You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We consider the stability problems of one dimensional SDEs when the diffusion coefficients satisfy the so called Nakao-Le Gall condition. The explicit rate of convergence of the stability problems are given by the Yamada-Watanabe method without the drifts. We also discuss the convergence rate for the SDEs driven by the symmetric $\alpha$ stable process. These stability rate problems are extended to the case where the drift coefficients are bounded and in $L^1$. It is shown that the convergence rate is invariant under the removal of drift method for the SDEs driven by the Wiener process.'
address:
- 'Sanwa Kagaku Kenkyusho Co., Ltd.'
- 'School of Computer Science and Engineering, The University of Aizu'
author:
- Hiroya Hashimoto
- Takahiro Tsuchiya
title: ' Convergence rate of stability problems of SDEs with (dis-)continuous coefficients '
---

Introduction
============

Consider the following sequence of one-dimensional stochastic differential equations, SDEs for short, $$\label{sde:stability}
X_n (t) = X_n (0)+\int_{0}^t b_n (X_n (s)) ds+\int_{0}^t \sigma_n (X_n (s)) dW_s,$$ and consider the solution $X$ given by $$\label{sde0}
X(t) = X(0)+\int_{0}^t b (X(s)) ds+\int_{0}^t \sigma (X(s)) dW_s,$$ where $\{ W_s \}_{s \geq 0 }$ is a Wiener process and $b_n : {\mathbb{R}}\rightarrow {\mathbb{R}}$ and $\sigma_n : {\mathbb{R}}\rightarrow {\mathbb{R}}$ for $n \in {\mathbb{N}}$ are coefficients which tend to $b$ and $\sigma$ respectively in some sense, as $n \rightarrow \infty$. The convergence of the sequence $\{ X_n \}_{n \in {\mathbb{N}}}$ to $X$ named as stability problems was introduced by Stroock and Varadhan in [@SV1979] to solve the martingale problems for unbounded coefficients $b$ and $\sigma$. The stability problem in the [*strong*]{} sense was treated by Kawabata-Yamada [@KY] in the case where the diffusion coefficients are Hölder continuous of exponent $\alpha \geq 1/2$. Le Gall investigated the stability problems, when the diffusion coefficients are positive and squared finite quadratic variation in [@LeGall1983]. Kaneko and Nakao showed in [@KN1988] that the pathwise uniqueness implies the stability property if the coefficients satisfy some condition on the modulus continuity or if the diffusion coefficient is positive definite.

The rate of convergence of the Euler-Maruyama scheme to the solution has been discussed by Deelstra and Delbaen in [@Del1998] and their results are considerably generalized by Gyöngy and Rásonyi in [@GR2011]. In their investigations, the Yamada-Watanabe method introduced in [@WY], [@YW] and [@Y1978] plays essential roles. Gyöngy and Rásonyi in [@GR2011] obtained the rate of convergence in the case where the diffusion coefficients are $(1/2 +  \gamma)$-Hölder continuous with the suitable drift coefficients in $L^1$. It is remarkable that the rate of convergence is given by $n^{-\gamma}$ where $ \gamma > 0$ and also is given by $(\log n)^{-1}$ in the case where $\gamma=0$.

These results suggest that the rate of convergence of the stability problems may depend also on the modulus continuity or irregularity of diffusion coefficients. Since the coefficient may be discontinuous under the Nakao-Le Gall condition, it seems to be very interesting to investigate the rate of convergence of the stability problems.

The present paper is organized as follows: Section \[sec:notation\] is devoted to the assumption and preliminaries. Our main result is shown in Section \[stability problem\]. Let us consider first drift-less system $$\begin{aligned}
X_n (t)&=X_n (0)+\int_{0}^t \sigma_n (X_n (s)) dW_s, \label{sde without drifts}  \\
X     (t)&=X     (0)   +\int_{0}^t \sigma (X(s)) dW_s, \label{sde without drifts 2}\end{aligned}$$ for $t \geq 0$ and $X_n (0) \equiv X (0)$. These diffusion coefficients satisfy the so called Nakao-Le Gall condition (See Definition $\ref{conditionC}$ below). Assume that there exists a positive constant $C_0$ such that $$\begin{aligned}
\int_{{\mathbb{R}}} | \sigma_n (x) - \sigma (x)  | dx \leq C_0 n^{-1} 
{\rm \ or \ }
\sup_{x \in{\mathbb{R}}} | \sigma_n (x) - \sigma (x)  |  \leq C_0 n^{-1},\end{aligned}$$ for all $\in {\mathbb{N}}$. It will be shown that there exist positive constants $C_p \ (p \geq 1)$ such that $$\begin{aligned}
{\mathbb{E}}\left[ \left| X (t) - X_n (t) \right| \right] \leq C_1 (\log n)^{-\frac{1}{2}} \ ( 0 \leq t \leq T )\end{aligned}$$ and $$\begin{aligned}
{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} \left| X (t) - X_n (t) \right|^p \right] \leq C_p (\log n)^{-\frac{p}{4(p+1)}} \end{aligned}$$ holds for $p > 1$ and $n>2$. In Section \[section:sas\], we also discuss the convergence rate of the stability problems in the case where the SDEs are driven by a symmetric $\alpha$ stable process without the drift for $\alpha \in (1,2)$. Finally, the stability problems with the drift coefficients will be discussed using the removal of drift method in Section \[Removal invariant\].

Assumptions and Preliminaries {#sec:notation}
=============================

On a probability space $(\Omega, \mathcal{F}, {\mathbb{P}})$ equipped with the filtration $\{ \mathcal{F}_t \}_{t \geq 0}$ satisfying the usual condition, let us consider the stochastic differential equations $(\ref{sde:stability})$ and $(\ref{sde0})$. For the simplify, we assume that $X (0) \equiv X_n (0) $. The following condition on the diffusion coefficients was proposed by Nakao [@N1972] and modified by Le Gall [@LeGall1983].

\[conditionC\] In this paper, we say that a real valued function $\sigma$ satisfies the Nakao-Le Gall condition and write $\sigma \in \mathcal{C}_{NL} (\epsilon, f)$ if $\sigma$ satisfies the following statements:

1.  There exists a positive real number $\epsilon$ such that $$\epsilon  \leq \sigma (x) 
    $$ holds for any $x$ in $\mathbb{R}$.

2.  There exists an increasing function $f$ such that $$\begin{aligned}
    |\sigma (x) - \sigma (y) |^2 \leq | f(x) - f(y) |\end{aligned}$$\[conditionC2\] holds for every $x$ and $y$ in $\mathbb{R}$.

In addition, if the function $f$ is bounded on an interval $I \subset {\mathbb{R}}$ as follows: $$\begin{aligned}
\| f \|_{I, \infty} :=\sup_{x \in I} |f(x)| < \infty, \end{aligned}$$ then we denote $\sigma \in  \mathcal{C}_{NL} (\epsilon, \| f \|_{I, \infty})$. In particular, in the case where $I={\mathbb{R}}$, we write $\sigma \in  \mathcal{C}_{NL} (\epsilon, \| f \|_{\infty})$ and $\| f \|_{\infty} :=\| f \|_{{\mathbb{R}}, \infty}$.

