We need to prove
$$ {\mathcal {B}}_{k}(\varepsilon_{1}, \varepsilon_{2}):= E \bigl\vert J_{\varepsilon _{1}}(k,f,t)-J_{\varepsilon_{2}}(k,f,t) \bigr\vert ^{2}\longrightarrow0, $$
(3.12)
for all and
\(t\geq0\) and
\(k=1,2\), as
\(\varepsilon_{1},\varepsilon _{2}\downarrow0\). Recall that
$$\Delta_{\varepsilon}(s,r)=E \bigl[f({W}_{s})f({W}_{r}) ({W}_{s+\varepsilon}-{W}_{s}) ({W}_{r+\varepsilon}-{W}_{r}) \bigr], $$
and denote
$$\Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r) =E \bigl[f({W}_{s})f({W}_{r}) ({W}_{s+\varepsilon_{1}}-{W}_{s}) ({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] $$
for all
\(\varepsilon,\varepsilon_{1},\varepsilon_{2}>0\) and
\(s,r\geq0\). Then we have
$$\begin{aligned} {\mathcal {B}}_{1}(\varepsilon_{1},\varepsilon_{2}) =& \frac{1}{\varepsilon_{1}^{4H}} \int_{0}^{t} \int_{0}^{t} Ef({W}_{s})f({W}_{r}) ({W}_{s+\varepsilon_{1}}-{W}_{s}) ({W}_{r+\varepsilon_{1}} -W_{r})\,d \eta_{r}\,d\eta _{s} \\ &{}-\frac{2}{\varepsilon_{1}^{2H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t} Ef({W}_{s})f({W}_{r}) ({W}_{s+\varepsilon _{1}}-{W}_{s}) ({W}_{r+\varepsilon_{2}} -{W}_{r})\,d\eta_{r}\,d\eta_{s} \\ &{}+\frac{1}{\varepsilon_{2}^{4H}} \int_{0}^{t} \int_{0}^{t} Ef({W}_{r})f({W}_{r}) ({W}_{s+\varepsilon_{2}}-{W}_{s}) ({W}_{r+\varepsilon_{2}} -{W}_{r})\,d\eta _{r}\,d\eta_{s} \\ =& \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t} \bigl\{ \Delta_{\varepsilon_{1}}(s,r) \varepsilon_{2}^{2H}- \Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r) \varepsilon_{1}^{2H} \bigr\} \,d\eta_{r}\,d \eta_{s} \\ &{}+ \frac{1}{\varepsilon_{1}^{2H}\varepsilon_{2}^{4H}} \int_{0}^{t} \int_{0}^{t} \bigl\{ \Delta_{\varepsilon_{2}}(s,r) \varepsilon_{1}^{2H} -\Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r) \varepsilon_{2}^{2H} \bigr\} \,d\eta_{r}\,d \eta_{s} \end{aligned}$$
for all
\(\varepsilon_{1},\varepsilon_{2}>0\) and
\(t\geq0\). Thus, to show that
\(\{J_{\varepsilon}(1,f,t),\varepsilon>0\}\) is a Cauchy sequence in
\(L^{2}(\Omega)\) we need to prove
$$ \lim_{\varepsilon_{i},\varepsilon_{j}\to0}\frac{1}{\varepsilon_{i}^{4H} \varepsilon_{j}^{2H}} \int_{0}^{t} \int_{0}^{t} \bigl\{ \Delta_{\varepsilon_{i}}(s,r) \varepsilon_{j}^{2H} - \Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r) \varepsilon_{i}^{2H} \bigr\} \,d\eta_{r}\,d \eta_{s}=0 $$
(3.13)
for all
\(i,j\in\{1,2\}\) and
\(i\neq j\). Without loss of generality one may assume that
\(\varepsilon_{1}>\varepsilon_{2}\) and by approximating we may also assume that
\(f\in C^{\infty}_{0}({\mathbb {R}})\). Denote
