In this section, we prove an averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in \(L^{p}\ (p\geq2)\) and in probability between the MSDEs and the corresponding averaged MSDEs.
For \(t\in[0, T]\), consider
$$\begin{aligned} dX^{\epsilon}_{t}+\epsilon\mathcal{A} \bigl(X^{\epsilon}_{t}\bigr)\,dt\ni\epsilon f\bigl(t,X^{\epsilon}_{t} \bigr)\,dt+\sqrt{\epsilon}g\bigl(t,X^{\epsilon}_{t} \bigr)\,dB_{t},\qquad X^{\epsilon}_{0}=x\in\overline{D( \mathcal{A})}. \end{aligned}$$
(3.1)
The standard form of (3.1) is defined as
$$\begin{aligned} X^{\epsilon}_{t}=X^{\epsilon}(0)+\epsilon \int_{0}^{t}f\bigl(s,X^{\epsilon }(s)\bigr)\,ds+ \sqrt{\epsilon} \int_{0}^{t}g\bigl(s,X^{\epsilon}_{s} \bigr)\,dB_{s}-\epsilon K(t),\quad t\in[0, T], \end{aligned}$$
(3.2)
and the corresponding averaged MSDEs of (3.2) are defined as
$$\begin{aligned} Y^{\epsilon}_{t}=Y^{\epsilon}(0)+\epsilon \int_{0}^{t}\overline {f}\bigl(Y^{\epsilon}(s) \bigr)\,ds+\sqrt{\epsilon} \int_{0}^{t}\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\,dB_{s}-\epsilon \overline{K}(t),\quad t\in[0, T]. \end{aligned}$$
(3.3)
Here \(\overline{f}:R^{d}\rightarrow R^{d}\) and \(\overline {g}:R^{d}\rightarrow R^{d}\) are measurable functions satisfying the non-Lipschitz conditions with respect to x as \(f(t,x)\) and \(g(t,x)\), \(Y^{\epsilon}(0)=X^{\epsilon}(0)=x\), and \(f,\overline{f}\), \(g,\overline{g}\) satisfy \(\mathbf{H2}\) and \(\mathbf{H3}\).
Now, we are in the position to investigate the relationship between the processes
\(X^{\epsilon}_{t}\)
and
\(Y^{\epsilon}_{t}\)
.
Theorem 3.1
Suppose that conditions
\(\mathbf{H1}\)-\(\mathbf{H4}\)
hold. Then, for a given arbitrarily small number
\(\delta>0\)
and for
\(\alpha\in(0, \frac{1}{2})\), there exists a number
\(\tilde{\epsilon}\in(0, \epsilon_{0}]\)
\((\epsilon_{0}=\frac{1}{16p^{2}})\)
such that, for all
\(\epsilon\in(0, \tilde{\epsilon})\)
and
\(p\geq1\), we have
$$E\Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\Bigr)\leq \delta. $$
Proof
Consider the difference \(X^{\epsilon}_{t}-Y^{\epsilon}_{t}\). From (3.2) and (3.3) we have
$$ \begin{aligned} X^{\epsilon}_{t}-Y^{\epsilon}_{t}={}& \epsilon \int_{0}^{t}\bigl[f\bigl(s,X^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr)\bigr]\,ds \\ &{}+\sqrt{\epsilon} \int_{0}^{t}\bigl[g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr]\,dB_{s}-\epsilon \bigl[K(t)-\overline{K}(t)\bigr]. \end{aligned} $$
By Itô’s formula [23],
$$ \begin{aligned} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}={}&{-}\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s}, dK(s)-d\overline {K}(s)\bigr\rangle \\ &{}+\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \\ &{}+\sqrt{\epsilon}2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \\ &{}+\epsilon p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ &{}+2\epsilon p(p-1) \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \end{aligned} $$
By Definition 2.2 and Lemma 2.1 we get
$$ \begin{aligned} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\leq{}&\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \\ &{}+\sqrt{\epsilon}2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \\ &{}+\epsilon p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ &{}+2\epsilon p(p-1) \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \end{aligned} $$
Then
$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\\ &\quad\leq \epsilon2pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ &\qquad{}+\sqrt{\epsilon}2pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \biggr\vert \\ &\qquad{}+\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \biggr\vert \\ &\qquad{}+2\epsilon p(p-1)E\sup_{0\leq t\leq T} \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \\ &\quad= I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned} $$
We now estimate \(I_{1},I_{2},I_{3}, I_{4}\) separately.