For the equation $(\ref{sde0})$, if $b$ is bounded measurable function and $\sigma \in \mathcal{C}_{NL} (\epsilon, \| f \|_{\infty})$, then there exists a unique strong solution $X$, see Theorem $1.3~(3)$ of the paper [@LeGall1983].

\[rem:disc\] If the Nakao-Le Gall condition holds, $\sigma \in \mathcal{C}_{NL} (\epsilon, f )$, we can select a sequence of smooth functions $\{ f_l \}_{l \in \mathbb{N}}$ such that for continuous points $x, y \in {\mathbb{R}}$, $$|\sigma (x) - \sigma (y) |^2 \leq | f_l (x) - f_l (y) | 
{\rm  \ and \ } f_l (x) \uparrow f (x) \ (l \rightarrow \infty) .$$ Then for an progressively measurable process $X$ we have that $$\int_{0}^{t} f(X(s)) ds = \lim_{l \to \infty} \int_{0}^{t} f_l (X(s)) ds$$ holds almost surely. We use this fact in the proof of Theorem \[Thm:main1\] through \[Thm:main3\].

Now, we shall introduce the Yamada-Watanabe method. Take a decreasing sequence $( a_m )_{ m \in {\mathbb{N}}}$ of positive numbers satisfying $\infty > a_1 > \cdots > a_m > \cdots \downarrow 0$ and $\int_{a_{m}}^{a_{m-1}} x^{-1} \, dx=m $. Let us consider a smooth symmetric around the origin function $\varphi_{m} (\cdot)$ with support in $(-a_{m-1}, -a_{m})$ and $(a_{m}, a_{m-1})$ such that $0 \leq \varphi_{m} (x) \leq {2} \left( x m \right)^{-1}$ holds and $\int_{{\mathbb{R}}}\varphi_{m} (y) dy=1 $. For all $x \in {\mathbb{R}}$ define $$u_m (x) = \int_{0}^{|x|} \int_0^y \varphi_m (z) \, dz \, dy.$$ Then we have $$\begin{aligned}
\label{Def:Yfunc}
|x| \geq u_m (x) \geq  -a_{m-1} + |x|
, \ \ 1 \geq | u'_m (x) | 
$$ and $u''_m(x) =\varphi_m (x)$ holds for $x\in {\mathbb{R}}$, see [@Y1978].

A strong convergence rate of the stability problems {#stability problem}
===================================================

In this section, we consider the convergence rate of the stability problems for the drift-less system $(\ref{sde without drifts})$ and $(\ref{sde without drifts 2})$ under the Nakao-Le Gall condition. The stability problem with the $L^1$-convergence was discussed by Le Gall [@LeGall1983]. In more detail, the explicit rate of the convergence in $L^p$ sup-norm is obtained as follows:

\[Thm:main1\] Let $T>0$ and $p \geq 1$ and let $X_n$ for $n \in {\mathbb{N}}$ be solutions of $(\ref{sde without drifts})$ and $X$ be a solution of $(\ref{sde without drifts 2})$ such that ${\mathbb{E}}\left| X (0) \right|^p < \infty $ and $X_n (0) \equiv X (0)$. Suppose that $\sigma$ and $\{ \sigma_n \}_{n \in {\mathbb{N}}}$ satisfy the Nakao-Le Gall condition, i.e., $\sigma, \sigma_n \in \mathcal{C}_{NL} (\epsilon, \| f\|_{\infty} )$. Assume one of the following stability rate conditions holds: there exists a positive constant $C_0$ such that $$\begin{aligned}
\label{con:L1} 
\int_{{\mathbb{R}}} | \sigma_n (x) - \sigma (x)  |^{2} dx \leq C_0 n^{-1},  \  n \in \mathbb{N}, \end{aligned}$$ or $$\begin{aligned}
\label{con:uni} 
\sup_{x \in {\mathbb{R}}} | \sigma_n (x) - \sigma (x)  |  \leq C_0 n^{-1},  \  n \in \mathbb{N}. \end{aligned}$$ Then there exist positive constants $C_p \ (p \geq 1)$ such that $$\begin{aligned}
{\mathbb{E}}\left[ \left| X (t) - X_n (t) \right| \right] \leq C_1 (\log n)^{-\frac{1}{2}} \ ( 0 \leq t \leq T )\end{aligned}$$ and $$\begin{aligned}
\label{th2:eq2}
{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} \left| X (t) - X_n (t) \right|^p \right] \leq C_p (\log n)^{-\frac{p}{4(p+1)}} \end{aligned}$$ holds for $p > 1$ and $n>2$.

The boundedness of $\sigma$ and $\sigma_n$ implies that $$\begin{aligned}
\label{Lp-property}
{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} |X(t)|^p \right] + \sup_{n \in {\mathbb{N}}}{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} |X_n (t)|^p  \right] 
< \infty. $$ Define $$\begin{aligned}
Y_n (t) :=  X (t) - X_n (t) . 
$$

Let $u_m$ be a function as defined in Section \[sec:notation\]. By $(\ref{Def:Yfunc})$ and then applying Itô’s formula to $u_m \left(Y_n ( \cdot ) \right) $

$$\begin{aligned}
\label{eq:9a}
| X (t) - X_n (t)  | &\leq a_{m-1} +u_m (Y_n (t) ) \\               &=a_{m-1} +M (t) +J(t), \notag$$

