilde{Z},\tilde{W},\tilde{L}) \end{equation*}

with the same initial distribution ( $Z(0) \overset{d}{=} \tilde{Z}(0)$ ), we have that (Z, L) and $(\tilde{Z},\tilde{L})$ have the same law.

Definition 5.3. We will say that pathwise uniqueness holds for Equation (5.1) if for every pair of solutions

\begin{equation*} (\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z,W,L) \; \text{ and } \; (\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},\tilde{Z},W,\tilde{L}) \end{equation*}

with common Brownian motion W and common filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$ , and such that $\mathbb{P}[Z(0) = \tilde{Z}(0)]=1$ , we have that

(5.5) \begin{equation} \mathbb{P}[Z(t) = \tilde{Z}(t) \;, L(t) = \tilde{L}(t) \; \text{ for every } 0 \leq t < \infty] = 1. \end{equation}

The relation between these two notions is stated in Proposition 5.1.

A second type of solution is the so-called strong solution. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space with a standard d-dimensional Brownian motion W defined there, together with a random variable $\xi$ taking values in $\mathcal{I}$ such that $\xi$ is independent of W. For the filtration $\{\mathcal{G}_t \;:\!=\; \sigma(\xi, W(s) ;\; 0 \leq s \leq t ) \;:\; 0 \leq t < \infty \}$ and the collection of null sets $\mathcal{N} \;:\!=\; \{ N \in \mathcal{F} \;|\; \mathbb{P}(N) = 0 \}$ , we define $\hat{\mathcal{F}}^W_t \;:\!=\; \sigma(\mathcal{G}_t \cup \mathcal{N})$ for $t \geq 0$ . Then $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P})$ satisfies the usual conditions. In addition, $\xi$ is $\hat{\mathcal{F}}^W_0$ -measurable and W is an $\{\hat{\mathcal{F}}^W_t\}$ -martingale. We call $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P},W,\xi)$ a setup.

In this article we will always emphasize the word ‘strong’ when working with a strong solution. Note that in Definitions 5.1 and 5.4, Z is one-dimensional, whereas W can be multidimensional. This formulation will be used in constructing the CLA, in the proofs of Theorems 3.2 and 3.4. In Lemma 5.1 we will show how to merge the d stochastic integrals of (5.4) into one.

A famous result of Yamada and Watanabe [Reference Yamada and Watanabe24] can be readily generalized from stochastic differential equations to SDERs. Accordingly, we state the following proposition without proof.

Along the same lines, we state the next technical result, which will be used in the sequel. The proof follows the same lines as that of Corollary 5.3.23 in Karatzas and Shreve [Reference Karatzas and Shreve10] and is therefore omitted.

Equation (2.9) is of the form of (5.1) with $b=\mu$ , $\rho=\frac{1}{\sqrt{r}}\sigma$ , $\alpha = \frac{1}{\sqrt{r}}$ , $\gamma$ as in (2.7) and $d=1$ . In addition, we will also consider solutions of

(5.6) \begin{equation} \begin{split} Z^r(t) &=Z^r(0)+\int_{0}^{t}\mu(Z^r(s))ds + \frac{1}{\sqrt{r}}\sum_{j=1}^{n}\int_{0}^{t}\ell_j\sqrt{\beta(Z^r(s),\ell_j)}dW_j(s) \\[5pt] &+ \frac{1}{\sqrt{r}}\int_{0}^{t}\gamma(Z^r(s))dL^r(s), \end{split} \end{equation}

for each $r\in \mathcal{R}$ , which corresponds to (5.1) with $b=\mu$ , $\rho_j = \frac{1}{\sqrt{r}}\ell_j\sqrt{\beta(\cdot,\ell_j)}$ for $j=1,\ldots,n$ , $\alpha = \frac{1}{\sqrt{r}}$ , $\gamma$ as in (2.7), and $d=n$ . These equations are needed for technical reasons, as a key step in constructing the coupling in Theorems 3.2 and 3.4.

In the following, we study equations of the form (5.1), with an eye towards (2.9) and (5.6). We start by studying uniqueness in the next section.

5.2. Uniqueness

To our knowledge, the conditions in this section for SDERs have not previously been given anywhere in the literature. We allow both the drift and dispersion coefficient to be more general than locally Lipschitz.

Theorem 5.1. Consider Equation (5.1) with coefficients $(b,\rho,\alpha,\gamma)$ . Suppose that for every compact set $\mathcal{K} \subseteq \mathcal{I}$ there exist strictly increasing functions $h_{\mathcal{K}},g_{\mathcal{K}}\;:\;[0,\infty) \longrightarrow [0,\infty)$ with $g_{\mathcal{K}}(0)=h_{\mathcal{K}}(0)=0$ , where $g_{\mathcal{K}}$ is continuous and concave, such that

(5.7) \begin{equation} \int_{(0,\varepsilon)} \frac{1}{g_{\mathcal{K}}(u)}du = \int_{(0,\varepsilon)} \frac{1}{h_{\mathcal{K}}^{2}(u)}du = \infty \qquad \textit{ for every } \varepsilon > 0 \end{equation}

and

(5.8) \begin{equation} |b(x)-b(y)| \leq g_{\mathcal{K}}(|x-y|), \qquad |\rho_i(x)-\rho_i(y)| \leq h_{\mathcal{K}}(|x-y|) \end{equation}

for every $x,y \in \mathcal{K}$ and $i=1,\dots,d$ . Then pathwise uniqueness holds for (5.1). In particular, it holds for $g_{\mathcal{K}}(u) = A_{\mathcal{K}}u$ and $h_{\mathcal{K}}(u) = A_{\mathcal{K}}u^{\alpha}$ with $\alpha \geq 1/2$ , where $A_{\mathcal{K}} > 0$ is a constant.

Yamada [Reference Yamada23] proved this result for the case $\mathcal{I} = [0,\infty)$ and $d=1$ , under the assumption $g_{\mathcal{K}}(u) = Au$ , where $A > 0$ is a constant, and with $h_{\mathcal{K}}$ being the same function for every compact set $\mathcal{K} \subseteq \mathcal{I}$ . The technique used in [Reference Yamada23] can be extended to the more general case described above, as we now show.

