Proof of Corollary 1. Let \(h: \mathbb{R}^d \to \mathbb{R}^d\) be a truncation function, and let \((B^h, C, \nu)\) be the characteristics of \(Y\) associated with \(h\). Denote \[B^h_t = \int_0^t b^h_s \,ds,\] where \(b^h\) is an \(\mathbb{R}^d\)-valued predictable process. By 7 and 13 , it is easy to check that \((b^h, c, \kappa)\) satisfy 9 . Then, Theorem 1 yields measurable functions \(\widehat{b}^h: \mathbb{R}_+ \times \mathcal{E} \to \mathbb{R}^d\), \(\widehat{c}: \mathbb{R}_+ \times \mathcal{E} \to \mathbb{S}_+^d\), and a Lévy transition kernel \(\widehat{\kappa}\) from \(\mathbb{R}_+ \times \mathcal{E}\) to \(\mathbb{R}^d\) such that \((\widehat{c}, \widehat{\kappa})\) and \((c, \kappa)\) satisfy 10 , thus 14 by approximation,⁴ and for Lebesgue-a.e.\(t \geq 0\), \[\widehat{b}^h(t, Z_t) = \mathbb{E}[b^h_t \,|\, Z_t].\] If we take a closer look at the construction of \(\widehat{\kappa}\), which is based on [5], Lemma 2.5 (with an obvious extension to \(\mathcal{E}\)-valued processes and transition kernels from \(\mathbb{R}_+ \times \mathcal{E}\) to \(\mathbb{R}^d\)), one may require \(\widehat{\kappa}\) to satisfy the following property due to 13 : \[\int_{\mathbb{R}^d} |\xi| \land |\xi|^2 \,\widehat{\kappa}(t, z, d\xi) < \infty,\quad \forall\, t \geq 0,\, z \in \mathcal{E}.\] Moreover, Theorem 1 yields a filtered probability space \((\widehat{\Omega}, \widehat{\mathcal{F}}, (\widehat{\mathcal{F}}_t)_{t \geq 0}, \widehat{\mathbb{P}})\), with right-continuous filtration, that supports an \(\mathcal{E}\)-valued random variable \(\widehat{Z}_0\) and an \(\mathbb{R}^d\)-valued Itô semimartingale \(\widehat{Y}\) with initial value \(0\) and characteristics \((\widehat{B}^h, \widehat{C}, \widehat{\nu})\) associated with \(h\), such that the one-dimensional marginal laws of \(\widehat{Z} = \Phi(\widehat{Z}_0, \widehat{Y})\) agree with \(Z\). Here \(\widehat{C}\) and \(\widehat{\nu}\) are defined as in 11 , and \[\widehat{B}^h_t = \int_0^t \widehat{b}^h(s, \widehat{Z}_s) \,ds.\]

It only remains to show \(\widehat{Y}\) is a special semimartingale with the desired canonical characteristics. From 13 , 14 and the fact that \(\widehat{Z}\) and \(Z\) have the same one-dimensional marginal laws, we get \[\widehat{\mathbb{E}}\biggl[\int_0^t \int_{\mathbb{R}^d} |\xi| \land |\xi|^2 \,\widehat{\kappa}(s, \widehat{Z}_s, d\xi) \,ds\biggr] = \mathbb{E}\biggl[\int_0^t \int_{\mathbb{R}^d} |\xi| \land |\xi|^2 \,\widehat{\kappa}(s, Z_s, d\xi) \,ds\biggr] < \infty,\quad \forall\, t \geq 0.\] Thus, [9], Proposition II.2.29(a) implies that \(\widehat{Y}\) is a special semimartingale. We define the measurable function \(\widehat{b}: \mathbb{R}_+ \times \mathcal{E} \to \mathbb{R}^d\) via \[\widehat{b}(t, z) \mathrel{\vcenter{:}}= \widehat{b}^h(t, z) + \int_{\mathbb{R}^d} (\xi - h(\xi)) \,\widehat{\kappa}(t, z, d\xi),\quad t \geq 0,\, z \in \mathcal{E},\] which we will show to be independent of the choice of \(h\). Then by 7 , the first canonical characteristic \(\widehat{B}\) of \(\widehat{Y}\) is given by \[\begin{align} \widehat{B}_t &= \widehat{B}^h_t + \int_{[0, t] \times \mathbb{R}^d} (\xi - h(\xi)) \,\widehat{\nu}(ds, d\xi)\\ &= \int_0^t \widehat{b}^h(s, \widehat{Z}_s) \,ds + \int_0^t \int_{\mathbb{R}^d} (\xi - h(\xi)) \,\widehat{\kappa}(s, \widehat{Z}_s, d\xi) \,ds = \int_0^t \widehat{b}(s, \widehat{Z}_s) \,ds. \end{align}\] Another application of 7 shows that for Lebesgue-a.e.\(t \geq 0\), \[\begin{align} \widehat{b}(t, Z_t) &= \widehat{b}^h(t, Z_t) + \int_{\mathbb{R}^d} (\xi - h(\xi)) \,\widehat{\kappa}(t, Z_t, d\xi)\\ &= \mathbb{E}[b^h_t \,|\, Z_t] + \mathbb{E}\biggl[\int_{\mathbb{R}^d} (\xi - h(\xi)) \,\kappa_t(d\xi) \,\bigg|\, Z_t\biggr] = \mathbb{E}[b_t \,|\, Z_t]. \end{align}\] This finishes the proof. ◻

Proof. The proof is verbatim the same as that of [1], Theorem 4.3 (pp.-1602, including all the related lemmas and cited results). We only need to replace their canonical space \(\Omega^{\mathcal{E}, d}\) by ours. The key facts are that \(\Omega^{\mathcal{E}, \mathcal{X}} = \mathcal{E} \times \mathcal{X}\) is a Polish space and \(\mathcal{X}\) is \(\Delta\)-stable, which guarantee that the same proof works. ◻

Proof. The proof of (i) (resp.(ii)) is exactly the same as in Step 2 (resp.Step 5) of the proof of [1], Theorem 7.1. In their proof, they have an explicit formula for each \(T_i\) in terms of \(U\), but the key point is the \(\sigma(U)\)-measurability of each \(T_i\). ◻

Proof. The proof is almost the same as in the proof of [1], Proposition 6.1, with \(C^{\mathcal{E}}\) replaced by \(D^{\mathcal{E}}\). Only two places need slight changes. First, since \(Z^\infty\) is càdlàg, (ii) implies that \(Z^m_t\) converges to \(Z^\infty_t\) in law for all but countably many \(t \geq 0\). Together with (i) and the right-continuity of the sample paths of \(Z^\infty\), we still obtain that \(Z^\infty_t\) and \(Z^1_t\) have the same law for all \(t \geq 0\). Secondly, in the proof of (v), one needs to check that the map \(D^{\mathcal{E}} \ni z \mapsto \int_0^{\cdot \land k} f^k(s, z(s)) \,ds \in C^d_0\) is continuous, where \(f^k: [0, k] \times \mathcal{E} \to \mathbb{R}^d\) is a bounded continuous function. Since convergence in the Skorokhod space implies pointwise convergence almost everywhere, the dominated convergence theorem then finishes the proof. ◻