where $M$ and $J$ are defined as follows: $$\begin{aligned}
 M  (t)  &=  \int_0^t u'_m (Y_n (s)) 
    \left\{  \sigma \left(X (s) \right) -  \sigma_n ( X_n (s) )   \right\} dW_s , \\ 
 J  (t)  &=  \frac{1}{2} \int_0^t u''_m (Y_n (s))  
    \left| \sigma \left(X (s) \right)  -  \sigma_n ( X_n (s) ) \right|^2 ds . \end{aligned}$$ Now we also define $J^\sigma$ and $J^Y$, $$\begin{aligned}
\label{eq:9b}
 J  (t)  &\leq  \int_0^t u''_m (Y_n (s))  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds  
 \\ &  \qquad   
 +   \int_0^t u''_m (Y_n (s))   \left| \sigma_n \left(X (s) \right)  - \sigma_n ( X_n (s) )  \right|^2 ds ,\notag 
  \\ &=: J^\sigma (t)  +  J^Y (t) , \  {\rm  say \ for \  } 0 \leq t \leq T \notag. \end{aligned}$$ Note that by $(\ref{eq:9b})$ $$\begin{aligned}
\label{eq:9f}
| X (t) - X_n (t)  |^p  
    &\leq \left| a_{m-1} + | M (t) |  +J^\sigma (t)  +  J^Y (t) \right|^p \\
    &\leq 4^{p-1} \left\{ a_{m-1}^p  +|M (t)|^p + \left| J^\sigma (t) \right|^p  + | J^Y (t)|^p \right\}. \notag $$ Then taking the sup-norm over the time interval $[0,T]$, we have $$\begin{aligned}
\label{eq:9f}
&\sup_{0 \leq t \leq T} | X (t) - X_n (t)  |^p \\
    &\leq 4^{p-1} \left\{ a_{m-1}^p 
    + \sup_{0 \leq t \leq T}|M (t)|^p +\left| J^\sigma (T) \right|^p   + | J^Y (T)|^p \right\}. \notag $$ Selecting the smooth functions $ f_l \  ( l \in {\mathbb{N}})$ in Remark \[rem:disc\], we have for $x, y \in {\mathbb{R}}$, $${f_l (x) - f_l (y)}= (x-y )\int_0^1 f_l^{'} (x + \theta (y-x)) d\theta .$$ Then we define and consider $ J^{Y}_l$ as follows: $$\begin{aligned}
 J^{Y}_l (t)     &:= \int_0^t u''_m (Y_n (s))   \left| f_l \left(X (s) \right)  - f_l ( X_n (s) )  \right| ds
\\&= \int_0^t \varphi_m (Y_n (s))  |Y_n(s)| 1_{\left( 0<|Y_n(s)|  \right)} \frac{\left|  f_l \left(X (s) \right) -  f_l \left(X_n  (s)  \right)  \right| }{|Y_n(s)|} ds    \notag
  \\ 
  &\leq
 {2}{m}^{-1}    \int_0^t  \int_0^1  f_l^{'} ( Z^{\theta}(s)) d\theta ds \end{aligned}$$ holds where for $\theta \in [0,1]$ we define $$Z(t) \equiv Z^{\theta}(t):= X (t)  + \theta \left( X_n ( t ) -X (t) \right).$$ The fact that the martingale part of $Z$ is expressed by $$\int_0^{t} \left\{ (1-\theta) \sigma \left( X ( s) \right) +\theta  \sigma_n \left( X_n ( s) \right)  \right\}  dW_s  =: \int_0^t \tilde{\sigma} (s)dW_s,$$ and $(1-\theta)\sigma(x) +\theta \sigma_n (x) \geq \epsilon$ for $x \in {\mathbb{R}}$ implies that we have $\langle Z, Z \rangle_t  \geq \epsilon^2 t $. Now the occupation times formula implies that we have $$\begin{aligned}
\int_0^t  \int_0^1   f_l^{'}  ( Z^{\theta }(s) ) d\theta ds  
&\leq  \epsilon^{-2}  \int_0^1  \int_{-\infty}^{\infty}   f_l^{'}(a)  L_t^a (Z^{\theta}) da d\theta \end{aligned}$$ where $L_t^a (Z)$ stands for the local time of $Z$ accumulated at $a$ until time $t$. By the Meyer-Tanaka formula and $L^p$ integrability of the solutions $X$ and $X_n$ of $(\ref{Lp-property})$, we have $$\begin{aligned}
\label{eq:9h}
c_{L} := \sup_{\theta \in [0,1]} \sup_{a \in {\mathbb{R}}}  {\mathbb{E}}[ \left(  L_T^a (Z^{\theta}) \right)^p ]  < \infty  . \end{aligned}$$ Then we obtain $$\begin{aligned}
{\mathbb{E}}[| J^Y_l (t)|^p ] 
    &\leq
    2^p (m\epsilon^2)^{-p}  {\mathbb{E}}[ \left(   \int_0^1  \int_{-\infty}^{\infty}   f_l^{'}(a)  L_t^a (Z^{\theta}) da d\theta  \right)^p   ] 
    \\&\leq 
    2^p (m\epsilon^2)^{-p} 
    \left(   \int_0^1  \int_{-\infty}^{\infty}   f_l^{'}(a)  da d\theta  \right)^{p-1}   
    {\mathbb{E}}[   \int_0^1  \int_{-\infty}^{\infty}   f_l^{'}(a)  \left(  L_t^a (Z^{\theta})  \right)^p da d\theta   ] 
    \\&\leq 
    2^p (m\epsilon^2)^{-p} 
    \| f_l \|^{p-1}_{\infty}   
    \int_0^1  \int_{-\infty}^{\infty}   f_l^{'}(a)  {\mathbb{E}}[  \left(  L_t^a (Z^{\theta})  \right)^p ] da d\theta    
    \\&\leq 
    2^p (m\epsilon^2)^{-p}  \| f_l \|^{p}_{\infty}  c_L.  \end{aligned}$$ By the monotone convergence theorem, we have $ J^{Y}_l \rightarrow  J^{Y}$ as $l \rightarrow \infty$. the $L^p$ estimate for $J^Y (T)$ in $(\ref{eq:9b})$, $$\begin{aligned}
{\mathbb{E}}\left| J^Y (T) \right|^p &\leq 2^p (m\epsilon^2)^{-p} \| f \|^{p}_{\infty}  c_L . 
$$

On the other hand, let us consider $J^\sigma$ in $(\ref{eq:9b})$. By the construction of $\varphi_m$ we have $$\begin{aligned}
J^\sigma (t) 
&= \int_0^t u''_m (Y_n (s))  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds 
\\ &\leq 2 (m a_m )^{-1}  \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds. \end{aligned}$$

If the convergence rate $(\ref{con:L1})$ holds, then using Hölder’s inequality we obtain $$\begin{aligned}
&\left( \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds \right)^p 
\\ &\leq \epsilon^{-2p}\left( \int_{{\mathbb{R}}}  \left| \sigma \left(a \right)  - \sigma_n ( a )  \right|  L_t^a (X) da \right)^{p}
\\ &\leq \epsilon^{-2p}\left( \int_{{\mathbb{R}}}  \left| \sigma \left(a \right)  - \sigma_n ( a )  \right| da \right)^{p-1}
    \int_{{\mathbb{R}}}  \left| \sigma \left(a \right)  - \sigma_n ( a )  \right| | L_t^a (X) |^p da
\\ &\leq \epsilon^{-2p}\left( C_0 n^{-1} \right)^{p-1}
    \int_{{\mathbb{R}}}  \left| \sigma \left(a \right)  - \sigma_n ( a )  \right| | L_t^a (X) |^p da,\end{aligned}$$ and then we obtain $$\begin{aligned}
{\mathbb{E}}\left( \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds \right)^p 
\leq \epsilon^{-2p}\left( C_0 n^{-1} \right)^{p-1} \times  C_0 n^{-1}  c_L
= \epsilon^{-2p}c_L \left( C_0 n^{-1} \right)^{p} ,
$$ where $L_t^a (X)$ is the local time of $X$ such that $$\begin{aligned}
 \sup_{a \in {\mathbb{R}}}  {\mathbb{E}}[ \left( L_t^a (Z^{0}) \right)^p ] =  \sup_{a \in {\mathbb{R}}} {\mathbb{E}}[ \left( L_t^a (X)\right)^p ] \leq c_{L}. \end{aligned}$$

On the other hand, the rate of convergence of $(\ref{con:uni})$ implies that $$\begin{aligned}
&\left( \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^2 ds \right)^p \leq (t C_0 n^{-1})^p. \end{aligned}$$