Proof. We first consider the case where $\mathcal{I} = [0,a]$ , for some $a > 0$ . Because of this, we only consider the functions $h_{\mathcal{I}}$ and $g_{\mathcal{I}}$ and omit the subscript $\mathcal{I}$ . Under the condition (5.7) we can construct an increasing sequence $\{\psi_m\}_{m \geq 1}$ of real-valued $\mathcal{C}^2(\mathbb{R})$ functions such that the following properties hold for every $m \geq 1$ (for an illustration of how to construct such an increasing sequence see Proposition 5.2.13 in Karatzas and Shreve [Reference Karatzas and Shreve10]):

1. (i) $\psi_m$ is even, $\psi_m \geq 0$ , $\psi_m(0)=0$ , $\psi_m(x) \leq \psi_{m+1}(x)$ , and $\lim_{m \to \infty} \psi_m(x)=|x|$ for every $x \in \mathbb{R}$ ;

2. (ii) $|\psi^{\prime}_m| \leq 1$ , $\psi^{\prime}_m(x) \geq 0$ for $x \geq 0$ , and $\psi^{\prime}_m(x) \leq 0$ for $x \leq 0$ ;

3. (iii) $\psi^{\prime\prime}_m \geq 0$ and $\psi^{''}_m(x)h^2(|x|) \leq \frac{2}{m}$ for every $x \in \mathbb{R}$ .

Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z_1,W,L_1)$ and $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z_2,W,L_2)$ be two solutions of (5.1) with common Brownian motion W, common probability space $(\Omega, \mathcal{F},\mathbb{P})$ , and common filtration $\{\mathcal{F}_t\}$ , and such that $\mathbb{P}[Z_1(0) = Z_2(0)]=1$ . We define $\Delta(t) \;:\!=\; Z_1(t) - Z_2(t)$ for $t \geq 0$ . For $m \geq 1$ , we apply Itô’s formula and obtain that $\mathbb{P}$ -a.s. for all $t \geq 0$ , $\psi_m(\Delta(t))$ equals

(5.9) \begin{equation} \begin{split} & \int_0^t\psi^{\prime}_m(\Delta(s))(b(Z_1(s)) -b(Z_2(s)))ds + \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\gamma(Z_1(s))dL_1(s) \\[5pt] &- \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\gamma(Z_2(s))dL_2(s) + \frac{1}{2} \sum_{i=1}^d\int_0^t\psi^{''}_m(\Delta(s))(\rho_i(Z_1(s)) -\rho_i(Z_2(s)))^2ds\\[5pt] &+ \sum_{i=1}^d\int_0^t\psi^{\prime}_m(\Delta(s))(\rho_i(Z_1(s)) -\rho_i(Z_2(s)))dW_i(s). \end{split} \end{equation}

Fix $t \geq 0$ . We claim that

(5.10) \begin{equation} \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\gamma(Z_1(s))dL_1(s) \leq 0, \qquad \mathbb{P}\text{-a.s.} \end{equation}

For this, observe first that

\begin{align*} \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\gamma(Z_1(s))dL_1(s) &= \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\mathbb{1}_{\{Z_1(s)=0\}}dL_1(s) \\[5pt] &- \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\mathbb{1}_{\{Z_1(s)=a\}}dL_1(s). \end{align*}

Now, for $0 \leq s \leq t$ , if $Z_1(s)=0$ , then $\Delta(s) = -Z_2(s) \leq 0$ and $\psi^{\prime}_m(\Delta(s)) \leq 0$ . So, $\mathbb{P}$ -a.s., we have $\alpha\int_0^t\psi^{\prime}_m(\Delta(s))\mathbb{1}_{\{Z_1(s)=0\}}dL_1(s) \leq 0$ . On the other hand, for $0 \leq s \leq t$ , if $Z_1(s)=a$ , then $\Delta(s) = a-Z_2(s) \geq 0$ and $\psi^{\prime}_m(\Delta(s)) \geq 0$ . Then, $\mathbb{P}$ -a.s., we have $-\alpha\int_0^t\psi^{\prime}_m(\Delta(s))\mathbb{1}_{\{Z_1(s)=a\}}dL_1(s) \leq 0$ . Thus (5.10) holds. In a similar fashion, we can show that $- \alpha\int_0^t\psi^{\prime}_m(\Delta(s))\gamma(Z_2(s))dL_2(s) \leq 0$ , $\mathbb{P}$ -a.s. Combining these results with the martingale property of the stochastic integrals and the bound of one on $|\psi^{\prime}_m|$ , using (5.8), on taking expectations in (5.9) we obtain

\begin{align*} \mathbb{E}[\psi_m(\Delta(t))] &\leq \mathbb{E}\left[\int_0^t\psi^{\prime}_m(\Delta(s))(b(Z_1(s)) -b(Z_2(s)))ds\right] \\[5pt] &+ \frac{1}{2}\mathbb{E}\left[\sum_{i=1}^d\int_0^t\psi^{''}_m(\Delta(s))(\rho_i(Z_1(s)) -\rho_i(Z_2(s)))^2ds\right] \\[5pt] &\leq \int_0^t\mathbb{E}[g(|\Delta(s)|)]ds + \frac{td}{m} \leq \int_0^t g(\mathbb{E}[|\Delta(s)|])ds + \frac{td}{m}, \end{align*}

where the last inequality follows from the concavity of g and Jensen’s inequality. On letting $m \to \infty$ and applying Lemma 1.4.1 in Agarwal and Lakshmikantham [Reference Agarwal and Lakshmikantham1] (related to Osgood’s criterion), we obtain that for every $t \geq 0$ , $\mathbb{E}[|\Delta(t)|] = 0$ . This implies that $\mathbb{P}[Z_1(t)=Z_2(t)\ \text{for every } t \geq 0] =1$ .

For $k=1,2$ define $Y_k(t) \;:\!=\; \int_{0}^{t}\gamma(Z_k(s))dL_k(s)$ , for $t \geq 0$ . Since both $(Z_1,L_1)$ and $(Z_2,L_2)$ satisfy (5.4) with common Brownian motion, and $Z_1=Z_2$ $\mathbb{P}$ -a.s. by the previous analysis, we must have $\mathbb{P}\left[Y_1(t)=Y_2(t)\ \text{for every } t \geq 0\right] =1$ . Next, observe that $\mathbb{P}$ -a.s. for every $t \geq 0$ and $k=1,2$ ,

$$\int_{0}^{t}\gamma(Z_k(s))dY_k(s) = \int_{0}^{t}\gamma^2(Z_k(s))dL_k(s) = \int_{0}^{t}\mathbb{1}_{\{Z_k(s) \in \partial \mathcal{I} \}}dL_k(s)= L_k(t),$$

where we used (5.2). We conclude that $\mathbb{P}[L_1(t)=L_2(t) \text{ for every } t \geq 0] =1$ .

For the case $\mathcal{I} = [0,\infty)$ , consider $K >0$ and define $T_K = \inf\{t \geq 0 \;|\; Z_1(t) \geq K \text{ or } Z_2(t) \geq K \}$ . This case follows from choosing $\mathcal{K}=[0,K]$ , constructing the family $\{\psi^{\mathcal{K}}_m\}_{m \geq 1}$ related to $h_{\mathcal{K}}$ , replacing t with $t \wedge T_K$ in (5.9), estimating in a similar manner to the previous case, and then letting $K \to \infty$ .