Proof. By Lemma 1, \(\Pi^m\) is indeed an extended partition, and the law of \(Z_t\) under \(\mathbb{P}\) agrees with the law of \(Z_t\) under \(\mathbb{P}^{\otimes \Pi^m}\) for each \(t \geq 0\) and \(m \in \mathbb{N}^*\). By the definition of \(\widehat{a}\) and Jensen’s inequality, we have \(\mathbb{E}[\int_0^t |\widehat{a}(s, Z_s)| \,ds] < \infty\), \(\forall\, t > 0\), thus \(\overline{A}\) is well-defined under \(\mathbb{P}\) and all \(\mathbb{P}^{\otimes \Pi^m}\). This implies that the collection \((\widehat{a}(\cdot, Z_\cdot), \mathrm{Leb}([0, t]) \otimes \mathbb{P}^{\otimes \Pi^m})_{m \in \mathbb{N}^*}\) is uniformly integrable, for every \(t > 0\). The rest of the proof follows exactly the same as in Step 7 of the proof of [1], Theorem 7.1. ◻

Proof. First we notice the following fact. Since \(t \mapsto \mu_t\) is continuous in the sense of weak convergence, we know that \(t \mapsto \mu_t(\mathbb{R}^d)\) is a continuous function. For \(0 \leq s < t\), by assumption \(\mu_t - \mu_s\) is a positive measure, so we have \[\label{eq:mu95cts} 0 \leq \mu_t(A) - \mu_s(A) \leq \mu_t(\mathbb{R}^d) - \mu_s(\mathbb{R}^d),\quad A \in \mathcal{B}(\mathbb{R}^d).\tag{17}\] This implies that \(t \mapsto \mu_t(A)\) is continuous for all \(A \in \mathcal{B}(\mathbb{R}^d)\).

For \(0 \leq s \leq t \leq \infty\) and \(A \in \mathcal{B}(\mathbb{R}^d)\), we define a set function \[\label{eq:nu95mu952} \nu([s, t) \times A) \mathrel{\vcenter{:}}= \mu_t(A) - \mu_s(A),\tag{18}\] where we use the convention \(\mu_\infty(A) \mathrel{\vcenter{:}}= \lim_{t \to \infty} \mu_t(A)\), which is well-defined by monotonicity (but can be infinite). We also define a collection of subsets of \(\mathbb{R}_+ \times \mathbb{R}^d\) via \[\mathcal{A} \mathrel{\vcenter{:}}= \Biggl\{\bigcup_{i = 1}^n ([s_i, t_i) \times A_i): 0 \leq s_i \leq t_i \leq \infty,\, A_i \in \mathcal{B}(\mathbb{R}^d),\, 1 \leq i \leq n,\, n \in \mathbb{N}\Biggr\},\] which is an algebra that generates \(\mathcal{B}(\mathbb{R}_+ \times \mathbb{R}^d)\). For each \(E \in \mathcal{A}\), we can write it in the form \(E = \bigcup_{i = 1}^n ([s_i, t_i) \times A_i)\), where the sets \([s_i, t_i) \times A_i\) are disjoint. Then, we define \[\nu(E) \mathrel{\vcenter{:}}= \sum_{i=1}^n \nu([s_i, t_i) \times A_i).\] It is straightforward to verify that \(\nu(E)\) is well-defined, i.e.it does not depend on how \(E\) is partitioned. So far we have defined a set function \(\nu\) on \(\mathcal{A}\), which satisfies \(\nu(\varnothing) = 0\) and is finitely additive. If we manage to show \(\nu\) is \(\sigma\)-additive on \(\mathcal{A}\), then by Carathéodory’s extension theorem, we could uniquely extend \(\nu\) to a measure on \(\mathcal{B}(\mathbb{R}_+ \times \mathbb{R}^d)\), and 16 follows from 18 and the continuity of \(\mu\). This would finish the proof.

It only remains to prove \(\nu\) is \(\sigma\)-additive on \(\mathcal{A}\). To prove this, it suffices to show the following statement: if \([s, t) \times A = \bigcup_{i=1}^\infty ([s_i, t_i) \times A_i)\), where \(0 \leq s < t \leq \infty, 0 \leq s_i < t_i \leq \infty, A, A_i \in \mathcal{B}(\mathbb{R}^d)\), and the sets \([s_i, t_i) \times A_i\) are disjoint, then \[\nu([s, t) \times A) = \sum_{i=1}^\infty \nu([s_i, t_i) \times A_i).\] One direction of inequality is simple. For each \(n \in \mathbb{N}^*\), we have \([s, t) \times A \supseteq \bigcup_{i=1}^n ([s_i, t_i) \times A_i)\). By the finiteness, it is easy to check that \(\nu([s, t) \times A) \geq \sum_{i=1}^n \nu([s_i, t_i) \times A_i)\). Sending \(n \to \infty\) proves the “\(\geq\)” direction. Conversely, let us first assume that \(s > 0\) and \(t < \infty\). Pick any \(\varepsilon > 0\). Using 18 and the continuity of \(t \mapsto \mu_t(A)\), one can find \(t^\prime \in (s, t)\) such that \(\nu([s, t^\prime) \times A) > \nu([s, t) \times A) - \varepsilon/4\). Next, since \(\mu_{t^\prime} - \mu_s\) is a finite positive Borel measure on \(\mathbb{R}^d\), by the regularity one can find a compact set \(K \subseteq A\) such that \(\nu([s, t^\prime) \times K) > \nu([s, t^\prime) \times A) - \varepsilon/4\). Combining these two steps gives us \[\label{eq:reg} \nu([s, t^\prime) \times K) > \nu([s, t) \times A) - \frac{\varepsilon}{2}.\tag{19}\] Similarly, for each \(i \in \mathbb{N}^*\), one can find \(s_i^\prime \in (0, s_i)\) and an open set \(U_i \supseteq A_i\) such that \[\label{eq:reg952} \nu([s_i^\prime, t_i) \times U_i) < \nu([s_i, t_i) \times A_i) + \frac{\varepsilon}{2^{i+1}}.\tag{20}\] Note that \([s, t^\prime] \times K\) is a compact set, and we have \([s, t^\prime] \times K \subseteq \bigcup_{i=1}^\infty ((s_i^\prime, t_i) \times U_i)\). Thus, we can extract a finite subcover \([s, t^\prime] \times K \subseteq \bigcup_{i=1}^n ((s_i^\prime, t_i) \times U_i)\), which leads to \([s, t^\prime) \times K \subseteq \bigcup_{i=1}^n ([s_i^\prime, t_i) \times U_i)\). By the finiteness and 19 , 20 , we obtain the estimate \[\begin{align} \nu([s, t) \times A) &< \nu([s, t^\prime) \times K) + \frac{\varepsilon}{2} \leq \sum_{i=1}^n \nu([s_i^\prime, t_i) \times U_i) + \frac{\varepsilon}{2} \leq \sum_{i=1}^\infty \nu([s_i^\prime, t_i) \times U_i) + \frac{\varepsilon}{2}\\ &\leq \sum_{i=1}^\infty \biggl(\nu([s_i, t_i) \times A_i) + \frac{\varepsilon}{2^{i+1}}\biggr) + \frac{\varepsilon}{2} = \sum_{i=1}^\infty \nu([s_i, t_i) \times A_i) + \varepsilon. \end{align}\] Sending \(\varepsilon \to 0\) finishes the proof for \(s > 0\) and \(t < \infty\). Finally, when \(s = 0\), for those \(i\) with \(s_i = 0\), the interval \([0, t_i)\) is relatively open in \(\mathbb{R}_+\), so we may take \(s_i^\prime = s_i = 0\), and the interval \([0, t_i)\) serves our purpose. When \(t = \infty\), we simply partition \([s, \infty)\) into countably many subintervals \([s, t^j)\) with all \(t^j < \infty\). We then apply what we just proved to each \([s, t^j) \times A\) and sum up the results. ◻

