Let us first introduce some additional notation. We will denote the error between the exact solution
X to (
1.5) and the numerical approximation
\({\mathcal {X}}\) defined in (
6.3) by
\(E(t):= X(t) - {\mathcal {X}}(t)\),
\(t \in [0,T]\). Furthermore, it will be convenient to split the error into two parts
$$\begin{aligned} E(t) = E_1(t) + E_2(t), \quad t \in [0,T], \end{aligned}$$
where
$$\begin{aligned} E_1(t)&:= \int _{0}^{t} \big ( {\overline{{\mathcal {H}}}}(s) - \eta (s) \big ) \,\mathrm {d}s + \int _{0}^{t} \big ( b(X(s)) - b({\overline{{\mathcal {X}}}}(s)) \big ) \,\mathrm {d}s, \end{aligned}$$
(6.9)
$$\begin{aligned} E_2(t)&:= G(t) - {\mathcal {G}}(t) \end{aligned}$$
(6.10)
\({\mathbf {P}}\)-almost surely for every
\(t \in (0,T]\). We expand the square of the norm of
E as
$$\begin{aligned} |E(t)|^2 = |E_1(t)|^2 + 2 \langle E_1(t),E_2(t)\rangle _{} + |E_2(t)|^2, \quad t \in [0,T]. \end{aligned}$$
(6.11)
In order to estimate the terms on the right-hand side of (
6.11) we first observe in (
6.9) that
\(E_1\) has absolutely continuous sample paths with
\(E_1(0)=0\). Hence we have
\(\frac{1}{2} \frac{\mathrm {d}}{\,\mathrm {d}t}|E_1(t)|^2 = \langle {\dot{E}}_1(t), E_1(t) \rangle \) for almost every
\(t \in [0,T]\). Therefore, after integrating from 0 to
\(t \in (0,T]\), we get
$$\begin{aligned} \frac{1}{2} |E_1(t)|^2= & {} \int _{0}^{t} \langle {\dot{E}}_1(s),E_1(s)\rangle _{} \,\mathrm {d}s\nonumber \\= & {} \int _{0}^{t} \langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s - \int _{0}^{t} \langle {\dot{E}}_1(s),E_2(s)\rangle _{} \,\mathrm {d}s. \end{aligned}$$
(6.12)
Furthermore, we also have that
$$\begin{aligned} \langle E_1(t),E_2(t)\rangle _{} = \Big \langle \int _0^t {\dot{E}}_1(s) \,\mathrm {d}s, E_2(t) \Big \rangle = \int _0^t \langle {\dot{E}}_1(s),E_2(t)\rangle _{} \,\mathrm {d}s. \end{aligned}$$
(6.13)
Thus, after combining (
6.12) and (
6.13) we obtain
$$\begin{aligned} \begin{aligned}&\frac{1}{2} |E_1(t)|^2 + \langle E_1(t),E_2(t)\rangle _{}\\&\quad = \int _0^t \langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s + \int _0^t \langle {\dot{E}}_1(s),E_2(t)-E_2(s)\rangle _{} \,\mathrm {d}s. \end{aligned}\nonumber \\ \end{aligned}$$
(6.14)
For the first integral on the right-hand side of (
6.14) we insert the derivative of
\(E_1\) and the definition of the error process
E. This yields, for almost every
\(s \in (0, T]\),
$$\begin{aligned} \langle {\dot{E}}_1(s),E(s)\rangle _{} = \langle {\overline{{\mathcal {H}}}}(s) - \eta (s),X(s) - {\mathcal {X}}(s)\rangle _{} + \langle b(X(s)) - b({\overline{{\mathcal {X}}}}(s)),X(s) - {\mathcal {X}}(s)\rangle _{}. \end{aligned}$$
After recalling the definition of
\({\mathcal {X}}\) we use Assumptions
4.1 and
6.3. Then, for almost every
\(s \in (t_{n-1},t_n]\) and all
\(n \in \{1,\ldots ,N\}\), we get
$$\begin{aligned}&\langle {\overline{{\mathcal {H}}}}(s) - \eta (s),X(s) - {\mathcal {X}}(s)\rangle _{}\\&\quad = \frac{t_n - s}{k}\langle \eta ^n - \eta (s) ,X(s) - X^{n-1}\rangle _{} + \frac{s - t_{n-1}}{k} \langle \eta ^n - \eta (s) ,X(s) - X^n\rangle _{}\\&\quad \le \gamma \frac{t_n - s}{k} \langle \eta ^n - \eta ^{n-1} ,X^n - X^{n-1}\rangle _{} - \frac{s - t_{n-1}}{k} \langle \eta (s)- \eta ^n,X(s) - X^n\rangle _{}\\&\quad \le \gamma \frac{t_n - s}{k} \langle \eta ^n - \eta ^{n-1} ,X^n - X^{n-1}\rangle _{}, \end{aligned}$$