5.3. Merging stochastic integrals

Equations (2.9) and (5.6) are related in the following way. Denote the dispersion coefficient of (5.6) by $\rho^r \;:\!=\; (\rho_1^r, \ldots,\rho_n^r)$ , where $\rho^r_j(x) \;:\!=\; \frac{\ell_j}{\sqrt{r}}\sqrt{\beta(x,\ell_j)}$ for $j=1,\dots,n$ and $x \in \mathcal{I}$ . Then

(5.11) \begin{equation} \sum_{j=1}^n(\rho^r_j(x))^2 = \left(\frac{1}{\sqrt{r}}\sigma(x)\right)^2 \qquad \text{ for } x \in \mathcal{I}. \end{equation}

The next lemma implies that we can merge the n stochastic integrals of (5.6) into a single stochastic integral in (2.9). We state this in the general context of Equation (5.1) and the equation

(5.12) \begin{equation} Z(t)=Z(0)+\int_{0}^{t}b(Z(s))ds + \int_{0}^{t}\Upsilon(Z(s))d\hat{W}(s) + \alpha\int_{0}^{t}\gamma(Z(s))dL(s), \end{equation}

where $\Upsilon(x) \;:\!=\; \Big(\sum_{i=1}^d \rho^2_i(x)\Big)^{1/2}$ for $x\in \mathcal{I}$ , and $\hat{W}$ is a one-dimensional standard Brownian motion. The following is similar to results in the literature on the representation of continuous local martingales, such as Theorem 3.4.2 in Karatzas and Shreve [Reference Karatzas and Shreve10].

Proof. The process $M = \{M(t),\; 0 \leq t < \infty\}$ , defined by

\begin{equation*} M(t) \;:\!=\; \sum_{i=1}^d \int_{0}^{t} \rho_i(Z(s))dW_i(s), \qquad t \geq 0, \end{equation*}

is a continuous $\{\mathcal{F}_t\}$ -local martingale such that $\mathbb{P}[M(0)=0] =1$ . Its quadratic variation process is given by $\langle M\rangle(t) = \int_{0}^{t}\Upsilon^2(Z(s))ds$ for $t \geq 0$ . Consider the process

(5.13) \begin{equation} \hat{W}(t) \;:\!=\; \int_{0}^{t} \frac{1}{\Upsilon(Z(s))}\mathbb{1}_{\{\Upsilon(Z(s)) \neq 0\}}dM(s) + \int_{0}^{t}\mathbb{1}_{\{\Upsilon(Z(s)) = 0\}}dW_1(s), \qquad t \geq 0. \end{equation}

Then $\hat{W} = \{\hat{W}(t),\; 0 \leq t < \infty \}$ is a continuous $\{\mathcal{F}_t\}$ -local martingale such that $\mathbb{P}[\hat{W}(0)=0]=1$ and with quadratic variation process given by

\begin{align*} \langle\hat{W}\rangle(t) &= \left\langle \int_{0}^{\bullet} \frac{1}{\Upsilon(Z(s))}\mathbb{1}_{\{\Upsilon(Z(s)) \neq 0\}}dM(s)\right\rangle(t) + \left\langle\int_{0}^{\bullet}\mathbb{1}_{\{\Upsilon(Z(s)) = 0\}}dW_1(s)\right\rangle(t) \\[5pt] &= \int_{0}^{t}\mathbb{1}_{\{\Upsilon(Z(s)) \neq 0\}}ds + \int_{0}^{t}\mathbb{1}_{\{\Upsilon(Z(s)) = 0\}}ds = t \end{align*}

for every $t \geq 0$ . By Lévy’s characterization, $\hat{W}$ is a standard Brownian motion and an $\{\mathcal{F}_t\}$ -martingale. Moreover, for $t \geq 0$ ,

\begin{align*} \int_{0}^{t}\Upsilon(Z(s))d\hat{W}(s) &= \int_{0}^{t}\Upsilon(Z(s))\frac{\mathbb{1}_{\{\Upsilon(Z(s)) \neq 0\}}}{\Upsilon(Z(s))}dM(s) \\[5pt] & \qquad \quad + \int_{0}^{t}\Upsilon(Z(s))\mathbb{1}_{\{\Upsilon(Z(s)) = 0\}}dW_1(s) \\[5pt] &= \int_{0}^{t}\mathbb{1}_{\{\Upsilon(Z(s)) \neq 0\}}dM(s). \end{align*}

Since $\langle M\rangle(t) = \int_{0}^{t}\Upsilon^2(Z(s))ds$ , it follows that $\int_{0}^{t}\mathbb{1}_{\{\Upsilon(Z(s)) = 0\}}dM(s)=0$ for every $t \geq 0$ . Hence $\int_{0}^{t} \Upsilon(Z(s))d\hat{W}(s) = M(t)$ for every $t \geq 0$ , and we can conclude the desired result.

5.4. Existence and uniqueness

Tanaka [Reference Tanaka22] studied SDERs like (5.1) in a more general setting. Namely, he considered the process Z to be multidimensional, taking values in a convex set and with reflection at the boundary given by the inward-pointing normal. The following is a direct consequence of Theorem 4.2 in [Reference Tanaka22] applied in the one-dimensional case.

Using this result as a starting point, we proceed to establish strong existence for Equations (2.9) and (5.6). We include strong uniqueness and uniqueness in law in the statement below, which comes from Corollary 5.1.

Proof. It suffices to prove existence for (5.6). Indeed, by Corollary 5.2, a solution to (5.6) with initial distribution $\nu$ will generate a solution to (2.9) with initial distribution $\nu$ . Then pathwise uniqueness (established in Corollary 5.1) and Proposition 5.1 enable us to conclude that strong existence holds for both (2.9) and (5.6).