Proof. For \(n \in \mathbb{N}^*\), define \(\mathcal{T}_n: D^d_0 \to [0, \infty]\) via \(\mathcal{T}_n(y) \mathrel{\vcenter{:}}= \inf\{t > 0: |y(t)| \geq n\}\), \(y \in D^d_0\), then define \(\Phi_n: D^d_0 \to D^d_0\) via \(\Phi_n(y) \mathrel{\vcenter{:}}= y^{\mathcal{T}_n(y)}\), \(y \in D^d_0\). Note that both \(\mathcal{T}_n\) and \(\Phi_n\) are measurable maps. Assumption (i) tells us that \(\Psi(X)\) is adapted to \((\mathcal{F}_t)_{t \geq 0}\), and \(\Psi(X^*)\) is adapted to \((\mathcal{F}^*_t)_{t \geq 0}\). Thus, \(\tau_n \mathrel{\vcenter{:}}= \mathcal{T}_n(\Psi(X))\) is an \((\mathcal{F}_t)_{t \geq 0}\)-stopping time, and \(\tau^*_n \mathrel{\vcenter{:}}= \mathcal{T}_n(\Psi(X^*))\) is an \((\mathcal{F}^*_t)_{t \geq 0}\)-stopping time. By assumption (ii), we have that \(|\Phi_n \circ \Psi(x)| \leq n + M\) for all \(x \in D^{\mathcal{E}}\), so both \(\Psi(X)^{\tau_n} = \Phi_n \circ \Psi(X)\) and \(\Psi(X^*)^{\tau^*_n} = \Phi_n \circ \Psi(X^*)\) are bounded processes. It is easy to check that \(\Phi_n \circ \Psi\) satisfies (i) and (ii). Therefore, it suffices to prove the lemma for the case where \(\Psi(x)(t)\) is uniformly bounded in \(x \in D^{\mathcal{E}}\) and \(t \geq 0\). In particular, by the continuity of \(F\), in this case both processes \(t \mapsto F(\Psi(X)_t)\) and \(t \mapsto F(\Psi(X^*)_t)\) are bounded.

Now assume that \(t \mapsto F(\Psi(X)_t)\) is a bounded martingale on \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})\), and we prove \(t \mapsto F(\Psi(X^*)_t)\) is a bounded martingale on \((\Omega^*, \mathcal{F}^*, (\mathcal{F}^*_t)_{t \geq 0}, \mathbb{P}^*)\). Let \(0 \leq s < t\). Our goal is to show \[\mathbb{E}^*[F(\Psi(X^*)_t) \boldsymbol{1}_F] = \mathbb{E}^*[F(\Psi(X^*)_s) \boldsymbol{1}_F],\quad \forall\, F \in \mathcal{F}^*_s.\] By Dynkin’s \(\pi\)-\(\lambda\) theorem, it suffices to take \(F\) of the form \(\{X^*_{s_1} \in A_1, ..., X^*_{s_n} \in A_n\}\), where \(0 \leq s_1 < \cdots < s_n \leq s\), \(A_1, ..., A_n \in \mathcal{B}(\mathcal{E})\) and \(n \in \mathbb{N}^*\). Then, by the definition of \(\mathbb{P^*}\) and the martingale property of \(\Psi(X)\) on \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})\), it follows that \[\begin{align} \mathbb{E}^*\bigl[F(\Psi(X^*)_t) \boldsymbol{1}_{\{X^*_{s_1} \in A_1, ..., X^*_{s_n} \in A_n\}}\bigr] &= \mathbb{E}\bigl[F(\Psi(X)_t) \boldsymbol{1}_{\{X_{s_1} \in A_1, ..., X_{s_n} \in A_n\}}\bigr]\\ &= \mathbb{E}\bigl[F(\Psi(X)_s) \boldsymbol{1}_{\{X_{s_1} \in A_1, ..., X_{s_n} \in A_n\}}\bigr]\\ &= \mathbb{E}^*\bigl[F(\Psi(X^*)_s) \boldsymbol{1}_{\{X^*_{s_1} \in A_1, ..., X^*_{s_n} \in A_n\}}\bigr], \end{align}\] which finishes the proof. ◻

Proof. First we take any subsequence \((m_k)_{k \in \mathbb{N^*}}\) of \(\mathbb{N}^*\). Consider the sequence of probability measures \((\mathbb{P}^{m_k} \circ (Y^{m_k}, Z^{m_k}, Z^{m_k}, f(Z^{m_k}))^{-1})_{k \in \mathbb{N}^*}\) on \(\mathcal{E}_1 \times \mathcal{E}_2 \times \mathcal{E}_2 \times \mathcal{E}_3\). This is a tight sequence, since it is a coupling of two tight sequences of probability measures on \(\mathcal{E}_1 \times \mathcal{E}_2\) and \(\mathcal{E}_2 \times \mathcal{E}_3\). Then, there exists a further subsequence \((m_{k_l})_{l \in \mathbb{N^*}}\) of \((m_k)_{k \in \mathbb{N^*}}\) and a probability measure \(\mu\) on \(\mathcal{E}_1 \times \mathcal{E}_2 \times \mathcal{E}_2 \times \mathcal{E}_3\), such that \(\mathbb{P}^{m_{k_l}} \circ (Y^{m_{k_l}}, Z^{m_{k_l}}, Z^{m_{k_l}}, f(Z^{m_{k_l}}))^{-1} \Rightarrow \mu\). Let \(\mu_{12}(dx_1, dx_2)\), \(\mu_{34}(dx_3, dx_4)\) denote the marginals of \(\mu\) on the first and last two coordinates, and let \(\mu(dx_1, dx_2, dx_3, dx_4) = \mu_{34|12}(x_1, x_2, dx_3, dx_4) \mu_{12}(dx_1, dx_2)\) be the disintegration. We know that \(\mu_{12} = \mathbb{P}^\infty \circ (Y^\infty, Z^\infty)^{-1}\) and \(\mu_{34} = \mathbb{P}^\infty \circ (Z^\infty, f(Z^\infty))^{-1}\). We also know from the Portmanteau theorem that \[\mu(\{x_2 = x_3\}) \geq \limsup_{l \to \infty} \mathbb{P}^{m_{k_l}}(Z^{m_{k_l}} = Z^{m_{k_l}}) = 1.\] This implies that \(\mu_{34|12}(x_1, x_2, dx_3, dx_4) = \delta_{(x_2, f(x_2))}(dx_3, dx_4)\), thus we conclude that \(\mu = \mathbb{P}^\infty \circ (Y^\infty, Z^\infty, Z^\infty, f(Z^\infty))^{-1}\).