$$\begin{aligned} Q_{j}(1,s,r,\varepsilon) :=& \varepsilon_{j}^{2H} E \bigl[({W}_{r+\varepsilon}-{W}_{r}) ({W}_{s+\varepsilon}-{W}_{s}) \bigr] \\ &{}-\varepsilon^{2H} E \bigl[({W}_{s+\varepsilon_{1}}-{W}_{s}) ({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr], \\ Q_{j}(2,s,r,\varepsilon) :=& \varepsilon_{j}^{2H} E \bigl[{W}_{r}({W}_{r+\varepsilon}-{W}_{r}) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon}-{W}_{s}) \bigr] \\ &{}-\varepsilon^{2H} E \bigl[B^{H}_{r} \bigl(B^{H}_{r+\varepsilon_{2}} -B^{H}_{r}\bigr) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr], \\ Q_{j}(3,s,r,\varepsilon) := & \varepsilon_{j}^{2H} E \bigl[{W}_{s}({W}_{r+\varepsilon}-{W}_{r}) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon}-{W}_{s}) \bigr] \\ &{}-\varepsilon^{2H} E \bigl[{W}_{s}({W}_{r+\varepsilon_{2}} -{W}_{r}) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr], \\ Q_{j}(4,s,r,\varepsilon) :=& \varepsilon_{j}^{2H} E \bigl[{W}_{r}({W}_{r+\varepsilon}-{W}_{r}) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon}-{W}_{s}) \bigr] \\ &{}-\varepsilon^{2H} E \bigl[{W}_{r}({W}_{r+\varepsilon_{2}} -{W}_{r}) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr], \\ Q_{j}(5,s,r,\varepsilon) :=& \varepsilon_{j}^{2H} E \bigl[{W}_{s}({W}_{r+\varepsilon}-{W}_{r}) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon}-{W}_{s}) \bigr] \\ &{}-\varepsilon^{2H} E \bigl[{W}_{s}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr], \end{aligned}$$
for
\(j\in\{1,2\}\) and
\(\varepsilon,\varepsilon_{1}, \varepsilon_{2},s,r>0\). From the proof of Statement (a) it follows that
$$\begin{aligned} \Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r) =& E \bigl[f({W}_{s})f({W}_{r}) \bigr] E \bigl[({W}_{r+\varepsilon _{2}}-{W}_{r}) ({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr] \\ &{}+E \bigl[f'({W}_{s})f'({W}_{r}) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr] E \bigl[{W}_{s}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] \\ &{}+E \bigl[f''({W}_{s})f \bigl(B^{H}_{r}\bigr) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr] E \bigl[{W}_{s}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] \\ &{}+E \bigl[f'({W}_{s})f'({W}_{r}) \bigr] E \bigl[{W}_{s}({W}_{s+\varepsilon _{1}}-W_{s}) \bigr] E \bigl[W_{r}({W}_{r+\varepsilon_{2}}-W_{r}) \bigr] \\ &{}+E \bigl[f({W}_{s})f''({W}_{r}) \bigr] E \bigl[{W}_{r}({W}_{s+\varepsilon _{1}}-{W}_{s}) \bigr] E \bigl[{W}_{r}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] \end{aligned}$$
and
$$\begin{aligned}& \varepsilon_{j}^{2H}\Delta_{\varepsilon_{i}}(s,r)- \varepsilon_{i}^{2H} \Delta_{\varepsilon_{1},\varepsilon_{2}}(s,r)\\& \quad= E \bigl[f({W}_{s})f({W}_{r}) \bigr]Q_{j}(1,s,r, \varepsilon_{i}) \\& \qquad{}+E \bigl[f({W}_{s})f''({W}_{r}) \bigr]Q_{j}(2,s,r,\varepsilon_{i})+ E \bigl[f''({W}_{s})f({W}_{r}) \bigr]Q_{j}(3,s,r,\varepsilon_{i}) \\& \qquad{}+E \bigl[f'({W}_{s})f'({W}_{r}) \bigr] \bigl(Q_{j}(4,s,r,\varepsilon _{i})+Q_{j}(5,s,r, \varepsilon_{i})\bigr) \end{aligned}$$
with
\(i\neq j\) and
\(i,j\in\{1,2\}\). Now, let us prove the convergence (
3.13) in three steps. We only need to show that (
3.13) holds with
\(j=2\) and
\(i=1\) by symmetry.