Then, under the stability rate condition $(\ref{con:L1})$ or $(\ref{con:uni})$, the term of $J^\sigma  $ is estimated in $L^p$ as follows: $$\begin{aligned}
{\mathbb{E}}[|J^\sigma (T) |^p ] 
    &\leq  2^p (m a_m )^{-p} \left( \epsilon^{-2p}  c_{L} +T^p  \right)(C_0 n^{-1})^{p}. \end{aligned}$$ In short, combining these estimates with $(\ref{eq:9b})$ we obtain $$\begin{aligned}
\label{eq:9e}  
{\mathbb{E}}\left| J (T) \right|^p \leq  2^p {m}^{-p}  c_{JY}   +  2^p (m a_m n )^{-p}  c_{J\sigma} , \end{aligned}$$ where we define $$\begin{aligned}
c_{JY} &:= \epsilon^{-2p}  \| f \|^{p}_{\infty}  c_L  
\\ 
c_{J\sigma} &:=  \left( \epsilon^{-2p}  c_{L} +T^p  \right) C_0^{p}. \end{aligned}$$

Now let us estimate $Y_n$ in the case of $p=1$, $$\begin{aligned}
| Y_n (t)  | 
    &\leq  a_{m-1} +M (t)  + J^\sigma (t)  +  J^Y (t). \end{aligned}$$ Then we obtain $$\begin{aligned}
\label{eq:9c}
{\mathbb{E}}\left[ \left| Y_n (t) \right| \right] & \leq a_{m-1}+
2c_{JY} {m}^{-1}   +  2c_{J\sigma} (m a_m n )^{-1}  =  A_{m,n}, \ {\rm \ say. }\end{aligned}$$ Here, in the Yamada-Watanabe method let us choose $( a_m )_{m \in {\mathbb{N}}}$ as $a_m = \exp \left( -m(m+1)/2 \right), \ a_0=1$ and select a sequence $(m_n)_{n \in {\mathbb{N}}}$ such that $$\begin{aligned}
\label{eq:9g}
\frac{1}{ a_{m_n} n}  \leq 1 \end{aligned}$$ holds for $n > 2$. Since we have that $ a_{m-1} \leq {1}/{m}$ for any $m \in {\mathbb{N}}$, there exists a finite positive number $c_{a}$ such that $A_{m_n,n}$ in $(\ref{eq:9c})$ is bounded by $$\begin{aligned}
\label{eq:9d}
{\mathbb{E}}\left[ \left| Y_n (t) \right| \right]  \leq A_{m_n,n} \leq   c_{a}    ( \log n)^{-\frac{1}{2}}. \end{aligned}$$

We shall obtain the $L^p$-estimate $(\ref{th2:eq2})$ from this $L^1$-estimate $(\ref{eq:9d})$. Let us estimate the quadratic process $\langle M \rangle$, $$\begin{aligned}
\langle M \rangle_T 
&\leq \int_0^t \left|  \sigma \left(X (s) \right) -  \sigma_n \left(X_n (s) \right)   \right|^2 ds \\
&\leq
\int_0^T  \left|  \sigma \left(X (s) \right) -  \sigma_n \left(X (s) \right)   \right|^2 ds 
+
\int_0^T  \left|  \sigma_n \left(X (s) \right) -  \sigma_n \left(X_n (s) \right)   \right|^2 ds 
\\
&=: \langle M^\sigma \rangle_T  + \langle M^Y \rangle_T, \ {\rm say}.\end{aligned}$$ By the same argument as we estimate ${\mathbb{E}}| J(T) |^p $ in $(\ref{eq:9e})$ with $c_{J\sigma}$, we have $$\begin{aligned}
{\mathbb{E}}[\langle M^\sigma \rangle_T^{\frac{p}{2}} ] 
\leq \sqrt{ {\mathbb{E}}[\langle M^\sigma \rangle_T^{{p}} ] }
&\leq   \sqrt{ c_{J\sigma}  n^{-p} }. 
$$ On the other hand, for any positive number $y$ we have $$\begin{aligned}
&\langle M^Y \rangle_T
=
\int_0^T  \left|  \sigma_n \left(X (s) \right) -  \sigma_n \left(X_n (s) \right)   \right|^{2} ds  
\\&
\leq \int_0^T  \left| f_l \left(X (s) \right) -  f_l \left(X_n (s) \right)   \right| 1_{(|Y_n (s)| >y)}ds \\&
+ \int_0^T  \left| f_l \left(X (s) \right) -  f_l \left(X_n (s) \right)     \right| 1_{(|Y_n (s) | \leq y)}ds \\
&\leq
2 \| f \|_{\infty}  \int_0^T  1_{(|Y_n (s)| >y)} ds 
+
y \int_0^T   \int_0^1   f_l' (Z^\theta (s) )   d\theta ds .$$ Therefore, putting $y=\left( \log n \right)^{-\frac{1}{2(p+1)}}$, there exists a positive number $c_{M}$ such that $$\begin{aligned}
&{\mathbb{E}}[\langle M^Y \rangle_t^{\frac{p}{2}}] \leq \sqrt{ {\mathbb{E}}[\langle M^Y \rangle_t^{p}] }
\\&\leq 
\sqrt{  2^{p} \max \left( (2 \| f \|_{\infty})^{p}  T^{p}  y^{-1} ( \log n )^{-\frac{1}{2}}, \  
y^{p}  { c_{JY} } \right)}
\leq c_{M} \left( \log n \right)^{-\frac{p}{4(p+1)}}
. $$ Then by combining these estimates with $(\ref{eq:9f})$ and the Burkholder-Davis-Gundy Inequality we observe that there exists a constant value $c_p$ such that $$\begin{aligned}
&{\mathbb{E}}[ \sup_{0 \leq t \leq T} | X (t) - X_n (t)  |^p  ]
    \\&\leq c_p \left\{ a_{m-1}^p 
    +{\mathbb{E}}[\langle M^{\sigma} \rangle_t^{\frac{p}{2}}] +{\mathbb{E}}[\langle M^Y \rangle_t^{\frac{p}{2}}] 
    +{\mathbb{E}}[  \left| J^\sigma (T) \right|^p ]   + {\mathbb{E}}[ | J^Y (T)|^p] \right\}.     \\&\leq c_p \left\{ a_{m-1}^p 
    +\sqrt{c_{J\sigma}  n^{-p}} +{\mathbb{E}}[\langle M^Y \rangle_t^{\frac{p}{2}}]     +2^p c_{JY} {m}^{-p}   +  2^p c_{J\sigma} (m a_m n )^{-p}  \right\}. $$ Since we have selected the sequence $(m_n)_{n \in {\mathbb{N}}}$ as in $(\ref{eq:9g})$, $$\begin{aligned}
&{\mathbb{E}}[ \sup_{0 \leq t \leq T} | X (t) - X_n (t)  |^p  ]
    \\&\leq c_p (\log n)^{-\frac{p}{4(p+1)}} \left\{ 
    1
    + \sqrt{c_{J\sigma}}   +c_{M}    +2^p c_{JY}   +  2^p c_{J\sigma}\right\}, $$ for $n>2$. Therefore we obtain the desired result.