Estimate of
\(I_{1}\). Using the trigonometric inequality, we have
$$ \begin{aligned} I_{1}={}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ \leq{}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon }(s)\bigr)-f \bigl(s,Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,ds \biggr\vert \\ &{}+2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,Y^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ ={}& I_{11}+I_{12}. \end{aligned} $$
For \(I_{11}\), using the non-Lipschitz condition of f and the Cauchy-Schwarz inequality, we have
$$ \begin{aligned} I_{11}\leq{}& 2\epsilon pE\sup _{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \biggr\vert \\ \leq{}& 2\epsilon p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-1}\rho_{1,\eta }\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$
For \(I_{12}\), using the Hölder and Young inequalities, we deduce
$$ \begin{aligned} I_{12}={}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,Y^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,ds \biggr\vert \\ \leq{}&\epsilon pE\sup_{0\leq t\leq T}\biggl| \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl( \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}+ \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2}\bigr)\,ds\biggr| \\ \leq{}& \epsilon pE\sup_{0\leq t\leq T} \int_{0}^{t}\biggl(\frac{2p-2}{2p} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\frac{1}{p}\biggr)\bigl( \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ \leq{}& \epsilon\bigl[(p-1)C_{T}^{(p, \Vert x \Vert )}+1\bigr]E\biggl(\sup _{0\leq t\leq T}t\frac {1}{t} \int_{0}^{t} \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}\,ds \biggr) \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds. \end{aligned} $$
Taking condition \(\mathbf{H 2}\), Lemma 2.2, and the Young inequality into account, we have
$$ \begin{aligned} I_{12}\leq{}& \epsilon\bigl[(p-1)C_{T}^{(p, \Vert x \Vert )}+1 \bigr]\sup_{0\leq t\leq T}\Bigl\{ t\varphi_{1}{(t)}\Bigl[1+E \Bigl(\sup_{0\leq s\leq T} \bigl\Vert Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\Bigr]\Bigr\} \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ \leq{}& \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds+\epsilon C\bigl(p,T, \Vert x \Vert \bigr) T. \end{aligned} $$
Finally, we have
$$ \begin{aligned} I_{1}\leq{}& \epsilon T C\bigl(p,T, \Vert x \Vert \bigr)+\epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ &{}+2\epsilon p \int_{0}^{T}E\sup_{0\leq u\leq s} \bigl\Vert X^{\epsilon}_{u}-Y^{\epsilon}_{u} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{u}-Y^{\epsilon}_{u} \bigr\Vert \bigr)\,ds. \end{aligned} $$
Estimate of
\(I_{2}\). Using the Burkholder-Davis-Gundy and Young inequalities, we have
$$ \begin{aligned} I_{2}={}&\sqrt{\epsilon}2pE\sup _{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon }_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,dB_{s} \biggr\vert \\ \leq{}& \sqrt{\epsilon}8p E\biggl\{ \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{4p-4} \bigl\vert \bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \bigr\vert ^{2}\,ds\biggr\} ^{\frac{1}{2}} \\ \leq{}& \sqrt{\epsilon}8p E\biggl\{ \int_{0}^{T}\sup_{0\leq s\leq T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds\biggr\} ^{\frac{1}{2}} \\ \leq{}& \sqrt{\epsilon}4p E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+ \sqrt{\epsilon}4pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon }_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ ={}&I_{21}+I_{22}. \end{aligned} $$
In the following, we estimate
$$ \begin{aligned} I_{22}=\sqrt{\epsilon}4pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds. \end{aligned} $$
Using conditions \(\mathbf{H1}\) and \(\mathbf{H3}\) and the Young inequality, we get:
$$ \begin{aligned} I_{22}\leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl( \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-g\bigl(s,Y^{\epsilon }_{s}\bigr) \bigr\Vert ^{2}+ \bigl\Vert g\bigl(s,Y^{\epsilon}_{s}\bigr)-\overline{g} \bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}& \sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\bigl(\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)+ \bigl\Vert g\bigl(s,Y^{\epsilon}_{s}\bigr)- \overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &{}+\sqrt{\epsilon}8pE \int_{0}^{T}\biggl[\frac{2p-2}{2p} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\frac{1}{p}\biggr]\bigl( \bigl\Vert g \bigl(s,Y^{\epsilon }_{s}\bigr)-\overline{g}\bigl(Y^{\epsilon}_{s} \bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &{}+\sqrt{\epsilon}C\bigl(p,T, \Vert x \Vert \bigr)E \biggl(\sup _{0\leq t\leq T}t\frac {1}{t} \int_{0}^{t} \bigl\Vert g\bigl(s,Y^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr) \bigr\Vert ^{2}\,ds\biggr) \\ \leq{}&\sqrt{\epsilon} T C_{2}\bigl(p,T, \Vert x \Vert \bigr)+\sqrt{ \epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$
Combing the estimates of \(I_{21}\) and \(I_{22}\), we conclude that