In order to prove existence for (5.6), first consider the case $\mathcal{I}=[0,a]$ . Under Assumption 3.1, Proposition 5.3 implies the existence of solutions to (5.6) for initial distributions charging a single point. Now, consider an initial distribution $\nu$ on $\mathcal{I}$ and a setup $(\Omega, \mathcal{F}, \{\hat{\mathcal{F}}^W_t\}, \mathbb{P}, W, \xi)$ , where W is an n-dimensional Brownian motion and $\xi$ has distribution $\nu$ . Then, for $r \in \mathcal{R}$ , for each integer $k \geq 1$ , let

$$\xi_k \;:\!=\; \sum_{m=0}^{\infty} \frac{m}{2^k}\mathbb{1}_{\left\{\frac{m}{2^k} \leq \xi < \frac{m+1}{2^k}\right\}}.$$

Then $\xi_k \nearrow \xi$ as $k \to \infty$ . By Proposition 5.2, for each pair of integers $k \geq 1$ and $0 \leq m \leq \lfloor a2^k \rfloor$ there is a strong solution $(Z^{r,m}_k,L^{r,m}_k)$ to Equation (5.6) on $(\Omega, \mathcal{F}, \{\hat{\mathcal{F}}^W_t\}, \mathbb{P}, W, \xi)$ , with $Z^{r,m}_k(0) = \frac{m}{2^k}$ . Then

(5.14) \begin{equation} Z^r_k(t) \;:\!=\; \sum_{m=0}^{\lfloor a2^k \rfloor } Z^{r,m}_k(t)\mathbb{1}_{\{\xi_k = \frac{m}{2^k}\}}, \quad L^r_k(t) \;:\!=\; \sum_{m=0}^{\lfloor a2^k \rfloor} L^{r,m}_k(t)\mathbb{1}_{\{\xi_k = \frac{m}{2^k}\}}, \qquad t \geq 0, \end{equation}

is a strong solution $(Z^r_k,L^r_k)$ on $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P},W)$ to Equation (5.6) with initial condition $\xi_k$ . The sequence $\{(Z^r_k,L^r_k,W)\}_{k \geq 1}$ can be proved to be $\mathcal{C}$ -tight, and moreover it can be proved that every subsequential weak limit generates a solution to (5.6) with initial distribution $\nu$ . By weak uniqueness, the sequence actually converges weakly and generates a (weak) solution to (5.6) with initial distribution $\nu$ . By Proposition 5.1, there exists a strong solution to (5.6) on $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P},W)$ .

For the case $\mathcal{I}=[0,\infty)$ , the coefficients in (5.6) are not necessarily bounded, and we cannot apply Proposition 5.3 directly. For any integer $k \geq 1$ , define $\mu_k(x) \;:\!=\; \mu(x \wedge k)$ and $\rho^r_{j,k}(x) \;:\!=\; \rho^r_j(x \wedge k)$ for $x \in \mathcal{I}$ and $1 \leq j \leq n$ , where $\rho^r_j(x) \;:\!=\; \frac{\ell_j}{\sqrt{r}}\sqrt{\beta(x,\ell_j)}$ for $x \in \mathcal{I}$ and $1 \leq j \leq n$ . Additionally, consider the following equation for $t \geq 0$ :

(5.15) \begin{equation} \begin{split} Z^r_k(t) &=Z^r_k(0)+\int_{0}^{t}\mu_k(Z^r_k(s))ds + \sum_{j=1}^{n}\int_{0}^{t}\rho^r_{j,k}(Z^r_k(s))dW_j(s) \\[5pt] & \qquad \qquad + \frac{1}{\sqrt{r}}\int_{0}^{t}\gamma(Z^r_k(s))dL^r_k(s). \end{split} \end{equation}

For each $k \geq 1$ , the coefficient $\mu_k$ is Lipschitz continuous and bounded, while the $\rho^r_{j,k}$ are Hölder continuous of order $1/2$ and bounded, for every $1 \leq j \leq n$ . For $x_0 \in \mathcal{I}$ , consider a setup $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P},W,\xi)$ where $\mathbb{P}[\xi=x_0]=1$ . By combining Theorem 5.1, Proposition 5.3, and Proposition 5.2 we obtain a sequence of processes $\{(Z^r_k,L^r_k)\}_{k \geq 1}$ where, for every $k \geq 1$ , $(Z^r_k,L^r_k)$ is a strong solution to (5.15) with initial condition $\xi$ on the setup $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^W_t\},\mathbb{P},W,\xi)$ . Following an approach similar to that of the proof of Theorem 10.6 in Chung and Williams [Reference Chung and Williams8], and by using the linear growth condition (3.7), we can show that $\mathbb{P}$ -a.s. the limits $Z^r(t) \;:\!=\; \lim_{k \to \infty} Z^r_k(t)$ and $L^r(t) \;:\!=\; \lim_{k \to \infty} L^r_k(t)$ exist for every $t \geq 0$ and define a strong solution to Equation (5.6) with initial condition $Z^r(0)=x_0$ .

For an arbitrary initial distribution $\nu$ , we prove existence using similar arguments to those used for the case of a bounded interval $\mathcal{I}$ .

5.5. Time change

We end this section with a result needed in the proofs of Theorems 3.2 and 3.4.

Lemma 5.2. Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P},Z^r,W,L^r)$ , $r \in \mathcal{R}$ , be a family of solutions of (5.6), and, extending the probability space if needed, let $\tilde{W}$ be an n-dimensional standard Brownian motion defined on $(\Omega,\mathcal{F},\mathbb{P})$ that is independent of $Z^r,W,L^r$ . Then, for each $r \in \mathcal{R}$ , there exists an n-dimensional standard Brownian motion $B^r= (B^r_1,$ $\ldots,B^r_n)$ defined on $(\Omega,\mathcal{F},\mathbb{P})$ such that, $\mathbb{P}$ -a.s., $Z^r(t)$ is equal to

(5.16) \begin{equation} Z^r(0) + \int_{0}^{t} \mu(Z^r(s))ds + \sum_{j=1}^n \frac{\ell_j}{r}B^r_j\left(r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds\right) + \frac{1}{\sqrt{r}}\int_{0}^{t}\gamma(Z^r(s))dL^r(s) \end{equation}

for every $t \geq 0$ . Furthermore, $\sigma(B) \subseteq \sigma(Z^r,W,\tilde{W},\mathcal{N})$ , where $\mathcal{N}$ is the collection of null sets in $(\Omega,\mathcal{F},\mathbb{P})$ .

The statement of Lemma 5.2 is similar to that of Theorem 6.5.3.b in Ethier and Kurtz [Reference Ethier and Kurtz9], except that here (5.16) has an additional reflection term. The underlying idea of the proof is similar; namely, a continuous local martingale can be time-changed to a Brownian motion. Moreover, for a vector of continuous local martingales with cross-variation equal to zero, we can time-change each component and obtain, by means of Knight’s theorem (Theorems V.1.9 and V.1.10 in Revuz and Yor [Reference Revuz and Yor19]), a multidimensional Brownian motion. For completeness, we provide a proof of Lemma 5.2.