0

\[YmethodOpt\] If one uses the Yamada-Watanabe method, it sufficient to have $$\begin{aligned}
\label{Ymethod1}
\lim_{m, n \to \infty } \frac{1}{ n a_m} < \infty \ {\rm \ for \ }  a_m =e^{-\frac{1}{2}m(m+1)}. \end{aligned}$$ Therefore, the convergence rate of the stability problem is estimated by $\log n$. $$\left| Y_n (t) \right| \leq  H_{\epsilon}  n^{-\frac{1}{2} +\delta}$$ holds for any $\delta$ with $0 < \delta < 1/2$, where $H_\epsilon $ is a finite random value.

This says that the convergence rate is strongly determined by the regularity of the coefficient.

SDEs driven by symmetric $\alpha$-stable processes {#section:sas}
==================================================

In this section, we introduce a symmetric $\alpha$ stable process $Z$ and the sequence of SDEs driven by $Z$: $$\begin{aligned}
X(t) &= X(0) + \int_0^t \sigma (X(s-)) dZ_s ,\label{sde:sas1}  \\
X_n (t) &= X_n (0) + \int_0^t \sigma_n (X_n (s-)) dZ_s\ \ ( n \in {\mathbb{N}}), \end{aligned}$$ where both $\{ \sigma_n \}_{n \in {\mathbb{N}}}$ and $\sigma$ satisfy the following Belfadli-Ouknine condition.

We say that a function $\sigma$ satisfies the Belfadli-Ouknine condition if $\sigma$ satisfies Definition $\ref{conditionC}$ $(1)$ and the modified condition $(2)'$ of Definition $\ref{conditionC}$: there exists an increasing function $f$ such that $$\begin{aligned}
| \sigma (x)- \sigma (y) |^{\alpha} &\leq | f(x) - f(y)| {\rm \ and \ }  \| f \|_{\infty} < \infty\end{aligned}$$ holds for any $x, y \in {\mathbb{R}}$.

Under the assumptions, the pathwise uniqueness holds for $X$ and also $X_n$, $n \in {\mathbb{N}}$, which is proven by Belfadli and Ouknine in [@BO2008]. Moreover, it is shown by Hashimoto [@Hashimoto2013] that the solution is realized by the stability problem . The local time is again the key to compute the strong convergence rates via stability problem. For the results of the local time for the symmetric $\alpha$ stable process see K. Yamada [@KYamada2002] and Salminen and Yor [@YorSalminen2007].

\[Thm:main2\] Let $\alpha$ be a positive number with $1< \alpha <2$. Assume $X_n (0) \equiv X (0)$ with ${\mathbb{E}}\left| X_n (0) \right|^{\alpha -1} < \infty $. If $\sigma$ and $\{ \sigma_n \}_{n \in {\mathbb{N}}}$ satisfy the Belfadli-Ouknine condition and either of the stability rate condition $(\ref{con:L1})$ or $(\ref{con:uni})$, then there exists a positive constant $C_2$ such that $$\begin{aligned}
{\mathbb{E}}\left[ \left| X (t) - X_n (t) \right|^{\alpha-1} \right] \leq  C_2  (\log n)^{-\frac{\alpha-1}{2}}\end{aligned}$$ holds for $0 \leq t \leq T$ and $n>2$.

Using Émery’s inequality in [@Emery1978], we have $${\mathbb{E}}\left[ \sup_{0 \leq t \leq T} |X(t)|^{\alpha -1} \right] + \sup_{n \in {\mathbb{N}}}{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} |X_n (t)|^{\alpha -1} \right] 
<\infty.$$ Define $Y_n (t) :=  X (t) - X_n (t) $. Let $\varphi_m $ be a function as defined in Section \[sec:notation\]. Define $$v_m = v \ast \varphi_m,$$ where $v(x)=|x|^{\alpha-1}$ and $\ast$ stands for the convolution. Then we have $$-(a_{m-1})^{\alpha -1} +v_m (x) \leq 
v(x) \leq  (a_{m-1})^{\alpha -1} +v_m (x) ,$$ holds for $x \in {\mathbb{R}}$, see [@Hashimoto2013].

By the Itô formula we have $$\begin{aligned}
| Y_n(t) |^{\alpha-1} &\leq (a_{m-1})^{\alpha-1} +v_m (Y_n (t) ) \\
              &=(a_{m-1})^{\alpha-1} +M (t) +J(t),  $$ where $M$ and $J$ are defined as follows: $$\begin{aligned}
 M  (t)  &=  \int_0^t u'_m (Y_n (s)) \left\{  \sigma \left(X (s) \right) -  \sigma_n ( X_n (s) )   \right\} dZ_s , \\
 J  (t)  &=   K_\alpha \int_0^t \varphi_m (Y_n (s))  \left| \sigma \left(X (s) \right)  -  \sigma_n ( X_n (s) ) \right|^\alpha ds,\end{aligned}$$ where we define a constant $K_{\alpha}=  -  \Gamma(\alpha) \cos ({\alpha \pi}/{2})/2 \geq 0$ for $\alpha \in (1,2)$ from the gamma function $\Gamma (\cdot )$.

Now we also define $J^\sigma$ and $J^Y$ such that $$\begin{aligned}
 J  (t)  &\leq  
2 K_\alpha \int_0^t \varphi_m (Y_n (s))   \left\{ \sigma \left(X (s) \right)  -  \sigma_n ( X (s) ) \right\}^\alpha ds  
 \\ &  \qquad    +
2 K_\alpha \int_0^t \varphi_m (Y_n (s))    \left| \sigma_n \left(X (s) \right)  -  \sigma_n ( X_n (s) ) \right|^\alpha ds , 
  \\ &=: 2 K_\alpha \left( J^\sigma (t)  +  J^Y (t) \right) , \  {\rm  say} . \end{aligned}$$

For $\theta \in [0,1]$, let us define $$V(t) \equiv V_t \equiv V^{\theta}(t):= X (t)  + \theta \left( X_n ( t ) -X (t) \right). 
$$ Notice that we have for $x, y \in {\mathbb{R}}$, $${f_l (x) - f_l (y)}= (x-y )\int_0^1 f_l^{'} (x + \theta (y-x)) d\theta .$$ Then we have that $$\begin{aligned}
 J^{Y}_l (t) 
 &:=\int_0^t \varphi_m (Y_n (s))  \left|  f_l \left(X (s) \right) -  f_l \left(X_n  (s) \right)  \right| ds    \notag
  \\ 
 &=
\int_0^t \varphi_m (Y_n (s))  |Y_n(s)| 1_{\left( 0<|Y_n(s)|  \right)} \frac{\left|  f_l \left(X (s) \right) -  f_l \left(X_n  (s)  \right)  \right| }{|Y_n(s)|} ds    \notag
  \\ 
  &\leq
 {2}{m}^{-1}    \int_0^t  \int_0^1  f_l^{'} ( V^{\theta}(s)) d\theta ds .\end{aligned}$$