where the second term in the last step is non-positive due to the monotonicity of
f (cf. Definition
2.1). Moreover, due to the Lipschitz continuity of
b, it follows for almost every
\(s \in (0,T]\) that
$$\begin{aligned}&\langle b(X(s)) - b({\overline{{\mathcal {X}}}}(s)),X(s) - {\mathcal {X}}(s)\rangle _{}\\&\quad = \langle b(X(s)) - b({\mathcal {X}}(s)),X(s) - {\mathcal {X}}(s)\rangle _{} + \langle b({\mathcal {X}}(s)) - b({\overline{{\mathcal {X}}}}(s)),X(s) - {\mathcal {X}}(s)\rangle _{}\\&\quad \le L_b |E(s)|^2 + L_b |{\mathcal {X}}(s) - {\overline{{\mathcal {X}}}}(s)| |E(s)| \le \frac{3}{2} L_b |E(s)|^2 + \frac{L_b}{2} |{\mathcal {X}}(s) - {\overline{{\mathcal {X}}}}(s)|^2, \end{aligned}$$
where we also made use of Young’s inequality. In addition, for every
\(n \in \{1,\ldots ,N\}\) and
\(s \in (t_{n-1},t_n]\), we have that
$$\begin{aligned} {\mathcal {X}}(s) - {\overline{{\mathcal {X}}}}(s) = \frac{s-t_{n-1}}{k} X^n + \frac{t_n-s}{k} X^{n-1} - X^n = - \frac{t_n-s}{k} \big (X^n - X^{n-1}\big ). \end{aligned}$$
Therefore,
$$\begin{aligned} \langle b(X(s)) - b({\overline{{\mathcal {X}}}}(s)),X(s) - {\mathcal {X}}(s)\rangle _{} \le \frac{3}{2} L_b |E(s)|^2 + \frac{L_b(t_n-s)^2}{2 k^2} |X^n - X^{n-1}|^2. \end{aligned}$$
Altogether, for every
\(t \in (t_{n-1}, t_n]\) and
\(n \in \{1,\ldots ,N\}\), we have shown that
$$\begin{aligned} \int _{t_{n-1}}^t \langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s&\le \frac{\gamma }{2} k \langle \eta ^n - \eta ^{n-1} ,X^n - X^{n-1}\rangle _{}\\&\quad + \frac{3}{2} L_b \int _{t_{n-1}}^{t} | E(s)|^2 \,\mathrm {d}s + \frac{L_b}{6} k |X^n - X^{n-1}|^2, \end{aligned}$$
where we also inserted that
\(\int _{t_{n-1}}^{t} (t_n - s) \,\mathrm {d}s \le \int _{t_{n-1}}^{t_n} (t_n - s) \,\mathrm {d}s = \frac{1}{2} k^2\) as well as
\(\int _{t_{n-1}}^{t} (t_n - s)^2 \,\mathrm {d}s \le \frac{1}{3} k^3\). It follows that, for every
\(n \in \{1,\ldots ,N\}\) and
\(t \in (t_{n-1},t_n]\),
$$\begin{aligned}&\int _0^t\langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s = \sum _{i = 1}^{n-1} \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s + \int _{t_{n-1}}^t \langle {\dot{E}}_1(s),E(s)\rangle _{} \,\mathrm {d}s \\&\quad \le \frac{\gamma }{2} k \sum _{i = 1}^n \langle \eta ^i - \eta ^{i-1} ,X^i - X^{i-1}\rangle _{} + \frac{L_b}{6} k \sum _{i = 1}^n |X^i - X^{i-1}|^2\\&\qquad + \frac{3}{2} L_b \int _0^t |E(s)|^2 \,\mathrm {d}s. \end{aligned}$$
Hence, together with Lemmas
5.4 and
6.2 this shows that
$$\begin{aligned} \int _0^t {\mathbf {E}}\big [\langle {\dot{E}}_1(s),E(s)\rangle _{}\big ] \,\mathrm {d}s&\le \frac{\gamma }{2} K_{\delta \eta } k^{\frac{1}{2}} + \frac{L_b}{3} K_X k + \frac{3}{2} L_b \int _0^t {\mathbf {E}}\big [|E(s)|^2\big ] \,\mathrm {d}s. \end{aligned}$$
(6.15)
Next, we give an estimate for the second integral on the right-hand side of (
6.14). For every
\(n \in \{1,\ldots ,N\}\) and
\(t \in (t_{n-1},t_n]\) we decompose the integral as follows