Proof. For $j=1,\ldots,n$ define

(5.17) \begin{equation} M^r_j(t) \;:\!=\; \int_{0}^{t}\sqrt{r\beta(Z^r(s),\ell_j)}dW_j(s), \qquad t \geq 0. \end{equation}

Then $M^r \;:\!=\; (M^r_1,\ldots,M^r_n)$ is a vector of continuous $\{\mathcal{F}_t\}$ -local martingales with $\langle{M^r_j}\rangle(t)=r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds$ for $t \geq 0$ , with $M^r_j(0)=0$ $\mathbb{P}$ -a.s., and with cross-variation $\langle{M^r_i,M^r_j}\rangle(t)=0$ for $i \neq j$ and $t \geq 0$ . By Proposition 1.26 in Chapter IV of Revuz and Yor [Reference Revuz and Yor19], $\lim_{t \to \infty} M^r_j(t)$ exists $\mathbb{P}$ -a.s. on $\{\langle M_j^r\rangle(\infty) < \infty\}$ , and we let $M^r_j(\infty)$ be a random variable that is $\mathbb{P}$ -a.s. equal to this limit on $\{\langle M_j^r\rangle(\infty) < \infty\}$ . Now, for $s \geq 0$ and $j=1,\ldots,n$ , define

\begin{equation*} T^r_j(s) \;:\!=\; \inf\{t \geq 0 \;|\; \langle{M^r_j}\rangle(t) > s \} \end{equation*}

and

(5.18) \begin{equation} B^r_j(s) \;:\!=\; \begin{cases} M^r_j(T^r_j(s)) & \text{ if } s < \langle{M^r_j}\rangle(\infty), \\[5pt] M^r_j(\infty) + \tilde{W}_j(s - \langle{M^r_j}\rangle(\infty)) & \text{ if } s \geq \langle{M^r_j}\rangle(\infty). \end{cases} \end{equation}

By Knight’s theorem (as in Theorems V.1.9 and V.1.10 in Revuz and Yor [Reference Revuz and Yor19]), the process $B^r\;:\!=\;(B^r_1,\ldots,B^r_n)$ is an n-dimensional Brownian motion. For $j=1,\ldots,n$ , $\mathbb{P}$ -a.s. we have $B^r_j\left(\langle{M^r_j}\rangle(t)\right)=M^r_j(t)$ for every $t \geq 0$ . We conclude by using these identities in (5.6).

6.1. Outline for the proofs of Theorems 3.2 and 3.4

1. (i) Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with an n-dimensional Brownian motion W defined there, for each $r \geq 8$ in $\mathcal{R}$ and $x^r_0 \in \overline{\mathcal{I}}^r$ , consider a strong solution $(Z^r,L^r)$ to Equation (5.6). Existence is guaranteed by Theorem 5.2.

2. (ii) Using the solution of (5.6), we construct a solution $(\Omega, \mathcal{F},\{\mathcal{F}_t\},\mathbb{P}, Z^r,W^r,L^r)$ to (2.9) and thereby obtain a CLA $Z^r$ . The construction is given in Lemma 5.1.

3. (iii) Using a time change, we obtain an n-dimensional Brownian motion $B^r$ , and $Z^r$ can be described as a solution to Equation (5.16) involving $B^r$ .

4. (iv) From the Brownian motion $B^r$ and independent uniform random variables, we construct independent Poisson processes $N^r_1,\ldots,N^r_n$ coupled appropriately to $B^r$ (see Theorem A.1).

5. (v) Using these Poisson processes, we generate the process $X^r$ as a solution to Equation (6.2), following Theorem 6.4.1 in Ethier and Kurtz [Reference Ethier and Kurtz9].

6. (vi) Having constructed both processes on the same space, we compare their paths. A critical property is the Lipschitz continuity of the Skorokhod map (see Appendix B).

6.2. Proof of Theorem 3.2

Let $\beta$ and $\{\epsilon^r\}_{r \in \mathcal{R}}$ be the functions associated to a given NDDF under Assumptions 2.1 and 3.1. Consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ on which we have a standard n-dimensional Brownian motion $W=\{W(t),\; 0 \leq t < \infty \}$ , a standard n-dimensional Brownian motion $\tilde{W}=\{\tilde{W}(t),\; 0 \leq t < \infty \}$ , and n double sequences $U^1 = (U^1_{i,j})_{i,j \geq 1},\ldots,U^n = (U^n_{i,j})_{i,j \geq 1}$ of independent and identically distributed (i.i.d.) random variables with common uniform distribution on (0,1), such that the elements $W,\tilde{W},U^1,\ldots,U^n$ are independent. Consider $\{\mathcal{F}^W_t\}$ to be the natural filtration for W, where $\mathcal{F}^W_t = \sigma(W(s) \:;\: 0 \leq s \leq t)$ for each $t \geq 0$ , and let $\mathcal{N}$ denote the null sets in $(\Omega,\mathcal{F},\mathbb{P})$ . Define the filtration $\{\mathcal{F}_t\}$ as $\mathcal{F}_t \;:\!=\; \sigma(\mathcal{F}^W_t \cup \mathcal{N})$ for $t \geq 0$ . This filtration is such that the quadruple $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$ satisfies the usual conditions.

Fix an $r \geq 8$ in $\mathcal{R}$ and $x^r_0 \in \overline{\mathcal{I}}^r$ . Define the random variable $\xi_{x^r_0}(\cdot) \;:\!=\;x^r_0$ . The $\sigma$ -algebra $\sigma(\xi_{x^r_0})$ is trivial and therefore independent from $\sigma(W(s) \;:\; s \geq 0)$ . As for Definition 5.4, we can construct the filtration $\{\hat{\mathcal{F}}^{W}_t\}$ and obtain a setup $(\Omega,\mathcal{F},\{\hat{\mathcal{F}}^{W}_t\},\mathbb{P},W,$ $\xi_{x^r_0})$ . Since $\sigma(\xi_{x^r_0})$ is trivial, $\{\hat{\mathcal{F}}^{W}_t\} = \{\mathcal{F}_t\}$ . By Theorem 5.2, there is a strong solution $(Z^r,L^r)$ of (5.6) with initial condition $\xi_{x^r_0}$ . By Corollary 5.2, there is a one-dimensional Brownian motion $W^r=\{W^r(t),\mathcal{F}_t,$ $0 \leq t < \infty \}$ on $(\Omega,\mathcal{F},\mathbb{P})$ such that $(\Omega, \mathcal{F},\{\mathcal{F}_t\}, \mathbb{P},Z^r,W^r,L^r)$ is a solution to (2.9). In other words, $Z^r$ is a CLA associated with $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P}, Z^r,W^r,L^r)$ . In addition, $\mathbb{P}$ -a.s., $Z^r(0) = x^r_0$ .