Define $$A_t := \int_0^{t} |H_s|^{\alpha}  ds {\rm \ where \ }
H_s :=  \sigma \left( X ( s) \right) +\theta \left(  \sigma_n \left( X_n ( s) \right)-  \sigma \left( X ( s) \right) \right) 
$$ By the time-change method described in [@BO2008] or [@Tsuchiya2007], we have $$\begin{aligned}
\int_0^t  f_l^{'} ( V_s ) ds 
=\int_0^t    f_l^{'} ( \tilde{Z}_{A_s}) |H_s|^{-\alpha} dA_s 
\leq \epsilon^{-\alpha} \int_0^t     f_l^{'} ( \tilde{Z}_{A_s})  dA_s ,\end{aligned}$$ where $V_s =: \tilde{Z}_{A_s}$. Making the random change of variable $s=\tau_u :=\inf \{ t \geq 0: A_t >u \}$, we obtain $$\begin{aligned}
\int_0^t  f_l^{'} ( V^{\theta}(s)) ds 
\leq \epsilon^{-\alpha} \int_0^{A_t}    f_l^{'} ( \tilde{Z}_u )du 
\leq \epsilon^{-\alpha} \int_0^{ \| \sigma \|_{\infty}^{\alpha} T}    f_l^{'} ( \tilde{Z}_u )du, \end{aligned}$$ where $\tilde{Z}$ is the symmetric $\alpha$ stable process with respect to the filtration $\mathcal{G}_t :=\mathcal{F}_{\tau^{-1}(t) }$. By the occupation time formula of the local time $ L_t^a $ of $\tilde{Z}$ we have $$\begin{aligned}
\int_0^{\tilde{T} }    f_l^{'} ( \tilde{Z}_u )du
=
\int_{-\infty}^{\infty}     f_l^{'} (a) L_{\tilde{T}}^a ( \tilde{Z} ) da, \end{aligned}$$ for $\tilde{T}:=\| \sigma \|_{\infty}^{\alpha} T$, for an example see [@Bertoinb1996]. Indeed, by the Tanaka-Meyer-K. Yamada formula in [@KYamada2002] implies that $$\begin{aligned}
L_t^a (\tilde{Z}) 
=  |\tilde{Z}_t -a|^{\alpha -1}- |a|^{\alpha-1} -N_t^a
&\leq \left| |\tilde{Z}_t -a|^{\alpha -1}- |a|^{\alpha-1} \right| -N_t^a 
\\ &\leq \left| |\tilde{Z}_t -a|- |a| \right|^{\alpha-1} -N_t^a
\\ &\leq | \tilde{Z}_t  |^{\alpha-1} -N_t^a, 
$$ where $N^a$ is a squared martingale such that $$\begin{aligned}
\langle N^a \rangle_t 
=c_\alpha \int_0^t \frac{ds}{| \tilde{Z}_s -a |^{2-\alpha}}. \end{aligned}$$ Then we obtain $$c_{\tilde{L}}:=\sup_{a \in {\mathbb{R}}} {\mathbb{E}}[ L_{\tilde{T}}^a (\tilde{Z}) ] < \infty.$$ By Jensen’s inequality for the concave function $x^{\alpha-1}$, we have $$\begin{aligned}
{\mathbb{E}}[| J^Y_l (t)|^{\alpha -1} ] &\leq 
    2^{\alpha -1} (m \epsilon^\alpha)^{ -(\alpha -1)}
    {\mathbb{E}}[ \left( \int_0^1 \int_{-\infty}^{\infty}     f_l^{'} (a) L_{\tilde{T}}^a ( \tilde{Z} ) da d\theta \right)^{\alpha-1}   ] 
    \\ &\leq 
    2^{\alpha -1}  (m \epsilon^\alpha)^{ -(\alpha -1)}
     \left( \int_0^1 \int_{-\infty}^{\infty}     f_l^{'} (a)  {\mathbb{E}}[  L_{\tilde{T}}^a ( \tilde{Z} )] da d\theta   \right)^{\alpha-1}   
    \\ &\leq 
    2^{\alpha -1}  (m \epsilon^\alpha)^{ -(\alpha -1)}
     \left( \|f \|_{\infty} c_{\tilde{L}} \right)^{\alpha-1} . \end{aligned}$$ On the other hand, let us consider $J^\sigma$, $$\begin{aligned}
J^\sigma (t) 
&= \int_0^t  \varphi_m (Y_n (s))  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^\alpha ds 
\\ &\leq 2 (m a_m )^{-1}  \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^\alpha ds. \end{aligned}$$ Note that $$\begin{aligned}
 \int_0^t  \left| \sigma \left(X (s) \right)  - \sigma_n ( X (s) )  \right|^\alpha ds
    \leq  \| \sigma_0 \|_{\infty}^{\alpha-1}  \epsilon^{-\alpha} 
    \int_{{\mathbb{R}}}  \left| \sigma \left(a \right)  - \sigma_n ( a )  \right| L_t^a (X) ds, \end{aligned}$$ where $L_t^a (X)$ is the local time of $X$ such that $$\sup_{a \in {\mathbb{R}}}  {\mathbb{E}}[  L_t^a (Z^{0}) ] =  \sup_{a \in {\mathbb{R}}} {\mathbb{E}}[L_t^a (X) ] \leq c_{\tilde{L}},$$ with $c_{\tilde{L}}$ as in $(\ref{eq:9h})$ in Section \[stability problem\], and we define $$\| \sigma_0 \|_{\infty} := \sup_{x \in {\mathbb{R}}} (| \sigma (x)| , |\sigma_n (x)| ) < \infty.$$ The stability rate condition $(\ref{con:L1})$ or $(\ref{con:uni})$ implies that $${\mathbb{E}}[|J^\sigma (t) |^{\alpha -1} ] \leq  2^{\alpha -1} (m a_m )^{-(\alpha -1)} 
\left( 
t^{\alpha-1}
+
\left( \|f \|_{\infty} c_{\tilde{L}} \right)^{\alpha-1} 
\right)\left(
 \| \sigma_0 \|_{\infty}
C_0 n^{-1} \right)^{\alpha -1}.$$

In short, we obtain $$\begin{aligned}
{\mathbb{E}}\left| J (t) \right|^{\alpha-1} \leq  
    (4 K_\alpha )^{\alpha-1} c_{JY} {m}^{-(\alpha-1)}   +  (4 K_\alpha )^{\alpha-1} 
    c_{J\sigma} (m a_m n)^{-(\alpha-1)}. \end{aligned}$$ where we define $$\begin{aligned}
c_{JY} &:= \left(  \epsilon^{-\alpha}  \| f \|_{\infty} c_{\tilde{L}} \right)^{\alpha-1}, \\ 
c_{J\sigma} &:=   
\left( t^{\alpha-1} + \left( \|f \|_{\infty} c_{\tilde{L}} \right)^{\alpha-1}  \right)
\left( \| \sigma_0 \|_{\infty} C_0  \right)^{\alpha -1}
$$