So far we have proved that for any subsequence of \((\mathbb{P}^m \circ (Y^m, Z^m, f(Z^m))^{-1})\), there exists a further subsequence that converges weakly to \(\mathbb{P}^\infty \circ (Y^\infty, Z^\infty, f(Z^\infty))^{-1}\). This implies the weak convergence of the whole sequence, and the proof is complete. ◻

Proof of Theorem 1. The existence of \(\widehat{b}\), \(\widehat{c}\) and \(\widehat{\kappa}\) satisfying 10 follows from [1], Proposition 5.1 and [5], Lemma 2.5 (which obviously extends to \(\mathcal{E}\)-valued processes and transition kernels from \(\mathbb{R}_+ \times \mathcal{E}\) to \(\mathbb{R}^d\)). Also, without loss of generality, we may assume that \(h\) is continuous. Otherwise, we can take a continuous truncation function \(\widetilde{h}\) and prove the theorem. With back-and-forth applications of 6 , we first compute the characteristics of \(Y\) associated with \(\widetilde{h}\), apply the theorem with \(\widetilde{h}\), then compute the characteristics of \(\widehat{Y}\) associated with \(h\). This argument is similar to the proof of Corollary 1.

Step 1: Canonical space and processes.We define the measure-valued process \(M\) on \(\Omega\) via \[M_t(A) \mathrel{\vcenter{:}}= \int_{[0, t] \times A} 1 \land |\xi|^2 \,\nu(ds, d\xi) = \int_0^t \int_A 1 \land |\xi|^2 \,\kappa_s(d\xi) \,ds,\quad t \geq 0,\, A \in \mathcal{B}(\mathbb{R}^d).\] Then, the random object \((Z_0, Y, B, C, M)\) takes values in \(\mathcal{E} \times D^d_0 \times C^d_0 \times C^{d^2}_0 \times C^{\mathcal{M}_+, d}_{0, \mathrm{i}}\) \(\mathbb{P}\)-a.s. In order to utilize the approximation results developed in [1], we need to use a randomized discretization of time, which leads to an extra dimension. Thus, we define our canonical space as \(\Omega^* \mathrel{\vcenter{:}}= [0, 1] \times \mathcal{E} \times D^d_0 \times C^d_0 \times C^{d^2}_0 \times C^{\mathcal{M}_+, d}_{0, \mathrm{i}}\). By viewing \(\mathcal{E}^* \mathrel{\vcenter{:}}= [0, 1] \times \mathcal{E}\) as a new Polish space, and noticing \(\mathcal{X}^* \mathrel{\vcenter{:}}= D^d_0 \times C^d_0 \times C^{d^2}_0 \times C^{\mathcal{M}_+, d}_{0, \mathrm{i}}\) is a \(\Delta\)-stable closed subset of \(D^{\mathcal{E}^{*\prime}}_0\) with \(\mathcal{E}^{*\prime} = \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}^{d^2} \times \mathcal{M}_+(\mathbb{R}^d)\), one can write \(\Omega^* = \Omega^{\mathcal{E}^*, \mathcal{X}^*} = \mathcal{E}^* \times \mathcal{X}^*\), so all the results established in Section 3 apply to \(\Omega^*\). The generic element of \(\Omega^*\) is denoted by \(\omega = (u, \varepsilon, \eta, \beta, \gamma, \mu)\), and the projections are denoted by \[U^*(\omega) \mathrel{\vcenter{:}}= u,\quad Z_0^*(\omega) \mathrel{\vcenter{:}}= \varepsilon,\quad Y^*(\omega) \mathrel{\vcenter{:}}= \eta,\quad B^*(\omega) \mathrel{\vcenter{:}}= \beta,\quad C^*(\omega) \mathrel{\vcenter{:}}= \gamma,\quad M^*(\omega) \mathrel{\vcenter{:}}= \mu.\] We also write \(x = (\eta, \beta, \gamma, \mu)\) and \(X = (Y^*, B^*, C^*, M^*)\). Let \(\mathcal{F}^* \mathrel{\vcenter{:}}= \sigma(U^*, Z_0^*, X)\) and \(\mathcal{F}^*_t \mathrel{\vcenter{:}}= \sigma(U^*, Z_0^*, X^t)\) for \(t \geq 0\). Denote \(\mathbb{F}^* = (\mathcal{F}^*_t)_{t \geq 0}\) and let \(\widetilde{\mathbb{F}}^* = (\widetilde{\mathcal{F}}^*_t)_{t \geq 0}\) be the right-continuous regularization of \(\mathbb{F}^*\), i.e.\(\widetilde{\mathcal{F}}^*_t \mathrel{\vcenter{:}}= \bigcap_{s > t} \mathcal{F}^*_s\), \(t \geq 0\).⁷ Unless otherwise stated, we always refer to the natural filtration \(\mathbb{F}^*\). When we work with characteristics, we will explicitly mention \(\widetilde{\mathbb{F}}^*\). We define a probability measure \(\mathbb{Q}\) on \(\Omega^*\), which is the product of the Lebesgue measure on \([0, 1]\) and the law of \((Z_0, Y, B, C, M)\) on \(\mathcal{E} \times D^d_0 \times C^d_0 \times C^{d^2}_0 \times C^{\mathcal{M}_+, d}_{0, \mathrm{i}}\) under \(\mathbb{P}\), i.e. \[\mathbb{Q} \mathrel{\vcenter{:}}= \mathrm{Leb}([0, 1]) \otimes (\mathbb{P} \circ (Z_0, Y, B, C, M)^{-1}).\] Then under \(\mathbb{Q}\), we know that \(U^* \sim \mathrm{Unif}([0, 1])\), \((Z_0^*, Y^*, B^*, C^*, M^*)\) has the same joint law as \((Z_0, Y, B, C, M)\), and it is independent of \(U^*\).