Since $\tilde{W}$ is independent of W, and for each $r \in \mathcal{R}$ , $(Z^r,L^r)$ is a strong solution for (5.6) driven by W, it follows that $\tilde{W}$ is independent of $W,Z^r,L^r$ . By Lemma 5.2, for each $r \in \mathcal{R}$ , we can construct an n-dimensional standard Brownian motion $B^r= (B^r_1,$ $\ldots,$ $B^r_n)$ such that, $\mathbb{P}$ -a.s., the time-change equation (5.16) holds.

We continue by coupling $B^r$ to a vector of n independent Poisson processes. First, note that by Lemma 5.2, $\sigma(B^r) \subseteq \sigma(W,\tilde{W},\mathcal{N})$ , and so $B^r$ is independent of $U^1 = (U^1_{i,j})_{i,j \geq 1},\ldots,$ $U^n = (U^n_{i,j})_{i,j \geq 1}$ . By Theorem A.1, there are n mutually independent unit-rate Poisson processes $N^r_1,\ldots,N^r_n$ defined on $(\Omega,\mathcal{F},\mathbb{P})$ such that for each $1 \leq j \leq n$ ,

(6.1) \begin{equation} \mathbb{P}\left[ \sup_{0 \leq s \leq t} |N^r_j(s)-s-B^r_j(s)| > (C + \theta)\log t + x \right] \leq Kt^{-\theta\lambda}e^{-\lambda x} \end{equation}

for every $t > 1$ , $x >0$ , and $\theta > 0$ , where $C,K,\lambda > 0$ are universal constants. Recall the definition of $\lambda^r(\cdot,\cdot)$ from (2.1). Consider the following equation:

(6.2) \begin{equation} X^r(t) = rx^r_0 + \sum_{j=1}^{n} \ell_jN^r_j\left(r\int_{0}^{t}\lambda^r\left(\frac{X^r(s)}{r},\ell_j\right)ds \right), \qquad t \geq 0. \end{equation}

By Theorem 6.4.1 of Ethier and Kurtz [Reference Ethier and Kurtz9], this equation has a unique solution $X^r=\{X^r(t),\; 0 \leq t < \infty \}$ on $(\Omega,\mathcal{F},\mathbb{P})$ , and the solution is a continuous-time Markov chain taking values in $\mathcal{I}^r$ with infinitesimal generator given by $Q^r$ . Then, $\overline{X}^r = \frac{1}{r}X^r$ satisfies

\begin{equation*} \overline{X}^r(t) = x^r_0 + \sum_{j=1}^{n} \frac{\ell_j}{r}N^r_j\left(r\int_{0}^{t}\lambda^r\left(\overline{X}^r(s),\ell_j\right)ds \right) \qquad \text{ for all } t \geq 0. \end{equation*}

Define $\nu^r(x) \;:\!=\; \sum_{j=1}^{n}\ell_j\epsilon^r(x,\ell_j)$ for $x \in \mathcal{I}$ , and $Y^r_j(t) \;:\!=\; N^r_j(t)-t $ for $1 \leq j \leq n$ and $t \geq 0$ . Then, for $t \geq 0$ ,

\begin{align*} \overline{X}^r(t) &= x^r_0 + \sum_{j=1}^{n} \frac{\ell_j}{r}\left[Y^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right) + r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right] \\[5pt] &= x^r_0 + \sum_{j=1}^{n} \frac{\ell_j}{r}\left[Y^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)- B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)\right] \\[5pt] &+ \sum_{j=1}^{n} \frac{\ell_j}{r}B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right) +\int_{0}^{t}\mu(\overline{X}^r(s))ds +\int_{0}^{t}\nu^r(\overline{X}^r(s))ds. \end{align*}

On the other hand, note that $Z^r(t) = \chi^r (t) + \frac{1}{\sqrt{r}}\int_{0}^{t}\gamma(Z^r(s))dL^r(s)$ for every $t \geq 0$ , where

(6.3) \begin{equation} \chi^r(t) \;:\!=\; x^r_0 + \int_{0}^{t} \mu(Z^r(s))ds + \sum_{j=1}^{n}\frac{\ell_j}{r}B^r_j\left(r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds\right), \quad t \geq 0. \end{equation}

Now, fix $T \geq 1$ . Unless indicated otherwise, the following inequalities and estimates hold $\mathbb{P}$ -a.s. for all $0 \leq t \leq T$ :

(6.4) \begin{equation} \begin{split} |\overline{X}^r(t)- \chi^r(t)| &\leq \underbrace{\frac{1}{r}\sum_{j=1}^{n} |\ell_j|\left|Y^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)- B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)\right|}_{I_1(t)} \\[5pt] &+ \underbrace{\frac{1}{r}\sum_{j=1}^{n}|\ell_j|\left|B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right) -B^r_j\left(r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds\right)\right|}_{I_2(t)}\\[5pt] &+ \underbrace{\left|\int_{0}^{t}\mu(\overline{X}^r(s))-\mu(Z^r(s))ds \right|}_{I_3(t)} +\underbrace{\left|\int_{0}^{t}\nu^r(\overline{X}^r(s))ds \right|}_{I_4(t)}. \end{split} \end{equation}

In the following, we estimate each element $I_1,\ldots, I_4$ using a different method. Define

(6.5) \begin{equation} M_{\beta} \;:\!=\; \sup_{1 \leq j \leq n} \sup_{x \in \mathcal{I}} \beta(x,\ell_j). \end{equation}

For $I_1$ we use (6.1). Note first that for $s \in [0,T]$ ,

(6.6) \begin{equation} r\lambda^r(\overline{X}^r(s),\ell_j) =r[\beta(\overline{X}^r(s),\ell_j) + \epsilon^r(\overline{X}^r(s),\ell_j)] \leq r[M_{\beta} +M_{\mathcal{I}}], \end{equation}

where $M_{\mathcal{I}}$ is defined in (3.2). Then, for $0 \leq t \leq T$ , $r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds \leq r[M_{\beta} +M_{\mathcal{I}}]T$ . By (6.1) and Lemma C.1, if we choose $v \;:\!=\; [M_{\beta} +M_{\mathcal{I}}]T$ , there exist a constant C(v) and nonnegative random variables $\{\eta^{r}_{j,1}\}_{j=1}^{n}$ such that for each $1 \leq j \leq n$ ,

\begin{equation*} \sup_{0 \leq u \leq rv} |Y^r_j(u)-B^r_j(u)| \leq C(v) \log r + \eta^r_{j,1} \end{equation*}

and

(6.7) \begin{equation} \mathbb{P}[\eta^{r}_{j,1} > x] \leq Kr^{-2}e^{-\lambda x} \qquad \text{ for every } x > 0. \end{equation}

Define the constants $C_{T,1} \;:\!=\; C([M_{\beta} +M_{\mathcal{I}}]T)$ and $K_{T,1}\;:\!=\; K$ . Then