Now let us estimate $|Y|^{\alpha-1}$, $$\begin{aligned}
| Y_n (t)  |^{\alpha-1} 
    &\leq  (a_{m-1})^{\alpha-1} +M (t)  + J^\sigma (t)  +  J^Y (t). \end{aligned}$$ Then we obtain $$\begin{aligned}
{\mathbb{E}}\left[ | Y_n (t)  |^{\alpha-1}  \right] & \leq (a_{m-1})^{\alpha-1} +
(4 K_\alpha )^{\alpha-1}\left( c_{JY} {m}^{-(\alpha-1)}   +  c_{J\sigma} (m a_m n )^{-(\alpha-1)}  \right)
=  A_{m,n}, {\rm \ say. }\end{aligned}$$ Here let us consider the one of the sequence $( a_m )_{m \in {\mathbb{N}}}$ such that $a_m = \exp \left( -m(m+1)/2 \right)$. Then select a sequence $(m_n)_{n \in {\mathbb{N}}}$ such that $$\frac{1}{ a_m n}  \leq 1$$ holds for for $n > 2$. Since we have that $ a_{m-1} \leq {1}/{m}$ for any $m \in {\mathbb{N}}$, there exists a finite positive number $c_a$ such that $${\mathbb{E}}\left[ |Y_n(t) |^{\alpha-1}  \right] \leq 
A_{m_n,n} \leq   c_a    ( \log n)^{-\frac{\alpha-1}{2}}.$$

Invariant property under removal drift {#Removal invariant}
======================================

In this section, we consider the SDEs of $(\ref{sde0})$ with the drifts of SDEs driven by the Wiener process $W_s$. Now we introduce the method of removal of the drifts. Suppose that there exists a strong solution $X_n$ and $X$ to the equation $(\ref{sde:stability})$ and $(\ref{sde0})$ and taking values on $I \equiv (l, k)$ for $-\infty \leq l < k \leq \infty$. Consider the coefficients $b, b_n$ and $\sigma, \sigma_n $ for $n \in {\mathbb{N}}$ such that the functions $ b_n  \sigma^{-2}_n$ and $b  \sigma^{-2}$ belong to $L^1(I)$, i.e., $$\begin{aligned}
\label{ass:cff1}
\sup_{n \in {\mathbb{N}}}\int_{I} | b_n (u) \sigma^{-2}_n (u) | du < \infty  {\rm \ and \ }
\int_{I} | b (u) \sigma^{-2} (u) du | < \infty. \end{aligned}$$ Now let us consider the scale functions given by $$\begin{aligned}
\label{Def: F and Fn}
s_n' (x) := \exp \left( -2 \int^x \frac{b_n (u) }{\sigma_n^2 (u)}  du\right)   {\rm \ and \ }  
s'     (x) :=\exp \left( -2 \int^x \frac{b     (u) }{\sigma^2     (u)}  du\right).  \end{aligned}$$ By Itô’s formula we have the drift-less processes $\bar{X}_n (t) := s_n (X_n (t))$ taking values on $s_n (I)$ and $\bar{X} (t) := s (X(t))$ taking values on $s (I)$ such that $$\begin{aligned}
\label{sde:without the drifts term}
\bar{X}_n (t) - \bar{X}_n (0)  &= \int_0^t \bar{\sigma} (\bar{X}_n (s) ) dW_s  \\
\bar{X} (t)     - \bar{X} (0)      &= \int_0^t \bar{\sigma}_n (\bar{X} (s) ) dW_s, \end{aligned}$$ where $$\begin{aligned}
\bar{\sigma}_n (\bar{x}): = (\sigma_n s'_n ) \circ s_n^{-1} (\bar{x}) = \sigma_n (s^{-1}_n (x) ) s'_n (s^{-1}_n (\bar{x}) )  &{\rm \ for \ }  \bar{x} \in s_n (I) ,\\
\bar{\sigma}     (x)        := (\sigma s' ) \circ s^{-1} (x) = \sigma      (s^{-1}    (x) ) s'    (s^{-1}      (x) )  &{\rm \ for \ }  \bar{x} \in s (I). \end{aligned}$$ The Nakao-Le Gall condition is invariant under the removal of drifts in the following sense.

\[remark: invariant\] If the diffusion coefficient $\sigma$ satisfies the Nakao-Le Gall’s condition, $\sigma \in \mathcal{C}_{NL} (\epsilon, \| f \|_{\infty})$, and the drift coefficient $b$ is bounded and in $L^1$, then so does $\bar{\sigma}$ on $s(I)$.

By the assumption $\sigma \in \mathcal{C}_{NL} (\epsilon, \| f \|_{\infty})$, we have $$\int_I | b(u) | |\sigma^{-2} (u) | du \leq \epsilon^{-2} \int_I | b(u) |   du < \infty.$$ In addition, the boundedness of $b$ implies that for $x \in I$ $$\exp \left(- 2 \| b \sigma^{-2}\|_{L^1}  \right)  \leq 
s' (x) \leq \exp \left( 2 \| b \sigma^{-2}\|_{L^1}  \right) .$$ Then there exists a positive number $\epsilon'$ such that $$0 < \epsilon' \leq \bar{\sigma} (\bar{x})$$ holds for $\bar{x} \in s(I)$.

On the other hand, using the boundedness of $b$ we have $$|s'' (x)| = |\left( -2 {b (x) }{\sigma^{-2} (x)} \right)  s' (x) | 
     \leq 2 \| b \sigma \|_{I, \infty} \exp \left( 2 \| b \sigma^{-2}\|_{L^1}  \right) <\infty$$ where $ \| b \sigma \|_{I, \infty} :=\sup_{u \in I} |b (u) \sigma (u) | $. Thus $s'$ is Lipschitz continuous. Therefore there exists a monotone function $f_b$ such that $$| \bar{\sigma} (\bar{x}) - \bar{\sigma} (\bar{y}) | \leq | f_b (x)- f_b (y) |,$$ holds for $x,y \in s(I)$.

This result suggests that the stability problem is independent of the drift coefficient itself. To be more precious, these convergence rates are also invariant under the removal drift as follows.

\[corollary:Ip\] Suppose that the diffusion coefficient $\sigma, \sigma_n $ for $n \in {\mathbb{N}}$ satisfy the Nakao-Le Gall’s condition, $\sigma, \sigma_n  \in \mathcal{C}_{NL} (\epsilon, \| f \|_{\infty})$, and the drift coefficient $b$ and $b_n$ are uniformly bounded and in $L^1$. If there exists a constant $C_0$ such that $$\begin{aligned}
\int_{x \in {\mathbb{R}}} | b_n (x)- b(x)  | + | \sigma_n (x) - \sigma (x)  | dx \leq C_0 n^{-1},\end{aligned}$$ then there exists a positive number $\bar{C_0}$ such that $$\int_{S_n} | \bar{\sigma} (\bar{x}) -  \bar{\sigma}_n (\bar{x}) |  d\bar{x} \leq \bar{C_0}n^{-1}.$$ where $S_n = s_n (I) \cap s (I) $.

