Denoting
\(Y_t^\varepsilon \,\,{:=}\,\,X_t^\varepsilon /\varepsilon \), we will make use of the decomposition [
29, Formula 5.8]
$$\begin{aligned} \begin{aligned} X_t^\varepsilon - X_s^\varepsilon&= -\int _s^t (\alpha \cdot V'(X_r^\varepsilon ))(1 + \varPhi '(Y^\varepsilon _r)) \,\mathrm {d}r\\&\quad +\sqrt{2\sigma } \int _s^t (1 + \varPhi '(Y_r^\varepsilon )) \,\mathrm {d}W_r - \varepsilon (\varPhi (Y^\varepsilon _t)-\varPhi (Y^\varepsilon _s)), \end{aligned} \end{aligned}$$
(37)
which is obtained applying the Itô formula to
\(\varPhi \), the solution of the cell problem (
5). Recall that by definition of
\(Z_t^\varepsilon \) we have
$$\begin{aligned} X_t^\varepsilon - Z_t^\varepsilon = \int _0^t k(t-s) (X_t^\varepsilon - X_s^\varepsilon ) \,\mathrm {d}s + e^{-t/\delta }X_t^\varepsilon . \end{aligned}$$
Plugging decomposition (
37) into the equation above, we obtain
$$\begin{aligned} X_t^\varepsilon - Z_t^\varepsilon = I_1^\varepsilon (t) + I_2^\varepsilon (t) + I_3^\varepsilon (t) + I_4^\varepsilon (t), \end{aligned}$$
where
$$\begin{aligned} \begin{aligned} I_1^\varepsilon (t)&\,\,{:=}\,\,-\int _0^t k(t-s)\int _s^t (\alpha \cdot V'(X_r^\varepsilon ))(1 + \varPhi '(Y^\varepsilon _r)) \,\mathrm {d}r \,\mathrm {d}s,\\ I_2^\varepsilon (t)&\,\,{:=}\,\,\sqrt{2\sigma }\int _0^t k(t-s) \int _s^t (1 + \varPhi '(Y_r^\varepsilon )) \,\mathrm {d}W_r \,\mathrm {d}s, \\ I_3^\varepsilon (t)&\,\,{:=}\,\,- \varepsilon \int _0^t k(t-s)(\varPhi (Y^\varepsilon _t)-\varPhi (Y^\varepsilon _s)) \,\mathrm {d}s,\\ I_4^\varepsilon (t)&= e^{-t / \delta }X_t^\varepsilon . \end{aligned} \end{aligned}$$
Let us analyze the terms above singularly. For
\(I_1^\varepsilon (t)\), one can show [
29, Proposition 5.8]
$$\begin{aligned} \int _s^t (\alpha \cdot V'(X_r^\varepsilon )) (1 + \varPhi '(Y^\varepsilon _r)) \,\mathrm {d}r = (t - s)(A \cdot V'(X_t^\varepsilon )) + R_1^\varepsilon (t-s), \end{aligned}$$
where the remainder
\(R_1^\varepsilon \) satisfies
$$\begin{aligned} \left( {{\mathbb {E}}}^{\varphi ^\varepsilon }\left|R_1^\varepsilon (t-s)\right|^p\right) ^{1/p} \le C(\varepsilon ^2 + \varepsilon (t-s)^{1/2}+(t-s)^{3/2}). \end{aligned}$$
(38)
Therefore, it holds
$$\begin{aligned} \begin{aligned} I_1^\varepsilon (t)&= -(A \cdot V'(X_t^\varepsilon )) \int _0^t k(t-s) (t-s)\,\mathrm {d}s + \int _0^t k(t-s)R_1^\varepsilon (t-s) \,\mathrm {d}s\\&= -\delta (A \cdot V'(X_t^\varepsilon )) + e^{-t/\delta }(t + \delta )(A \cdot V'(X_t^\varepsilon )) + {\widetilde{R}}_1^\varepsilon (t), \end{aligned} \end{aligned}$$
where we exploited the equality
$$\begin{aligned} \int _0^t k(t-s) (t-s) \,\mathrm {d}s = \delta - e^{-t/\delta }(t + \delta ), \end{aligned}$$
and where
$$\begin{aligned} {\widetilde{R}}_1^\varepsilon (t) \,\,{:=}\,\,\int _0^t k(t-s)R_1^\varepsilon (t-s) \,\mathrm {d}s. \end{aligned}$$
Now, Lemma
B.1, inequality (
38) and Lemma
B.2 yield for all
\(p \ge 1\)$$\begin{aligned} \begin{aligned} {{\mathbb {E}}}^{\varphi ^\varepsilon }\left|{\widetilde{R}}_1^\varepsilon (t)\right|^p&\le C \int _0^t k(t-s){{\mathbb {E}}}^{\varphi ^\varepsilon } \left|R_1^\varepsilon (t-s)\right|^p \,\mathrm {d}s\\&\le C \int _0^t k(t-s)(\varepsilon ^{2p} + \varepsilon ^p(t-s)^{p/2}+(t-s)^{3p/2}) \,\mathrm {d}s\\&\le C \left( \varepsilon ^{2p} + \varepsilon ^p \delta ^{p/2} + \delta ^{3p/2} \right) , \end{aligned} \end{aligned}$$
where
C is a positive constant independent of
\(\varepsilon \) and
\(\delta \). Therefore, for
\(\delta \) sufficiently small, we get