According to Lemma 5, for each \(\omega \in \Omega^*\), there exists a Borel measure \(\lambda^*(\omega; \cdot)\) on \(\mathbb{R}_+ \times \mathbb{R}^d\) such that⁸ \[\lambda^*(\omega; [0, t] \times A) = M^*_t(\omega; A),\quad \forall\, t \geq 0,\, A \in \mathcal{B}(\mathbb{R}^d).\] By Dynkin’s \(\pi\)-\(\lambda\) theorem, it is easy to see that \(\lambda^*\) is a random measure, i.e.\(\lambda^*(\omega; E)\) is measurable in \(\omega\) for each fixed \(E \in \mathcal{B}(\mathbb{R}_+ \times \mathbb{R}^d)\). Moreover, for each \(0 \leq r < s\), \(F \in \widetilde{\mathcal{F}}^*_r\), and \(A \in \mathcal{B}(\mathbb{R}^d)\), let \(W(\omega, u, \xi) = \boldsymbol{1}_{F \times (r, s] \times A}(\omega, u, \xi)\). Then, the process \[(W * \lambda^*)_t = \boldsymbol{1}_F (M^*_{s \land t}(A) - M^*_{r \land t}(A))\] is continuous (recall 17 ) and adapted to \(\widetilde{\mathbb{F}}^*\), thus predictable. By another application of Dynkin’s \(\pi\)-\(\lambda\) theorem, this suffices to show that \(\lambda^*\) is a predictable random measure (with respect to \(\widetilde{\mathbb{F}}^*\)). Then, we define \(\nu^*(ds, d\xi) \mathrel{\vcenter{:}}= \boldsymbol{1}_{\{\xi \neq 0\}} (1 \land |\xi|^2)^{-1} \lambda^*(ds, d\xi)\), which is again a predictable random measure. It follows that for every measurable function \(f: \mathbb{R}^d \to \mathbb{R}\) satisfying \(|f(\xi)| \leq C (1 \land |\xi|^2)\), \(\forall\, \xi \in \mathbb{R}^d\), for some constant \(C > 0\), \[\label{eq:nu95M} (f * \nu^*)_t = \int_{\mathbb{R}^d} \frac{f(\xi)}{1 \land |\xi|^2} \,M^*_t(d\xi),\quad t \geq 0,\tag{21}\] and this process is continuous as \(|(f * \nu^*)_t - (f * \nu^*)_s| \leq C |M^*_t(\mathbb{R}^d) - M^*_s(\mathbb{R}^d)|\).

Define the process \(Y^*(h) \mathrel{\vcenter{:}}= Y^* - \sum_{s \leq \cdot} (\Delta Y^*_s - h(\Delta Y^*_s))\), which has bounded jumps as \(\Delta Y^*(h) = h(\Delta Y^*)\). Define the process \(\widetilde{C}^* \mathrel{\vcenter{:}}= C^* + (hh^\top) * \nu^*\).⁹ Let \(\mu^{Y^*}(dt, d\xi) \mathrel{\vcenter{:}}= \sum_{s > 0} \boldsymbol{1}_{\{\Delta Y^*_s \neq 0\}} \delta_{(s, \Delta Y^*_s)}(dt, d\xi)\) denote the integer-valued random measure associated with the jumps of \(Y^*\). We may also define their counterparts \(Y(h)\), \(\widetilde{C}\) and \(\mu^Y\) on the original probability space \(\Omega\). Then, using Lemma 6 and 21 , one can show the following processes are local martingales on \((\Omega^*, \mathcal{F}^*, \mathbb{F}^*, \mathbb{Q})\):

1. \(Y^*(h) - B^*\),

2. \((Y^*(h) - B^*) (Y^*(h) - B^*)^\top - \widetilde{C}^*\),

3. \(f * \mu^{Y^*} - f * \nu^*\), where \(f: \mathbb{R}^d \to \mathbb{R}\) is measurable and satisfies \(|f(\xi)| \leq C (1 \land |\xi|^2)\), \(\forall\, \xi \in \mathbb{R}^d\), for some constant \(C > 0\).

Since all these processes are càdlàg, it is easy to see that they are local martingales with respect to the right-continuous regularized filtration \(\widetilde{\mathbb{F}}^*\). We also note that \(C^*_t - C^*_s\) takes values in \(\mathbb{S}^d_+\) for all \(0 \leq s \leq t\), \(\mathbb{Q}\)-a.s. Thus, according to [9], Theorem II.2.21, \(Y^*\) is a semimartingale with characteristics triplet \((B^*, C^*, \nu^*)\) (associated with \(h\)) on the filtered probability space \((\Omega^*, \mathcal{F}^*, \widetilde{\mathbb{F}}^*, \mathbb{Q})\).

Next, we take a sequence of functions \((f_k)_{k \in \mathbb{N}^*}\) which is a class \(C_1(\mathbb{R}^d)\) (recall Definition 11 and Remark 11). This countable collection \((f_k)\) is convergence determining, thus measure determining. Without loss of generality, we may include functions \((h_i h_j)_{i, j=1}^d\) to the sequence \((f_k)\) and keep the same notation, where \(h_i\) is the \(i\)-th component of \(h\). For each \(i, j = 1, ..., d\), let \(k(i, j)\) be the index such that \(h_i h_j = f_{k(i, j)}\). Although \(h_i h_j\) does not vanish around \(0\), it is a continuous function satisfying \(|h_i h_j| \leq C(1 \land \lvert \cdot \rvert^2)\) for some constant \(C > 0\). Also, adding a finite number of functions does no harm to our following arguments. We define the process \[\label{eq:G42} G^*_{k, t} \mathrel{\vcenter{:}}= (f_k * \nu^*)_t = \int_{\mathbb{R}^d} \frac{f_k(\xi)}{1 \land |\xi|^2} \,M^*_t(d\xi).\tag{22}\] We also define its counterpart \(G_k\) on the original space \(\Omega\). In particular, \(G_k\) has the form of a Riemann integral: \(G_{k, t} = \int_0^t \int_{\mathbb{R}^d} f_k(\xi) \,\kappa_s(d\xi) \,ds\). By the definition of \(k(i, j)\), we have \[\label{eq:C95G} \widetilde{C}^*_{ij} = C^*_{ij} + G^*_{k(i, j)},\quad \widetilde{C}_{ij} = C_{ij} + G_{k(i, j)}.\tag{23}\] We define the \(\mathbb{R}^d\)-valued predictable process \(b^* = (b^*_i)\), the \(\mathbb{R}^{d^2}\)-valued predictable process \(c^* = (c^*_{ij})\), and the real-valued predictable processes \(g^*_k\), \(k \in \mathbb{N}^*\), via \[\begin{align} b^*_{i, t} &\mathrel{\vcenter{:}}= \boldsymbol{1}_{\mathbb{R}} \biggl(\liminf_{n \to \infty} \frac{B^*_{i, t} - B^*_{i, (t - 1/n)^+}}{1/n}\biggr),\\ c^*_{ij, t} &\mathrel{\vcenter{:}}= \boldsymbol{1}_{\mathbb{R}} \biggl(\liminf_{n \to \infty} \frac{C^*_{ij, t} - C^*_{ij, (t - 1/n)^+}}{1/n}\biggr),\\ g^*_{k, t} &\mathrel{\vcenter{:}}= \boldsymbol{1}_{\mathbb{R}} \biggl(\liminf_{n \to \infty} \frac{G^*_{k, t} - G^*_{k, (t - 1/n)^+}}{1/n}\biggr). \end{align}\] Since \(B\), \(C\), \((G_k)\) are absolutely continuous \(\mathbb{P}\)-a.s., we know that \(B^*\), \(C^*\), \((G^*_k)\) are absolutely continuous \(\mathbb{Q}\)-a.s. Consequently, \[\label{eq:AC95Qas} \begin{align} \mathbb{Q}\biggl(\int_0^t (|b^*_s| + |c^*_s| + |g^*_{k, s}|) \,ds < \infty,\, B^*_t &= \int_0^t b^*_s \,ds,\, C^*_t = \int_0^t c^*_s \,ds,\\ G^*_{k, t} &= \int_0^t g^*_{k, s} \,ds,\, \forall\, k \in \mathbb{N}^*,\, t \geq 0\biggr) = 1. \end{align}\tag{24}\]