Step I. The convergence
$$ \lim_{\varepsilon_{1},\varepsilon_{2}\to0}\frac{1}{\varepsilon _{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t} E \bigl[f({W}_{s})f({W}_{r}) \bigr]Q_{2}(1,s,r,\varepsilon_{1}) \,d\eta_{r}\,d \eta_{s}=0 $$
(3.14)
holds. Clearly, by Cauchy’s inequality we have
$$\begin{aligned} \bigl\vert E\bigl[({W}_{s+\varepsilon_{i}}-{W}_{s})({W}_{r+\varepsilon_{j}} -{W}_{r})\bigr]\bigr\vert \leq & \sqrt{E\bigl[({W}_{s+\varepsilon_{i}}-{W}_{s})^{2} E({W}_{r+\varepsilon_{j}} -{W}_{r})^{2}\bigr]} \\ \leq& \bigl(\lambda^{2}\varepsilon_{i}^{1-2H}+ \nu^{2}\bigr) \bigl(\lambda^{2}\varepsilon_{j}^{1-2H}+ \nu^{2}\bigr) \varepsilon_{i}^{H} \varepsilon_{j}^{H} \leq C \frac{\varepsilon_{i}^{2H+\theta} \varepsilon_{j}^{2H} }{\vert s-r\vert ^{2H+\theta}} \end{aligned}$$
for
\(0<\vert s-r\vert <\varepsilon_{i}\wedge\varepsilon_{j}\leq1\) and
\(0<\theta<1-2H\), where
\(i,j\in\{1,2\}\). It follows from (
2.9) with
\(\alpha=\frac{2H+\theta}{2-2H}\) that
$$\bigl\vert E\bigl[({W}_{s+\varepsilon_{1}}-{W}_{s}) ({W}_{r+\varepsilon_{1}} -{W}_{r})\bigr]\bigr\vert \leq\frac{C\varepsilon_{1}^{4H+\theta}}{ \vert s-r\vert ^{2H+\theta}} $$
and
$$\bigl\vert E\bigl[({W}_{s+\varepsilon_{1}} -{W}_{s}) ({W}_{r+\varepsilon_{2}} -{W}_{r})\bigr]\bigr\vert \leq \frac{C\varepsilon_{1}^{2H+\theta}\varepsilon_{2}^{2H}}{ \vert s-r\vert ^{2H +\theta}} $$
for all
\(\vert s-r\vert >0\) and
\(0<\theta<1-2H\), which gives
$$\frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(1,s,r,\varepsilon_{1}) \bigr\vert \leq \frac{C\varepsilon_{1}^{\theta}}{ \vert s-r\vert ^{2H +\theta}} \longrightarrow0\quad(\varepsilon_{1}, \varepsilon_{2}\to0) $$
for all
\(r,s>0\) and
\(0<\theta<1-2H\).
On the other hand, from the above proof we have also
$$\begin{aligned} \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(1,s,r,\varepsilon_{1}) \bigr\vert \leq & \frac{1}{\varepsilon_{1}^{4H}} \bigl\vert E\bigl[({W}_{s+\varepsilon_{1}}-{W}_{s}) ({W}_{r+\varepsilon_{1}} -{W}_{r})\bigr]\bigr\vert \\ &{}+\frac{1}{\varepsilon_{1}^{2H}\varepsilon_{2}^{2H}} \bigl\vert E\bigl[({W}_{s+\varepsilon_{1}} -{W}_{s}) ({W}_{r+\varepsilon_{2}} -{W}_{r})\bigr]\bigr\vert \\ \leq & \frac{C}{\vert s-r\vert ^{2H}} \end{aligned}$$
for all
\(\vert s-r\vert >0\) and
\(\varepsilon_{1},\varepsilon_{2}>0\), and
$$\int_{0}^{t} \int_{0}^{t}\frac{1}{\vert s-r\vert ^{2H}} \bigl\vert E \bigl[ f({W}_{s})f({W}_{r}) \bigr]\bigr\vert \, d \eta_{r}\,d\eta_{s} \leq C\Vert f\Vert ^{2}_{\mathbb {H}} $$
for any
\(0<\varepsilon_{1},\varepsilon_{2}<1\). Thus, Lebesgue’s dominated convergence theorem implies that the convergence (
3.14) holds.
Step II. The convergence
$$\begin{aligned} \lim_{\varepsilon_{1},\varepsilon_{2}\to0}\frac{1}{\varepsilon _{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t}&E \bigl[f({W}_{s})f''({W}_{r}) \bigr] Q_{2}(2,s,r,\varepsilon _{1})\,d\eta_{r}\,d \eta_{s}=0 \end{aligned}$$
(3.15)
holds. By Lemma
2.4, we have
$$\frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(2,s,r,\varepsilon_{1}) \bigr\vert \leq2 $$
and
$$\int_{0}^{t} \int_{0}^{t}\bigl\vert E \bigl[ f({W}_{s})f''({W}_{r}) \bigr] \bigr\vert \,d\eta_{r}\,d\eta_{s} \leq C \Vert f\Vert ^{2}_{\mathbb {H}} $$
for
\(\varepsilon_{1},\varepsilon_{2}>0\). On the other hand, by Lemma
2.4 and the fact
$$ b^{\gamma}-a^{\gamma}\leq b^{\gamma-\theta}(b-a)^{\theta} $$
(3.16)
with
\(b>a>0\) and
\(1\geq\theta\geq\gamma\geq0\), we have