$$\begin{aligned} \int _{0}^{t} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s= & {} \sum _{i = 1}^{n-1} \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s\nonumber \\&\quad + \int _{t_{n-1}}^t \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s. \end{aligned}$$
(6.16)
For every
\(i \in \{1,\ldots ,n-1\}\) we then add and subtract
\(E_2(t_i)\) in the second slot of the inner product in the first term on the right-hand side of (
6.16). This gives
$$\begin{aligned} \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s&= \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t) - E_2(t_{i})\rangle _{} \,\mathrm {d}s\\&\quad + \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t_{i}) - E_2(s)\rangle _{} \,\mathrm {d}s. \end{aligned}$$
After inserting the definition of
\(E_2\) from (
6.10) the first integral is then equal to
$$\begin{aligned}&\int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t) - E_2(t_{i})\rangle _{} \,\mathrm {d}s = \Big \langle \int _{t_{i-1}}^{t_i} {\dot{E}}_1(s) \,\mathrm {d}s, E_2(t) - E_2(t_{i}) \Big \rangle \\&\quad = \langle E_1(t_i) - E_1(t_{i-1}),E_2(t) - E_2(t_{i})\rangle _{}\\&\quad = \langle E_1(t_i) - E_1(t_{i-1}),G(t) - {\mathcal {G}}(t) - ( G(t_{i}) - {\mathcal {G}}(t_i)) \rangle _{} \\&\quad = \Big \langle E_1(t_i) - E_1(t_{i-1}), \int _{t_i}^{t} g(X(s)) \,\mathrm {d}W(s) \Big \rangle \\&\qquad - \Big \langle E_1(t_i) - E_1(t_{i-1}), \int _{t_i}^{t_{n-1}} g({\underline{{\mathcal {X}}}}(s)) \,\mathrm {d}W(s) + \frac{t - t_{n-1}}{k} g(X^{n-1}) \varDelta W^n \Big \rangle \end{aligned}$$
for all
\(i, n \in \{1,\ldots ,N\}\),
\(i < n\), and
\(t \in (t_{n-1},t_n]\). Since
\(E_1(t_i) - E_1(t_{i-1}) = E(t_i) - E(t_{i-1}) - (E_2(t_i) - E_2(t_{i-1}))\) is square-integrable and
\({\mathcal {F}}_{t_i}\)-measurable it therefore follows that
$$\begin{aligned} {\mathbf {E}}\Big [ \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t) - E_2(t_{i})\rangle _{} \,\mathrm {d}s \Big ] = 0 \end{aligned}$$
for all
\(n \in \{1,\ldots ,N\}\),
\(t \in (t_{n-1},t_n]\), and
\(t_i < t\). Hence, after taking expectations in (
6.16), we arrive at
$$\begin{aligned}&{\mathbf {E}}\Big [ \int _{0}^{t} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s \Big ]\\&\quad = \sum _{i = 1}^{n-1} {\mathbf {E}}\Big [ \int _{t_{i-1}}^{t_i} \langle {\dot{E}}_1(s),E_2(t_i) - E_2(s)\rangle _{} \,\mathrm {d}s \Big ]\\&\qquad + {\mathbf {E}}\Big [ \int _{t_{n-1}}^t \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s \Big ]\\&\quad \le \sum _{i = 1}^{n} {\mathbf {E}}\Big [ \int _{t_{i-1}}^{t_i} |{\dot{E}}_1(s)| |E_2(t_i) - E_2(s)| \,\mathrm {d}s \Big ]. \end{aligned}$$
Inserting the definitions (
6.9) and (
6.10) of
\(E_1\) and
\(E_2\) and applying Hölder’s inequality with
\(\rho = \max (2,p)\) and
\(\frac{1}{\rho }+\frac{1}{\rho '}=1\), we get
$$\begin{aligned}&{\mathbf {E}}\Big [ \int _{0}^{t} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s \Big ]\\&\quad \le \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ \big (|\eta ^i - \eta (s)| + |b(X(s)) - b(X^i)| \big )\\&\qquad \times \big (| G(t_i) - G(s)| + |{\mathcal {G}}(t_i) - {\mathcal {G}}(s)| \big ) \big ] \,\mathrm {d}s\\&\quad \le \sum _{i = 1}^{n} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |\eta ^i - \eta (s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | G(t_i) - G(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}\\&\qquad + \sum _{i = 1}^{n} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |\eta ^i - \eta (s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | {\mathcal {G}}(t_i) - {\mathcal {G}}(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}\\&\qquad + \sum _{i = 1}^{n} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |b(X(s)) - b(X^i)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | G(t_i) - G(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}\\&\qquad + \sum _{i = 1}^{n} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |b(X(s)) - b(X^i)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} \Big ( \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | {\mathcal {G}}(t_i) - {\mathcal {G}}(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}\\&\quad =: \varGamma _1 + \varGamma _2 + \varGamma _3 + \varGamma _4. \end{aligned}$$
In the following, we will estimate
\(\varGamma _1\),
\(\varGamma _2\),
\(\varGamma _3\), and
\(\varGamma _4\) separately. For
\(\varGamma _1\) we obtain after an application of Hölder’s inequality for sums that
$$\begin{aligned} \varGamma _1&\le \Big ( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |\eta ^i - \eta (s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} \Big ( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | G(t_i) - G(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}\\&\le \Big (\Big ( k \sum _{i = 1}^{n} {\mathbf {E}}\big [ |\eta ^i|^{\rho '} \big ] \Big )^{\frac{1}{\rho '}} + \Big ( \int _{0}^{t_n} {\mathbf {E}}\big [ |\eta (s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}}\Big )\\&\quad \times \Big ( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | G(t_i) - G(s)|^\rho \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho }}. \end{aligned}$$
If
\(p \in [2,\infty )\) then
\(\rho = p\) and
\(\rho ' = q\). In this case all integrals appearing are finite due to the bounds in Theorem
4.7 and Lemma
5.4. Moreover, if
\(p \in (1,2)\) then
\(\rho = \rho ' = 2 < q\). Then it follows from further applications of Hölder’s inequality and Jensen’s inequality that
$$\begin{aligned} k \sum _{i = 1}^{n} {\mathbf {E}}\big [ |\eta ^i|^{2} \big ] \le T^{\frac{q-2}{2}} \Big ( k \sum _{i = 1}^{n} {\mathbf {E}}\big [ |\eta ^i|^{q} \big ] \Big )^{\frac{2}{q}} \end{aligned}$$
as well as
$$\begin{aligned} \int _{0}^{t_n} {\mathbf {E}}\big [ |\eta (s)|^{2} \big ] \,\mathrm {d}s \le T^{\frac{q-2}{2}} \Big ( \int _{0}^{t_n} {\mathbf {E}}\big [ |\eta (s)|^{q} \big ] \,\mathrm {d}s \Big )^{\frac{2}{q}}. \end{aligned}$$
Hence, we arrive at the same conclusion. If
\(p = 1\) then the processes
\((\eta (t))_{t \in [0,T]}\) and
\((\eta ^n)_{n \in \{1,\ldots ,N\}}\) are globally bounded due to the bound on
f in Assumption
4.1. Using Lemma
6.1 we see that
$$\begin{aligned} \left( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | G(t_i) - G(s)|^\rho \big ] \,\mathrm {d}s \right) ^{\frac{1}{\rho }} \le K_{\rho } k^{\frac{1}{2}} \left( \int _0^{t_n} \big ( 1 + {\mathbf {E}}\big [ | X(s)|^\rho \big ] \big ) \,\mathrm {d}s \right) ^{\frac{1}{\rho }}. \end{aligned}$$
Altogether, this yields