Set \(Z^* = \Phi(Z_0^*, Y^*)\). Note that the joint law of \((Y^*, Z^*, B^*, C^*, (G^*_k))\) under \(\mathbb{Q}\) agrees with the joint law of \((Y, Z, B, C, (G_k))\) under \(\mathbb{P}\). Thus, 8 and 24 imply that for Lebesgue-a.e.\(t \geq 0\), the joint law of \((Y^*_t, Z^*_t, b^*_t, c^*_t, (g^*_{k, t}))\) under \(\mathbb{Q}\) agrees with the joint law of \((Y_t, Z_t, b_t, c_t, (\int_{\mathbb{R}^d} f_k(\xi) \,\kappa_t(d\xi)))\) under \(\mathbb{P}\). It follows that \[\label{eq:exp95id} \mathbb{E}^{\mathbb{Q}}\biggl[\int_0^t f(Y^*_s, Z^*_s, b^*_s, c^*_s, (g^*_{k, s})) \,ds\biggr] = \mathbb{E}\biggl[\int_0^t f\biggl(Y_s, Z_s, b_s, c_s, \biggl(\int_{\mathbb{R}^d} f_k(\xi) \,\kappa_s(d\xi)\biggr)\biggr) \,ds\biggr],\tag{25}\] for all \(t > 0\) and measurable \(f: \mathbb{R}^{\mathbb{N}} \to \mathbb{R}\) such that either side (and then both sides) of 25 is well-defined. In particular, 9 and 25 yield \[\label{eq:int95cond} \mathbb{E}^{\mathbb{Q}}\biggl[\int_0^t (|b^*_s| + |c^*_s| + |g^*_{k, s}|) \,ds\biggr] < \infty,\quad \forall\, k \in \mathbb{N}^*,\, t > 0.\tag{26}\] Recall that at the beginning of this proof, we have already established the existence of \(\widehat{b}\), \(\widehat{c}\) and \(\widehat{\kappa}\) satisfying 10 . For each \(k \in \mathbb{N}^*\), define \(\widehat{g}_k: \mathbb{R}_+ \times \mathcal{E} \to \mathbb{R}\) via \[\label{eq:ghat} \widehat{g}_k(t, z) \mathrel{\vcenter{:}}= \int_{\mathbb{R}^d} f_k(\xi) \,\widehat{\kappa}(t, z, d\xi),\quad t \geq 0,\, z \in \mathcal{E}.\tag{27}\] Then, by 10 , 12 , 25 and using [1], Lemma 5.2 twice, we deduce that for Lebesgue-a.e.\(t \geq 0\), \[\label{eq:cond95exp953} \begin{align} \widehat{b}(t, Z^*_t) &= \mathbb{E}^{\mathbb{Q}}[b^*_t \,|\, Z^*_t],\\ \widehat{c}(t, Z^*_t) &= \mathbb{E}^{\mathbb{Q}}[c^*_t \,|\, Z^*_t],\\ \widehat{g}_k(t, Z^*_t) &= \mathbb{E}^{\mathbb{Q}}[g^*_{k, t} \,|\, Z^*_t],\quad \forall\, k \in \mathbb{N}^*. \end{align}\tag{28}\] From 26 and Jensen’s inequality, we get \[\label{eq:int95cond952} \mathbb{E}^{\mathbb{Q}}\biggl[\int_0^t (|\widehat{b}(s, Z^*_s)| + |\widehat{c}(s, Z^*_s)| + |\widehat{g}_k(s, Z^*_s)|) \,ds\biggr] < \infty,\quad \forall\, k \in \mathbb{N}^*,\, t > 0.\tag{29}\]

Step 2: Extended partitions.For \(m \in \mathbb{N}^*\), set \(N(m) \mathrel{\vcenter{:}}= m^2\). Define the stopping times \(T^m_0 \mathrel{\vcenter{:}}= 0\), \(T^m_i \mathrel{\vcenter{:}}= (U^* + i-1) / m\), \(i = 1, ..., N(m)\), and \(T^m_{N(m)+1} \mathrel{\vcenter{:}}= \infty\). Set \(\mathcal{G}^m_0 = \mathcal{H}^m_0 \mathrel{\vcenter{:}}= \mathcal{F}^*_0 = \sigma(U^*, Z_0^*)\), \(\mathcal{G}^m_i \mathrel{\vcenter{:}}= \sigma(U^*, Z^*_{T^m_i})\), \(i = 1, ..., N(m)\), and \(\mathcal{H}^m_i \mathrel{\vcenter{:}}= \mathcal{G}^m_{i-1} \lor \sigma(\Delta(X^{T^m_i}, T^m_{i-1}))\), \(i = 1, ..., N(m)+1\). Then, by Lemma 1 (i), \(\Pi^m \mathrel{\vcenter{:}}= (T^m_i, \mathcal{G}^m_i)_{i=0}^{N(m)}\) is an extended partition.

Step 3: Concatenated probability measures.According to Theorem 5, for each \(m \in \mathbb{N}^*\), there exists a unique probability measure \(\mathbb{Q}^m \mathrel{\vcenter{:}}= \mathbb{Q}^{\otimes \Pi^m}\) on \((\Omega^*, \mathcal{F}^*)\) such that

1. \(\mathbb{Q}^m(A) = \mathbb{Q}(A)\), for all \(A \in \mathcal{H}^m_i\), \(i = 0, ..., N(m)+1\),

2. \(\mathbb{Q}^m(B \,|\, \mathcal{F}^*_{T^m_i}) = \mathbb{Q}(B \,|\, \mathcal{G}^m_i)\), for all \(B \in \mathcal{H}^m_{i+1}\), \(i = 0, ..., N(m)\).

From Step 1, we already know that the following processes are local martingales under \(\mathbb{Q}\):

1. \(Y^*(h) - B^*\),

2. \((Y^*(h) - B^*) (Y^*(h) - B^*)^\top - \widetilde{C}^*\),

3. \(f_k * \mu^{Y^*} - f_k * \nu^*\), \(k \in \mathbb{N}^*\).

By Lemma 2 and Lemma 3, the processes above are also local martingales under \(\mathbb{Q}^m\). Here we may use either \(\mathbb{F}^*\) or \(\widetilde{\mathbb{F}}^*\) due to the right-continuity of sample paths. Also, by Proposition 6 (ii), \(C^*_t - C^*_s\) takes values in \(\mathbb{S}^d_+\) for all \(0 \leq s \leq t\), \(\mathbb{Q}^m\)-a.s. Thus, applying [9], Theorem II.2.21, we deduce that \(Y^*\) is a semimartingale with characteristics triplet \((B^*, C^*, \nu^*)\) (associated with \(h\)) on \((\Omega^*, \mathcal{F}^*, \widetilde{\mathbb{F}}^*)\) under each \(\mathbb{Q}^m\).