$$\begin{aligned} \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(2,s,r,\varepsilon_{1})\bigr\vert =& \frac{1}{\varepsilon_{1}^{4H} \varepsilon_{2}^{2H}} \bigl\vert E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr]\bigr\vert \\ &{}\times\bigl\vert \varepsilon_{2}^{2H}E \bigl[{W}_{r}({W}_{r+\varepsilon_{1}}-{W}_{r}) \bigr] - \varepsilon^{2H}_{1} E \bigl[{W}_{r}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr]\bigr\vert \\ =& \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr]\bigr\vert \\ &{}\times C\bigl\vert \varepsilon_{2}^{2H} \bigl((r+ \varepsilon_{1})^{2H} -r^{2H} \bigr) - \varepsilon_{1}^{2H} \bigl((r+\varepsilon_{2})^{2H}-r^{2H} \bigr) \bigr\vert \\ \leq & Cr^{2H-\theta}\varepsilon_{1}^{\theta-2H}\longrightarrow0 \quad(\varepsilon_{1},\varepsilon_{2}\to0) \end{aligned}$$
(3.17)
for all
\(2H<\theta\leq1\) and
\(r>0\). Thus, the convergence (
3.15) follows from the Lebesgue dominated convergence theorem. Similarly, we can introduce the convergence
$$\begin{aligned} \lim_{\varepsilon_{1},\varepsilon_{2}\to0}\frac{1}{\varepsilon _{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t}&E \bigl[f''({W}_{s})f({W}_{r}) \bigr] Q_{2}(3,s,r,\varepsilon _{1})\,d\eta_{r}\,d \eta_{s}=0. \end{aligned}$$
(3.18)
Step III. The convergence
$$ \lim_{\varepsilon_{1},\varepsilon_{2}\to0} \frac{1}{\varepsilon _{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t} \bigl(Q_{2}(4,s,r, \varepsilon_{1})+Q_{2}(5,s,r,\varepsilon_{1}) \bigr)E \bigl[ f'({W}_{s})f'({W}_{r}) \bigr]\,d\eta_{r}\,d\eta_{s}=0 $$
(3.19)
holds. From Step II we have
$$\begin{aligned} \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(4,s,r,\varepsilon_{1}) \bigr\vert \leq & \frac{1}{\varepsilon_{1}^{4H} \varepsilon_{2}^{2H}}\bigl\vert E \bigl[{W}_{s}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr]\bigr\vert \\ &{} \times\bigl\vert \varepsilon_{2}^{2H} E \bigl[{W}_{r}({W}_{r+\varepsilon_{1}}-{W}_{r}) \bigr] - \varepsilon^{2H}_{1} E \bigl[{W}_{r}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr]\bigr\vert \\ \leq & Cr^{2H-\theta}\varepsilon_{1}^{\theta-2H}\longrightarrow0 \quad(\varepsilon_{1},\varepsilon_{2}\to0) \end{aligned}$$
and
$$\begin{aligned} \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \bigl\vert Q_{2}(5,s,r,\varepsilon_{1}) \bigr\vert =& \frac{1}{\varepsilon_{1}^{4H} \varepsilon_{2}^{2H}} \bigl\vert E \bigl[{W}_{r}({W}_{s+\varepsilon_{1}}-{W}_{s}) \bigr]\bigr\vert \\ &{}\times\bigl\vert \varepsilon_{2}^{2H}E \bigl[{W}_{s}({W}_{r+\varepsilon_{1}}-{W}_{r}) \bigr] - \varepsilon^{2H}_{1}E \bigl[{W}_{s}({W}_{r+\varepsilon_{2}}-{W}_{r}) \bigr] \bigr\vert \\ \leq & \bigl(r^{2H-\theta}+\vert s-r\vert ^{2H-\theta} \bigr) \varepsilon^{\theta-2H}\longrightarrow0 \end{aligned}$$
for all
\(2H<\theta\leq1\) and
\(\vert s-r\vert >0\), as
\(\varepsilon_{1},\varepsilon_{2}\to0\). On the other hand, we also have
$$\begin{aligned}& \frac{1}{\varepsilon_{1}^{4H}\varepsilon_{2}^{2H}} \int_{0}^{t} \int_{0}^{t} \bigl\vert E \bigl[f'({W}_{s})f'({W}_{r}) \bigr]\bigr\vert \bigl\vert Q_{2}(4,s,r,\varepsilon _{1})+Q_{2}(5,s,r,\varepsilon_{1}) \bigr\vert \,d \eta_{r}\,d\eta_{s} \\& \quad\leq4 \int_{0}^{t} \int_{0}^{t} \bigl\vert E \bigl[ f'({W}_{s})f'({W}_{r}) \bigr] \bigr\vert \,d\eta_{r}\,d\eta_{s} \leq C\Vert f\Vert ^{2}_{\mathbb {H}} \end{aligned}$$
for all
\(\varepsilon_{1},\varepsilon_{2}>0\). Thus, Lebesgue dominated convergence theorem implies that the convergence (
3.19) holds.