$$ \begin{aligned} I_{2}\leq{}& \sqrt{\epsilon}4p E\sup _{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+\sqrt{\epsilon} T C_{2}\bigl(p,T, \Vert x \Vert \bigr) \\ &{}+\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$
Estimate of
\(I_{3}\). Note that
$$ \begin{aligned} I_{3}=\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \biggr\vert . \end{aligned} $$
Using the same estimate as for \(I_{22}\), we have
$$I_{3}\leq\epsilon TC_{3}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon2pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. $$
Estimate of
\(I_{4}\). Using the Cauchy-Schwarz inequality, the term \(I_{4}\) has the same form with \(I_{3}\) with a different constant:
$$I_{4}\leq\epsilon TC_{4}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon4p(p-1)E \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. $$
Combing the estimates of \(I_{1},I_{2} \mbox{ and } I_{3},I_{4}\), we have
$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \\ &\quad\leq \epsilon TC_{1}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon C_{1}(p) \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+2\epsilon p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert \bigr)\,dt \\ &\qquad{}+\sqrt{\epsilon}4p E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+\sqrt{\epsilon} C_{2}\bigl(p,T, \Vert x \Vert \bigr) \\ &\qquad{}+\sqrt{\epsilon}8p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\epsilon TC_{3}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon 2p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho ^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\epsilon TC_{4}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon 4p(p-1) \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho ^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$
Taking \(\sqrt{\epsilon}4p<1\), that is, \(\epsilon<\frac{1}{16p^{2}}\), we have
$$ \begin{aligned}& E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\\ &\quad\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p}+\frac{\epsilon C_{1}(p)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+\frac{2\epsilon p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon }_{t} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert \bigr)\,dt \\ &\qquad{}+\frac{\sqrt{\epsilon}8p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\frac{\epsilon 2p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\frac{\epsilon 4p(p-1)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$
By Lemma 2.4 and the concavity of the function \(\rho_{1,\eta}\) we have
$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \\ &\quad\leq\frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p}+\frac{\epsilon C_{1}(p)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+\frac{\sqrt{\epsilon}C_{6}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \int_{0}^{T}\rho_{1,\eta}\Bigl(E\sup _{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\,dt \\ &\quad\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \\ &\qquad{}+\frac{\sqrt{\epsilon}C_{7}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\rho_{1,\eta}\Bigl(E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\,dt. \end{aligned} $$
Note that, for sufficiently small ϵ, we have \(g(0)=\frac{\sqrt{\epsilon}C_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt {\epsilon}4p}\leq\eta<\frac{1}{e}\), and from Lemma 2.3 and Example 2.1 we get the following estimate:
$$ E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon}4p} \exp^{(1-\ln\eta)\exp\{-\frac{\sqrt{\epsilon} TC_{7}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon}4p}\}}. $$
Choose \(\alpha\in(0,\frac{1}{2})\) such that, for every \(t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]\subseteq[0, T]\), we have
$$E \Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt {\epsilon})]} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \Bigr)\leq C\epsilon^{\alpha}, $$
where
$$ C=C_{5}\bigl(p, \Vert x \Vert ,T,\epsilon\bigr) \exp^{\{(1-\ln\eta)\exp\{-\epsilon ^{\alpha}(C_{7}(p, \Vert x \Vert ,T,\epsilon)\}\}}. $$
Consequently, given any number \(\delta>0\), we can choose \(\tilde{\epsilon}\in(0, \epsilon_{0}]\) (\(\epsilon_{0}=\frac{1}{16p^{2}}\)) such that, for each \(\epsilon\in(0, \tilde{\epsilon})\) and every \(t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]\),
$$E\Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\Bigr)\leq \delta, $$
which completes the proof of the theorem. □
Using the Chebyshev-Markov inequality, we can also get the convergence in probability.
Theorem 3.2
Suppose that conditions
\(\mathbf{H1}\)-\(\mathbf{H4}\)
hold. Then, for a given arbitrarily small number
\(\theta>0\)
and for
\(\alpha\in(0, \frac{1}{2})\), there exists a number
\(\tilde{\epsilon}\in(0, \epsilon_{0}]\)
\((\epsilon_{0}=\frac{1}{16p^{2}})\)
such that, for all
\(\epsilon\in(0, \tilde{\epsilon})\)
and
\(p\geq 1\), we have
$$\lim_{\epsilon\rightarrow0}\mathbb{P}\Bigl(\sup_{t\in[0, \epsilon ^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}>\theta\Bigr)=0. $$