We also note that by Proposition 6 (i) and the definition of \(b^*\), \(c^*\), \((g^*_k)\), for each \(m \in \mathbb{N}^*\), \[\label{eq:AC95Qmas} \begin{align} \mathbb{Q}^m\biggl(\int_0^t (|b^*_s| + |c^*_s| + |g^*_{k, s}|) \,ds < \infty,\, B^*_t &= \int_0^t b^*_s \,ds,\, C^*_t = \int_0^t c^*_s \,ds,\\ G^*_{k, t} &= \int_0^t g^*_{k, s} \,ds,\, \forall\, k \in \mathbb{N}^*,\, t \geq 0\biggr) = 1. \end{align}\tag{30}\]

Step 4: Tightness and convergence.By 23 , 24 , 26 , 30 , and Corollary 2, we know that each one of the following collections of probability measures is tight:

1. \((\mathbb{Q}^m \circ (B^*)^{-1})_{m \in \mathbb{N}^*}\) (on \(C^d_0\)),

2. \((\mathbb{Q}^m \circ (\widetilde{C}^*)^{-1})_{m \in \mathbb{N}^*}\) (on \(C^{d^2}_0\)),

3. \((\mathbb{Q}^m \circ (G^*_k)^{-1})_{m \in \mathbb{N}^*}\) (on \(C^1_0\)), for each \(k \in \mathbb{N}^*\).

Moreover, let \(\varepsilon > 0\), \(T > 0\), \(a > 0\). Take \(p(a) \in \mathbb{Q}\) such that \(2/a < p(a) < 3/a\), so we have \(p(a) \to 0\) as \(a \to \infty\). Let \(k(a) \in \mathbb{N}^*\) be the index such that \(f_{k(a)} = (p(a) \lvert \cdot \rvert - 1)^+ \land 1\). It is easy to check that \(f_{k(a)} \geq \boldsymbol{1}_{\{\lvert \cdot \rvert > a\}}\), and \(f_{k(a)} \to 0\) pointwisely as \(a \to \infty\). Then, applying Proposition 7 to \(G^*_{k(a)}\) and \(g^*_{k(a)}\), we get \[\label{eq:tight95est} \begin{align} &\mathbb{E}^{\mathbb{Q}^m}[\nu^*([0, T] \times \{\xi: |\xi| > a\})] = \mathbb{E}^{\mathbb{Q}^m}\biggl[\int_{\{\xi: |\xi| > a\}} \frac{1}{1 \land |\xi|^2} \,M^*_T(d\xi)\biggr]\\ &\quad\quad\quad\quad\leq \mathbb{E}^{\mathbb{Q}^m}\biggl[\int_{\mathbb{R}^d} \frac{f_{k(a)}(\xi)}{1 \land |\xi|^2} \,M^*_T(d\xi)\biggr] = \mathbb{E}^{\mathbb{Q}^m}\bigl[G^*_{k(a), T}\bigr] = \mathbb{E}^{\mathbb{Q}^m}\biggl[\int_0^T g^*_{k(a), s} \,ds\biggr]\\ &\quad\quad\quad\quad= \mathbb{E}^{\mathbb{Q}}\biggl[\int_0^T g^*_{k(a), s} \,ds\biggr] = \mathbb{E}^{\mathbb{Q}}\bigl[G^*_{k(a), T}\bigr] = \mathbb{E}^{\mathbb{Q}}\biggl[\int_{\mathbb{R}^d} \frac{f_{k(a)}(\xi)}{1 \land |\xi|^2} \,M^*_T(d\xi)\biggr]. \end{align}\tag{31}\] By 9 , we have \(\mathbb{E}^{\mathbb{Q}}[M^*_T(\mathbb{R}^d)] = \mathbb{E}[M_T(\mathbb{R}^d)] < \infty\). Thus, 31 and the dominated convergence theorem yield that \[\label{eq:tight95est2} \lim_{a \to \infty} \sup_{m \in \mathbb{N}^*}\mathbb{Q}^m(\nu^*([0, T] \times \{\xi: |\xi| > a\}) > \varepsilon) \leq \lim_{a \to \infty} \frac{1}{\varepsilon} \mathbb{E}^{\mathbb{Q}}\biggl[\int_{\mathbb{R}^d} \frac{f_{k(a)}(\xi)}{1 \land |\xi|^2} \,M^*_T(d\xi)\biggr] = 0.\tag{32}\] Therefore, given 32 and the tightness of (i-iii) above, [9], Theorem VI.4.18 tells us that the collection of measures \((\mathbb{Q}^m \circ (Y^*)^{-1})_{m \in \mathbb{N}^*}\) on \(D^d_0\) is tight. Since \(Z_0^*\) has the same law under every \(\mathbb{Q}^m\) (recall \(Z_0^*\) is \(\mathcal{H}^m_0 = \mathcal{F}^*_0\)-measurable), the collection of measures \((\mathbb{Q}^m \circ (Z_0^*, Y^*)^{-1})_{m \in \mathbb{N}^*}\) on \(\widehat{\Omega} \mathrel{\vcenter{:}}= \mathcal{E} \times D^d_0\) is tight. Passing to a convergent subsequence if necessary, with an abuse of notation, we may assume \(\mathbb{Q}^m \circ (Z_0^*, Y^*)^{-1}\) converges weakly to a Borel probability measure \(\widehat{\mathbb{P}}\) on \(\widehat{\Omega}\), as \(m \to \infty\).

On the space \(\widehat{\Omega} = \mathcal{E} \times D^d_0\), we denote the projections by \((\widehat{Z}_0, \widehat{Y})\). Let \(\widehat{\mathcal{F}} \mathrel{\vcenter{:}}= \sigma(\widehat{Z}_0, \widehat{Y})\) be the Borel \(\sigma\)-algebra, and \(\widehat{\mathcal{F}}_t \mathrel{\vcenter{:}}= \sigma(\widehat{Z}_0, \widehat{Y}^t)\) for \(t \geq 0\). Set \(\widehat{Z} \mathrel{\vcenter{:}}= \Phi(\widehat{Z}_0, \widehat{Y})\). The continuous mapping theorem then yields that the law of \((Y^*, Z^*)\) on \(D^d_0 \times D^{\mathcal{E}}\) under \(\mathbb{Q}^m\) converges weakly to the law of \((\widehat{Y}, \widehat{Z})\) on \(D^d_0 \times D^{\mathcal{E}}\) under \(\widehat{\mathbb{P}}\), i.e.\(\mathbb{Q}^m \circ (Y^*, Z^*)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{Z})^{-1}\).