$$\begin{aligned} \varGamma _1 \le C_{\varGamma _1} k^{\frac{1}{2}} \end{aligned}$$
for a suitable constant
\(C_{\varGamma } \in (0,\infty )\), which is independent of
k. To estimate
\(\varGamma _2\), we argue analogously as in the case for
\(\varGamma _1\) to obtain that
$$\begin{aligned} \varGamma _2&\le \left( \left( k \sum _{i = 1}^{n} {\mathbf {E}}\big [ |\eta ^i|^{\rho '} \big ] \right) ^{\frac{1}{\rho '}} + \Big ( \int _{0}^{t} {\mathbf {E}}\big [ |\eta (s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}}\right) \\&\quad \times \left( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | {\mathcal {G}}(t_i) - {\mathcal {G}}(s)|^\rho \big ] \,\mathrm {d}s \right) ^{\frac{1}{\rho }}. \end{aligned}$$
The first factor is bounded as we saw in the case for
\(\varGamma _1\). Furthermore, using Lemma
6.1, we have that
$$\begin{aligned} \left( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ | {\mathcal {G}}(t_i) - {\mathcal {G}}(s)|^\rho \big ] \,\mathrm {d}s \right) ^{\frac{1}{\rho }} \le K_{\rho } k^{\frac{1}{2}} \left( k \sum _{i =1}^{n} \big ( 1 + {\mathbf {E}}\big [ | X^{i-1}|^\rho \big ] \big ) \,\mathrm {d}s \right) ^{\frac{1}{\rho }}. \end{aligned}$$
Due to the a priori bound (
5.4), it follows that there exists a constant
\(C_{\varGamma _2} \in (0,\infty )\), which does not depend on
k such that
$$\begin{aligned} \varGamma _2 \le C_{\varGamma _2} k^{\frac{1}{2}}. \end{aligned}$$
The estimates
\(\varGamma _3\) and
\(\varGamma _4\) follow analogously with the only new term that appears is of the form
$$\begin{aligned}&\Big ( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |b(X(s)) - b(X^i)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}}\\&\quad \le L_b \Big ( \sum _{i = 1}^{n} \int _{t_{i-1}}^{t_i} {\mathbf {E}}\big [ |X(s) - X^i|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}}\\&\quad \le L_b \Big ( \int _{0}^{t_n} {\mathbf {E}}\big [ |X(s)|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}} + L_b \Big ( k \sum _{i = 1}^{n} {\mathbf {E}}\big [|X^i|^{\rho '} \big ] \,\mathrm {d}s \Big )^{\frac{1}{\rho '}}, \end{aligned}$$
which is bounded due to Theorem
4.7 and the a priori bound (
5.4). Therefore, there exist constants
\(C_{\varGamma _3}, C_{\varGamma _4} \in (0,\infty )\) such that
$$\begin{aligned} \varGamma _3 \le C_{\varGamma _3} k^{\frac{1}{2}} \quad \text {and} \quad \varGamma _4 \le C_{\varGamma _4} k^{\frac{1}{2}}. \end{aligned}$$
Hence, we obtain
$$\begin{aligned} {\mathbf {E}}\Big [ \int _{0}^{t} \langle {\dot{E}}_1(s),E_2(t) - E_2(s)\rangle _{} \,\mathrm {d}s \Big ] \le (C_{\varGamma _1} + C_{\varGamma _2} + C_{\varGamma _3} + C_{\varGamma _4}) k^{\frac{1}{2}} =: C_{\varGamma } k^{\frac{1}{2}} . \end{aligned}$$
(6.17)
After taking expectations in (
6.11) and inserting (
6.14), (
6.15), (
6.17) as well as (
6.4) from Lemma
6.1, we obtain for every
\(t \in (0,T]\) that
$$\begin{aligned}&{\mathbf {E}}\big [ |E(t)|^2 \big ] \\&\quad \le \gamma K_{\delta \eta } k^{\frac{1}{2}} + \frac{2 L_b}{3} K_X k + 2 C_\varGamma k^{\frac{1}{2}} + K_G k + \big ( 3 L_b + 2 L_g^2 \big ) \int _0^t |E(s)|^2 \,\mathrm {d}s. \end{aligned}$$
The assertion then follows from an application of Gronwall’s lemma, see for example, [
11, Appendix B].
\(\square \)