of Theorem 2.11 We begin by considering, for all
\(n\geq 1\), the particular approximations
$$ M_{n}(t):= \frac{1}{\sigma \sqrt{n}}\sum _{i=1}^{\lfloor nt \rfloor }\xi _{i}, $$
for
\(t\in \mathbb{I}\), of the driving Brownian motion
\(B\) in the dynamics of
\(X\). Here the
\(\xi _{i}\) satisfy Assumption
2.10, and so do the
\(\zeta _{i}\) in the construction of
\(Y_{n}\) from (
2.7). While each pair
\(\xi _{i}\) and
\(\zeta _{i}\) are correlated, they form an i.i.d. sequence
\(((\zeta _{i},\xi _{i}))_{i\geq 1}\) across the pairs. In particular, it is straightforward to see that each
\(M_{n}\) is a martingale on
\(\mathbb{I}\) for the filtration
\((\mathcal{F}^{n}_{t})\) defined by
\(\mathcal{F}^{n}_{t}:=\sigma (\zeta _{i},\xi _{i}:i=1,\ldots ,\lfloor nt \rfloor )\). Moreover, we have
$$ \mathbb{E}\bigl[[M_{n}]_{t}\big] = \lfloor nt \rfloor \frac{1}{\sigma ^{2}n } \mathbb{E}[\xi _{1}^{2}] =\frac{\lfloor nt \rfloor }{n} \leq 1 $$
for
\(t\in \mathbb{I}\) and for all
\(n\geq 1\). Consequently, we can simply take
\(\tau _{n}^{\gamma }\equiv +\infty \) for all
\(\gamma >0\) and
\(n\geq 1\) to satisfy the required control on the integrators
\(N_{n}:=M_{n}\) in Theorem
3.12. By Ethier and Kurtz [
25, Theorem 7.1.4], the
\(M_{n}\) converge in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\) weakly to a Brownian motion
\(B\). Now fix
\(\alpha \in (-\frac{1}{2}, \frac{1}{2})\) and define a sequence
\((H_{n})_{n \geq 1}\) of càdlàg processes on
\(\mathbb{I}\) by setting
\(H_{n}(1):=\Phi (\mathcal {G}^{\alpha }Y_{n})(1)\) and
\(H_{n}(t):=\Phi (\mathcal {G}^{\alpha }Y_{n})(t_{k-1})\) for
\(t\in [t_{k-1},t_{k})\) and
\(k=1,\ldots ,n\). In view of Theorem
3.11, the Arzelà–Ascoli characterisation of tightness (see [
13, Theorem 8.2]) for the space
\((\mathcal {C}(\mathbb{I}),\|\cdot \|_{\infty} )\) allows us to conclude that the
\(H_{n}\) converge in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\) weakly to
\(H:=\Phi (\mathcal {G}^{\alpha }Y)\). Furthermore, recalling the definition of
\(Y_{n}\) in (
2.7), each
\(H_{n}\) is adapted to the filtration
\((\mathcal{F}^{n}_{t})\) introduced above. By Corollary
3.6, we readily deduce that there is joint weak convergence in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal{D}})\times (\mathcal {D}(\mathbb{I}),d_{\mathcal{D}}) \times (\mathcal {D}(\mathbb{I}),d_{\mathcal{D}}) \) of
\((Y_{n},H_{n},M_{n})\) to
\((Y,H,B)\), where
\(Y\) satisfies (
2.4) for a Brownian motion
\(W\) with
\([ W, B ]_{t} = \rho t\) for all
\(t\in \mathbb{I}\). As noted in [
46], the Skorokhod topology on
\(\mathcal {D}(\mathbb{I},\mathbb{R}^{2})\) is stronger than the product topology on
\(\mathcal {D}(\mathbb{I})\times \mathcal {D}(\mathbb{I})\), but here it automatically follows that we have weak convergence in
\((\mathcal {D}(\mathbb{I}, \mathbb{R}^{2}),d_{\mathcal {D}} )\) of the pairs
\((H_{n},M_{n})\) to
\((H,B)\) by standard properties of the Skorokhod topology (see e.g. [
25, Theorem 3.10.2]), since the limiting pair
\((H,B)\) is continuous. Consequently, we are in a position to apply Theorem
3.12. To this end, observe that
$$ \big((H_{n})_{-} \bullet M_{n}\big)(t) = \sum _{k=1}^{\lfloor nt \rfloor } H_{n}(t_{k}-) \big(M_{n}(t_{k})-M_{n}(t_{k}-)\big) =\frac{1}{\sigma \sqrt{n}}\sum _{k=1}^{ \lfloor nt \rfloor } \Phi (\mathcal {G}^{\alpha }Y_{n})(t_{k-1})\xi _{k}, $$
which is precisely the second term on the right-hand side of (
2.8). Therefore, Theorem
3.12 gives that the stochastic integral
\(H \bullet M = \sqrt{\Phi (\mathcal {G}^{\alpha }Y)}\bullet B\) is in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\) the weak limit of the second term on the right-hand side of (
2.8). For the first term on the right-hand side of (
2.8), we have
\(-\frac{1}{2}\int _{0}^{\cdot }H_{n}(s) \mathrm {d}s\) converging weakly to
\(-\frac{1}{2}\int _{0}^{\cdot }H(s)\mathrm {d}s\) by the continuous mapping theorem, as the integral is a continuous operator from
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\) to itself. Since there is weak convergence in
\((\mathcal {D}(\mathbb{I},\mathbb{R}^{2}),d_{\mathcal {D}})\) of the pairs
\((H_{n},(H_{n})_{-} \bullet M_{n})\) to
\((H, H \bullet B)\), the sum of the two terms on the right-hand side of (
2.8) are then also weakly convergent in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\). Recalling that the limit
\(Y\) satisfies (
2.4) for a Brownian motion
\(W\) such that
\(W\) and
\(B\) are correlated with parameter
\(\rho \), we hence conclude that
\(X\) converges in
\((\mathcal {D}(\mathbb{I}),d_{\mathcal {D}})\) weakly to the desired limit. □