Step 5: Agreement of one-dimensional laws.From Step 4, we know that the law of \(Z^*\) on \(D^{\mathcal{E}}\) under \(\mathbb{Q}^m\) converges weakly to the law of \(\widehat{Z}\) on \(D^{\mathcal{E}}\) under \(\widehat{\mathbb{P}}\). Since \(\widehat{Z}\) is càdlàg, we know there exists a countable set \(N \subset \mathbb{R}_+\) such that \(\widehat{\mathbb{P}}(\widehat{Z}_t = \widehat{Z}_{t-}) = 1\) for every \(t \notin N\). In other words, the projection map \(D^{\mathcal{E}} \ni z \mapsto z(t) \in \mathcal{E}\) is continuous \((\widehat{\mathbb{P}} \circ \widehat{Z}^{-1})\)-a.s.for every \(t \notin N\). Thus, the continuous mapping theorem implies that the law of \(Z^*_t\) on \(\mathcal{E}\) under \(\mathbb{Q}^m\) converges weakly to the law of \(\widehat{Z}_t\) on \(\mathcal{E}\) under \(\widehat{\mathbb{P}}\), for every \(t \notin N\).

On the other hand, by Lemma 1 (ii), the law of \(Z^*_t\) under \(\mathbb{Q}^m\) agrees with the law of \(Z^*_t\) under \(\mathbb{Q}\) for every \(m \in \mathbb{N}^*\). This gives us \(\widehat{\mathbb{P}} \circ (\widehat{Z}_t)^{-1} = \mathbb{Q} \circ (Z^*_t)^{-1} = \mathbb{P} \circ (Z_t)^{-1}\), for every \(t \notin N\). Finally, by the right-continuity of the sample paths of \(\widehat{Z}\) and \(Z\), we have that \(\widehat{Z}_s \to \widehat{Z}_t\) in law and \(Z_s \to Z_t\) in law as \(s \downarrow t\), for every \(t \geq 0\). Therefore, we conclude that \(\widehat{\mathbb{P}} \circ (\widehat{Z}_t)^{-1} = \mathbb{P} \circ (Z_t)^{-1}\) for every \(t \geq 0\). This proves item (ii) of the theorem.

Step 6: Characteristics of the limit.It remains to show that on the filtered probability space \((\widehat{\Omega}, \widehat{\mathcal{F}}, (\widehat{\mathcal{F}}_t)_{t \geq 0}, \widehat{\mathbb{P}})\) (strictly speaking, one needs to replace \((\widehat{\mathcal{F}}_t)_{t \geq 0}\) by its right-continuous regularization), \(\widehat{Y}\) is a semimartingale with characteristics triplet \((\widehat{B}, \widehat{C}, \widehat{\nu})\) (defined in 11 ) associated with \(h\). We define the following processes on \(\widehat{\Omega}\) (recall 27 ): \[\widehat{G}_{k, t} \mathrel{\vcenter{:}}= (f_k * \widehat{\nu})_t = \int_0^t \int_{\mathbb{R}^d} f_k(\xi) \,\widehat{\kappa}(s, \widehat{Z}_s, d\xi) \,ds = \int_0^t \widehat{g}_k(s, \widehat{Z}_s) \,ds,\quad k \in \mathbb{N}^*.\] We also define the process \(\widehat{C}^\prime \mathrel{\vcenter{:}}= \widehat{C} + (hh^\top) * \widehat{\nu}\). Similar to 23 , one has \(\widehat{C}^\prime_{ij} = \widehat{C}_{ij} + \widehat{G}_{k(i,j)}\). If we manage to prove the following:

1. \(\mathbb{Q}^m \circ (Y^*, B^*, \widetilde{C}^*)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{B}, \widehat{C}^\prime)^{-1}\),

2. \(\mathbb{Q}^m \circ (Y^*, G^*_k)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{G}_k)^{-1}\), for each \(k \in \mathbb{N}^*\),

then applying [9], Theorem IX.2.4 (here we need \(h\) to be continuous), we would finish the proof of item (i) of the theorem.

On the other hand, on the space \(\Omega^*\) we define the following processes: \[\overline{B}_t \mathrel{\vcenter{:}}= \int_0^t \widehat{b}(s, Z^*_s) \,ds,\quad \overline{C}_t \mathrel{\vcenter{:}}= \int_0^t \widehat{c}(s, Z^*_s) \,ds,\quad \overline{G}_{k, t} \mathrel{\vcenter{:}}= \int_0^t \widehat{g}_k(s, Z^*_s) \,ds,\quad k \in \mathbb{N}^*.\] We also define the \(\mathbb{R}^{d^2}\)-valued process \(\overline{C}^\prime\) via \(\overline{C}^\prime_{ij} \mathrel{\vcenter{:}}= \overline{C}_{ij} + \overline{G}_{k(i, j)}\). Recall that in Step 4 we showed \(\mathbb{Q}^m \circ (Y^*, Z^*)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{Z})^{-1}\), and in Step 5 we showed \(\mathbb{Q}^m \circ (Z^*_t)^{-1} = \mathbb{Q} \circ (Z^*_t)^{-1}\) for all \(m \in \mathbb{N}^*\) and \(t \geq 0\). Using 29 , Proposition 8 and Lemma 7, we know these processes are well-defined \(\mathbb{Q}\)-a.s.and \(\mathbb{Q}^m\)-a.s.for each \(m \in \mathbb{N^*}\), and we get the following weak convergence:

1. \(\mathbb{Q}^m \circ (Y^*, \overline{B}, \overline{C}^\prime)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{B}, \widehat{C}^\prime)^{-1}\),

2. \(\mathbb{Q}^m \circ (Y^*, \overline{G}_k)^{-1} \Rightarrow \widehat{\mathbb{P}} \circ (\widehat{Y}, \widehat{G}_k)^{-1}\), for each \(k \in \mathbb{N}^*\).

Thus, to conclude the proof, we need to show as \(m \to \infty\): \(\mathbb{Q}^m \circ (Y^*, B^*, \widetilde{C}^*)^{-1}\) and \(\mathbb{Q}^m \circ (Y^*, \overline{B}, \overline{C}^\prime)^{-1}\) have the same limit; \(\mathbb{Q}^m \circ (Y^*, G^*_k)^{-1}\) and \(\mathbb{Q}^m \circ (Y^*, \overline{G}_k)^{-1}\) have the same limit. To do this, it suffices to show that for any \(\varepsilon > 0\) and \(t > 0\), \[\label{eq:ucp} \begin{align} \lim_{m \to \infty} \mathbb{Q}^m\biggl(\max_{s \leq t} |B^*_s - \overline{B}_s| \geq \varepsilon\biggr) &= 0,\\ \lim_{m \to \infty} \mathbb{Q}^m\biggl(\max_{s \leq t} |\widetilde{C}^*_s - \overline{C}^\prime_s| \geq \varepsilon\biggr) &= 0,\\ \lim_{m \to \infty} \mathbb{Q}^m\biggl(\max_{s \leq t} |G^*_{k, s} - \overline{G}_{k, s}| \geq \varepsilon\biggr) &= 0,\quad \forall\, k \in \mathbb{N}^*. \end{align}\tag{33}\] Given 26 and 28 (thus 29 as well), we know that 33 is the consequence of Lemma 4, and we are done. ◻