(6.8) \begin{equation} \begin{split} I_1(t) &= \frac{1}{r}\sum_{j=1}^{n} |\ell_j|\left|Y^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)- B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right)\right| \\[5pt] &\leq \frac{1}{r}\sum_{j=1}^{n} |\ell_j|\sup_{0 \leq u \leq r(M_{\beta} +M_{\mathcal{I}})T} |Y^r_j(u)-B^r_j(u)| \\[5pt] &\leq \frac{1}{r}\sum_{j=1}^{n} |\ell_j|(C_{T,1}\log r + \eta^{r}_{j,1}). \end{split} \end{equation}

For $I_2$ , we use modulus-of-continuity estimates for Brownian motion. For $1 \leq j \leq n$ , define

(6.9) \begin{equation} \Lambda_{j}^r(z) \;:\!=\; \sup_{0 \leq u \leq rv} \sup_{0 \leq s \leq z \log r} |B^r_j(u+s)-B^r_j(u)|, \qquad 0 \leq z \leq rv, \end{equation}

(6.10) \begin{equation} \overline{\Lambda}^r_{j} \;:\!=\; \sup_{0 \leq z \leq rv} (z+1)^{-1/2}\Lambda^r_{j}(z). \end{equation}

By the proof of Theorem 11.3.1 in Ethier and Kurtz [Reference Ethier and Kurtz9], there exist nonnegative random variables $\{\eta^r_{j,2}\}_{j=1}^{n}$ and constants $C_{T,2}, K_{T,2} > 0$ depending only on $M_{\beta},M_{\mathcal{I}}$ and T such that for every $1 \leq j \leq n$ ,

(6.11) \begin{equation} \overline{\Lambda}^r_{j} \leq C_{T,2}\log r + \eta^r_{j,2} \qquad \mathbb{P}\text{-a.s.} \end{equation}

and

(6.12) \begin{equation} \mathbb{P}[\eta^r_{j,2} > x] \leq K_{T,2}r^{-2}\exp\!\left[-\frac{x^2}{18\log r}\right] \qquad \text{ for every } x > 0. \end{equation}

To derive these estimates, we observe from (3.11)–(3.17) in [Reference Ethier and Kurtz9] that these equations can apply to any Brownian motion; furthermore, $\overline{\beta}_lT$ in [Reference Ethier and Kurtz9] can be replaced by our v, and the constants $K_l$ and $C_l$ in [Reference Ethier and Kurtz9] can be taken to not depend on l. (In this proof, we have dropped a term from the estimate (3.17) in [Reference Ethier and Kurtz9] when writing (6.12), since this term would not improve our bounds.) Now, for each $1 \leq j \leq n$ , for every $0\leq t \leq T$ , we have

\begin{align*} &\left|B^r_j\left(r\int_{0}^{t}\lambda^r(\overline{X}^r(s),\ell_j)ds\right) -B^r_j\left(r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds\right)\right| \\[5pt] &\leq \Lambda^r_{j}\left(\frac{r}{\log r}\int_{0}^{t}|\lambda^r(\overline{X}^r(s),\ell_j) -\beta(Z^r(s),\ell_j) |ds\right), \end{align*}

where we used (6.6) and the fact that $r\int_{0}^{t}\beta(Z^r(s),\ell_j)ds \leq rv$ for $t \in [0,T]$ . By (6.10), for every $0 \leq z \leq rv$ , we have that $\Lambda^r_j(z) \leq \overline{\Lambda}^r_{j}(1+z)^{1/2}$ . Now, since $r \geq 8$ , we have $\log r \geq 2$ and

\begin{equation*} \frac{r}{\log r}\int_{0}^{t}|\lambda^r(\overline{X}^r(s),\ell_j) -\beta(Z^r(s),\ell_j) |ds \leq \frac{r}{\log r}2[M_{\beta} + M_{\mathcal{I}}]T \leq r[M_{\beta} + M_{\mathcal{I}}]T = rv. \end{equation*}

Therefore,

\begin{align*} &\Lambda^r_{j}\left(\frac{r}{\log r}\int_{0}^{t}|\lambda^r(\overline{X}^r(s),\ell_j) -\beta(Z^r(s),\ell_j) |ds\right) \\[5pt] &\leq \overline{\Lambda}^r_{j}\left(1 + \frac{r}{\log r}\int_{0}^{t}|\lambda^r(\overline{X}^r(s),\ell_j) -\beta(Z^r(s),\ell_j) |ds \right)^{1/2} \\[5pt] &\leq \overline{\Lambda}^r_{j}\left(1 + \frac{r}{\log r}\int_{0}^{t}\left(|\beta(\overline{X}^r(s),\ell_j) -\beta(Z^r(s),\ell_j) | + |\epsilon^r(\overline{X}^r(s),\ell_j)|\right)ds \right)^{1/2} \\[5pt] &\leq \overline{\Lambda}^r_{j}\left(1 + \frac{r}{\log r}\int_{0}^{t}A|\overline{X}^r(s) -Z^r(s)| + \frac{M_{\mathcal{I}}}{r}ds \right)^{1/2} \\[5pt] &\leq \left(C_{T,2}\log r + \eta^{r}_{j,2}\right)\left(1 + \frac{TM_{\mathcal{I}}}{\log r} + \frac{Ar}{\log r}\int_{0}^{t}|\overline{X}^r(s) -Z^r(s)|ds \right)^{1/2}, \end{align*}

where in the third inequality we used (3.2) and the fact that all $\beta(\cdot,\ell_j)$ are Lipschitz continuous with Lipschitz constant $A > 0$ , and in the last inequality we used (6.11). Hence,

(6.13) \begin{equation} I_2(t) \leq \frac{1}{r}\sum_{j=1}^{n}|\ell_j| \left(C_{T,2}\log r + \eta^{r}_{j,2}\right)\left(1 + \frac{TM_{\mathcal{I}}}{\log r} + \frac{Ar}{\log r}\int_{0}^{t}|\overline{X}^r(s) -Z^r(s)|ds \right)^{1/2}. \end{equation}

We estimate $I_3$ using the Lipschitz property of $\mu$ . The Lipschitz constant for $\mu$ will be called $A_\mu$ , and it can be taken as $A\sum_{j=1}^{n}|\ell_j|$ . Then, for $0 \leq t \leq T$ ,

(6.14) \begin{equation} I_3(t) \leq \int_{0}^{t}|\mu(\overline{X}^r(s))- \mu(Z^r(s))|ds \leq A_{\mu}\int_{0}^{t}|\overline{X}^r(s)- Z^r(s)|ds. \end{equation}