$$\begin{aligned} \left( {{\mathbb {E}}}^{\varphi ^\varepsilon }\left|I_1^\varepsilon (t)\right|^{p}\right) ^{1/p} \le C \left( \delta + \varepsilon ^2 + \varepsilon \delta ^{1/2} + t e^{-t/\delta } \right) . \end{aligned}$$
We now consider the second term. Let us introduce the notation
$$\begin{aligned} Q_t^\varepsilon \,\,{:=}\,\,\int _0^t \left( 1 + \varPhi '(Y_r^\varepsilon )\right) \,\mathrm {d}W_r, \end{aligned}$$
and therefore rewrite
$$\begin{aligned} I_2^\varepsilon (t) = \sqrt{2\sigma } \int _0^t k(t-s)(Q_t^\varepsilon - Q_s^\varepsilon ) \,\mathrm {d}s. \end{aligned}$$
An application of the Itô formula to
\(u(s, Q_s^\varepsilon )\) where
\(u(s, x) = k(t-s)x\) yields
$$\begin{aligned} \begin{aligned} I_2^\varepsilon (t)&= \sqrt{2\sigma } \left( Q_t^\varepsilon \int _0^t k(t-s) \,\mathrm {d}s - Q_t^\varepsilon + \delta \int _0^t k(t-s)\left( 1 + \varPhi '(Y_s^\varepsilon )\right) \,\mathrm {d}W_s\right) \\&= \delta B_t^\varepsilon - \sqrt{2\sigma } e^{-t/\delta } Q_t^\varepsilon \,\,{=:}\,\,\delta B_t^\varepsilon - R_2^\varepsilon (t), \end{aligned} \end{aligned}$$
(39)
where
\(B_t^\varepsilon \) is defined in (
27). For the remainder
\(R_2^\varepsilon (t)\), let us remark that for all
\(p \ge 1\) it holds
$$\begin{aligned} \begin{aligned} \left( {{\mathbb {E}}}\left|Q_t^\varepsilon \right|^p\right) ^2 \le {{\mathbb {E}}}\left|Q_t^\varepsilon \right|^{2p} \le Ct^{p-1}\int _0^t {{\mathbb {E}}}\left|1 + \varPhi '(Y_r^\varepsilon )\right|^{2p} \,\mathrm {d}r \le C t^p, \end{aligned} \end{aligned}$$
where we applied Jensen’s inequality, an estimate for the moments of stochastic integrals [
19, Formula (3.25), p. 163] and the boundedness of
\(\varPhi \). Therefore, we have
$$\begin{aligned} \left( {{\mathbb {E}}}^{\varphi ^\varepsilon }\left|R_2^\varepsilon (t)\right|^p \right) ^{1/p} \le C \sqrt{t} e^{-t/\delta }. \end{aligned}$$
(40)
In order to obtain bound (
28) on
\(B_t^\varepsilon \), let us remark that from (
39) it holds for a constant
\(C > 0\) depending only on
p$$\begin{aligned} \left( {{\mathbb {E}}}\left|B_t^\varepsilon \right|^p\right) ^{1/p}\le C \delta ^{-1}\left( {{\mathbb {E}}}\left|I_2^\varepsilon (t)\right|^p\right) ^{1/p} + C \delta ^{-1} \left( {{\mathbb {E}}}\left|R_2^\varepsilon (t)\right|^p\right) ^{1/p}. \end{aligned}$$
The second term is bounded exponentially fast with respect to
t and
\(\delta \) due to (
40). For the first term, applying Lemma
B.1, inequality [
19, Formula (3.25), p. 163] and Lemma
B.2 we obtain for a constant
\(C > 0\) independent of
\(\delta \) and
t$$\begin{aligned} \begin{aligned} {{\mathbb {E}}}\left|I_2^\varepsilon (t)\right|^p&\le C \int _0^t k(t-s) {{\mathbb {E}}}\left|Q_t - Q_s\right|^p \,\mathrm {d}s \\&\le C\int _0^t k(t-s)(t-s)^{p/2} \,\mathrm {d}s \le C\delta ^{p/2}. \end{aligned} \end{aligned}$$
Therefore, it holds for
\(\delta \) sufficiently small
$$\begin{aligned} \left( {{\mathbb {E}}}\left|B_t^\varepsilon \right|^p\right) ^{1/p}\le C \delta ^{-1/2}, \end{aligned}$$
which proves bound (
28). Let us now consider
\(I_3^\varepsilon (t)\). Since
\(\varPhi \) is bounded, we simply have
$$\begin{aligned} \left|I_3^\varepsilon (t)\right| \le C \varepsilon , \end{aligned}$$
almost surely. Finally, due to [
29, Corollary 5.4], we know that
\(X_t^\varepsilon \) has bounded moments of all orders and therefore
$$\begin{aligned} \left( {{\mathbb {E}}}^{\varphi ^\varepsilon } \left|I_4^\varepsilon (t)\right|^p\right) ^{1/p} \le C e^{-t/\delta }, \end{aligned}$$
which concludes the proof.
\(\square \)