Finally, for $I_4$ we have

(6.15) \begin{equation} I_4(t) = \left|\int_{0}^{t}\nu^r(\overline{X}^r(s))ds \right| \leq \int_{0}^{t}|\nu^r(\overline{X}^r(s))|ds \leq \int_{0}^{t}\frac{M_{\mathcal{I}}}{r}\sum_{j=1}^{n}|\ell_j|ds \leq \frac{1}{r}\Big(TM_{\mathcal{I}}\sum_{j=1}^{n}|\ell_j|\Big). \end{equation}

For $s,t \in [0,T]$ , let

$$\zeta^r(s) \;:\!=\; \frac{r\left|\overline{X}^r(s)-Z^r(s)\right|}{\log r}$$

and

$$\overline{\zeta}^r(t) = \sup_{0 \leq s \leq t} \zeta^r(s)= \frac{r}{\log r} \left\lVert{\overline{X}^r-Z^r}\right\rVert_t.$$

Since T is fixed, let us write $\overline{\zeta}^r\;:\!=\; \overline{\zeta}^r(T)$ . Combining (6.4) with (6.8), (6.13), (6.14), and (6.15), for each $0 \leq t \leq T$ we obtain

\begin{align*} \frac{r|\overline{X}^r(t)-\chi^r(t)|}{\log r} &\leq \frac{1}{\log r}\sum_{j=1}^{n}|\ell_j| (C_{T,1} \log r + \eta^r_{j,1}) \\[5pt] &+\frac{1}{\log r}\sum_{j=1}^{n}|\ell_j| \left(C_{T,2}\log r + \eta^r_{j,2}\right)\left(1 + \frac{TM_{\mathcal{I}}}{\log r} + A\int_{0}^{t}\zeta^r(s)ds \right)^{1/2}\\[5pt] &+A_{\mu}\int_{0}^{t}\zeta^r(s)ds + \frac{1}{\log r}\left(TM_{\mathcal{I}}\sum_{j=1}^{n}|\ell_j|\right). \end{align*}

Hence, $\mathbb{P}$ -a.s. for each $0 \leq t \leq T$ ,

(6.16) \begin{equation} \frac{r|\overline{X}^r(t)-\chi^r(t)|}{\log r} \leq G^r + A_{\mu}\int_{0}^{t}\overline{\zeta}^r(s)ds, \end{equation}

where

\begin{equation*} G^r \;:\!=\; \sum_{j=1}^{n}|\ell_j| \left(C_{T,1} + \frac{TM_{\mathcal{I}}}{\log r}+\frac{\eta^r_{j,1}}{\log r}\right) +\sum_{j=1}^{n}|\ell_j|\Big(C_{T,2} + \frac{\eta^r_{j,2}}{\log r}\Big)\Bigg(1 + \frac{TM_{\mathcal{I}}}{\log r} + AT\overline{\zeta}^r\Bigg)^{1/2}. \end{equation*}

Notice that $G^r$ does not depend on t. Since the right-hand side of (6.16) is monotone in t, it follows that $\mathbb{P}$ -a.s. for all $t \in [0,T]$ ,

(6.17) \begin{equation} \sup_{0 \leq u \leq t}\frac{r|\overline{X}^r(u)-\chi^r(u)|}{\log r} = \frac{r}{\log r}\lVert{\overline{X}^r-\chi^r}\rVert_t \leq G^r + A_{\mu}\int_{0}^{t}\overline{\zeta}^r(s)ds. \end{equation}

Now consider the Skorokhod map $\Gamma_{0,a}$ on [0, a] defined in Section B. From the discussion there, we observe that $\mathbb{P}$ -a.s. $\Gamma_{0,a}(\chi^r) =Z^r$ . Also, we note that since $\overline{X}^r(t) \in \mathcal{I}$ for every $t \geq 0$ , we have $\Gamma_{0,a}(\overline{X}^r)=\overline{X}^r$ . By the Lipschitz continuity of the Skorokhod map (B.4), $\mathbb{P}$ -a.s. for all $0 \leq t \leq T$ ,

\begin{equation*} \lVert\overline{X}^r-Z^r\rVert_t = \lVert\Gamma_{0,a}(\overline{X}^r)-\Gamma_{0,a}(\chi^r)\rVert_t \leq 4 \lVert\overline{X}^r-\chi^r\rVert_t. \end{equation*}

Consequently, $\mathbb{P}$ -a.s. for every $0 \leq t \leq T$ ,

\begin{equation*} \overline{\zeta}^r(t) = \frac{r}{\log r}\lVert{\overline{X}^r-Z^r}\rVert_t \leq \frac{4r}{\log r}\lVert{\overline{X}^r-\chi^r}\rVert_t \leq 4G^r + 4A_{\mu}\int_{0}^{t}\overline{\zeta}^r(s)ds. \end{equation*}

Then, by Gronwall’s inequality, $\overline{\zeta}^r(t) \leq e^{4A_{\mu}t}4G^r$ for all $t \in [0,T]$ . In particular, $\mathbb{P}$ -a.s.,

\begin{align*} &\overline{\zeta}^r \leq e^{4A_{\mu}T}4G^r = 4e^{4A_{\mu}T}\sum_{j=1}^{n}|\ell_j|\left(C_{T,1} + \frac{TM_{\mathcal{I}}}{\log r}+\frac{\eta^r_{j,1}}{\log r}\right) \\[5pt] &+4e^{4A_{\mu}T}(AT)^{1/2}\left(\frac{1}{AT}+ \frac{M_{\mathcal{I}}}{A\log r} + \overline{\zeta}^r\right)^{1/2}\sum_{j=1}^{n}|\ell_j|\left(C_{T,2} + \frac{\eta^r_{j,2}}{\log r}\right). \end{align*}

This is an inequality of the type $\gamma \leq a + (c+ \gamma)^{1/2}b$ , which implies $\gamma \leq c+ 2a + b^2$ . Thus, $\mathbb{P}$ -a.s.,

\begin{align*} \overline{\zeta}^r &\leq \frac{1}{AT}+ \frac{M_{\mathcal{I}}}{A\log r} + 8e^{4A_{\mu}T}\sum_{j=1}^{n}|\ell_j|\left(C_{T,1} + \frac{TM_{\mathcal{I}}}{\log r}+\frac{\eta^r_{j,1}}{\log r}\right) \\[5pt] &+ 16e^{8A_{\mu}T}AT\left(\sum_{j=1}^{n}|\ell_j|\left(C_{T,2} + \frac{\eta^r_{j,2}}{\log r}\right)\right)^2. \end{align*}

Define $\lVert\ell\rVert_{\infty} = \sup_{1 \leq j \leq n}|\ell_j|$ . Since $\log r \geq 2$ , we have $\mathbb{P}$ -a.s.