Let us observe $$\begin{aligned}
& \left| \bar{\sigma}_n (\bar{x}) - \bar{\sigma} (\bar{x}) \right| 
    \\ & \leq
|\sigma_n (s^{-1}_n (\bar{x}) ) |
    \left(  |s'_n (s^{-1}_n (\bar{x}) )  - s' (s^{-1}_n    (\bar{x}) ) | + | s' (s^{-1}_n (\bar{x}) ) -  s' (s^{-1} (\bar{x}) ) |  \right) 
    \\ & \ \  +
    |s' (s^{-1} (\bar{x}))| 
    \left( | \sigma_n (s^{-1}_n (\bar{x}) ) -  \sigma (s^{-1}_n (\bar{x}) ) | +| \sigma (s^{-1}_n (\bar{x}) )  - \sigma  (s^{-1}    (\bar{x}) )  |\right) 
    \\ & \leq
    2 \| f  \|_{\infty}
     \left(  |s'_n (s^{-1}_n (\bar{x}) )  - s' (s^{-1}_n    (\bar{x}) ) | + | s' (s^{-1}_n (\bar{x}) ) -  s' (s^{-1}(\bar{x}) ) |  \right)
    \\ & \ \  +
    \| s' ( s^{-1})  \|_{\infty} \left( | \sigma_n (s^{-1}_n (\bar{x}) ) -  \sigma (s^{-1}_n (\bar{x}) ) | +| \sigma (s^{-1}_n (\bar{x}) )  - \sigma      (s^{-1}    (\bar{x}) )  |\right)
$$ By the definition of the scale functions $s_n$ and $s$ we have $$\begin{aligned}
&s(x) - s_n (x) = \int^{x} s' (y) - s_n' (y) dy
\\&= \int^{x} \exp \left (-2  \int^{y} b(u) \sigma^{-2} (u) du \right)  dy- \exp \left (-2  \int^{y}  b_n (u) \sigma^{-2}_n (u) du \right)  dy 
\\&= \int^{x} \left(  \log s'(y)- \log s_n' (y) \right)
\left(\int_0^1 \exp \left( \theta \log s'(y) + (1-\theta )\log s_n' (y) \right) d\theta  \right) 
dy 
\\&= \int^{x} \left(  \log \frac{ s'(y)}{  s_n' (y) } \right)
\left(\int_0^1 {\left( s'(y) \right)}^\theta  \left( s_n' (y)\right)^{1-\theta } d\theta \right)  dy .\end{aligned}$$ Now put $c_{F1}$ and $c_{F2}$ as follows: $$\begin{aligned}
c_{s1} &:= \min \left\{ \exp \left(- 2 \sup_{n \in {\mathbb{N}}} \|  b_n \sigma_n^{-2} \|_{L^1}  \right),  \exp \left(- 2 \| b \sigma^{-2}\|_{L^1}  \right) \right\} , 
\\
c_{s2} &:=\max \left\{ \exp \left( 2  \sup_{n \in {\mathbb{N}}}\|  b_n \sigma_n^{-2} \|_{L^1}  \right),  \exp \left(2 \| b \sigma^{-2}\|_{L^1}  \right) \right\} . \end{aligned}$$ By the assumption, we have $$\begin{aligned}
\left| \log \frac{ s'(y)}{  s_n' (y) } \right|
    &\leq
    \int_{I} | b_n (u) \sigma^2 (u)- \sigma_n^2 (u) b (u)  |   \sigma^{-2} (u)  \sigma_n^{-2} (u)  du 
    \\& \leq
    \int_{I}  | b_n (u)- b (u) | \sigma^{-2}_n (u)  
    +    |\sigma^2 (u) - \sigma_n^2 (u)| | b (u) |   \sigma^{-2} (u)  \sigma_n^{-2} (u)  du .
$$ Then by the rate of the stability convergence rate condition we have $$\begin{aligned}
\sup_{x \in {\mathbb{R}}}| s'(x) - s'_n(x) |  &\leq c_{s2} c_{b,\sigma} n^{-1},
$$ where $c_{b,\sigma}: =\| \sup_{n \in {\mathbb{N}}}   n  \log ( s'  /  s_n' )\|_{L^1} < \infty$.

For $\bar{x} = s^{-1} (y) \left( y \in s(I) \right)$ and $\bar{x} = s_n^{-1} (y_n ) \left( y_n  \in s_n (I) \right)$ we have $$\begin{aligned}
(s^{-1})'(\bar{x}) - (s_n^{-1})'(\bar{x})= \frac{1}{s' (y)} - \frac{1}{s'_n (y_n)} =\frac{ s'_n (y_n) - s' (y)}{s' (y) s'_n (y_n)}. \end{aligned}$$ Then it implies that $$\begin{aligned}
\sup_{\bar{x} \in {\mathbb{R}}} | (s^{-1})' (\bar{x}) - (s_n^{-1})'(\bar{x}) | \leq c_{s2} {c_{s1}^{-2} }c_{b,\sigma} n^{-1}=:c'_{b,\sigma}n^{-1}. \end{aligned}$$

Since we have $$s' (s^{-1}_n (\bar{x}) ) -  s' (s^{-1} (\bar{x}) )   = ( s^{-1}_n (\bar{x}) -  s^{-1} (\bar{x}) ) 
\left( \int_0^1 \left( -2 {b}{\sigma^{-2}} s' \right) (\theta s^{-1}_n (\bar{x}) +(1-\theta)s^{-1} (\bar{x})) d\theta \right),$$ then we obtain that $$\begin{aligned}
&\int_{S_n} \left| s' (s^{-1}_n (\bar{x}) ) -  s' (s^{-1} (\bar{x}) )  \right| d \bar{x} 
\\&=\int_{S_n} \left|  s^{-1}_n (\bar{x}) -  s^{-1} (\bar{x}) \right| 
\left| \int_0^1 \left( -2 {b}{\sigma^{-2}} s' \right) (\theta s^{-1}_n (\bar{x}) +(1-\theta)s^{-1} (\bar{x})) d\theta \right| d \bar{x}
    \\&\leq 
    2 n^{-1} c'_{b,\sigma}
    \times \int_0^1 \int_{S_n}  | \left({b}{\sigma^{-2}} \right) (\theta s^{-1}_n (\bar{x}) +(1-\theta)s^{-1} (\bar{x})) | d\bar{x} d\theta
    \\&\leq 
    2 n^{-1} c'_{b,\sigma}
    \times \int_0^1 \int_{S_n}  | \left({b}{\sigma^{-2}} \right) (g_\theta (\bar{x})) | g'_{\theta} (\bar{x}) d\bar{x} d\theta
    \\&\leq 2 n^{-1} c'_{b,\sigma} \| {b}{\sigma^{-2}}  \|_{L^1(I)} < \infty\end{aligned}$$ where we define $$g_\theta (\bar{x}) :=\theta s^{-1}_n (\bar{x})  +(1-\theta) (s^{-1}    (\bar{x}) )$$ and note that $ c_{s2}^{-1} \leq g'_{\theta} (\bar{x}) \leq c_{s1}^{-1}$.

Considering the transformation of variable $y_n = s_n^{-1} (\bar{x})$, we have $$\begin{aligned}
&\int_{S_n}  | \sigma_n (s^{-1}_n (\bar{x}) ) -  \sigma (s^{-1}_n (\bar{x}) ) | d \bar{x}
\leq c_{s2}
\int_{S_n}  | \sigma_n (y_n ) -  \sigma (y_n) | d y_n \leq c_{s2} C_0 n^{-1} .  \end{aligned}$$

Using the sequence of $\{ f_l \}_{l \in {\mathbb{N}}} $ in Remark \[rem:disc\], we have $$\begin{aligned}
&\int_{S_n} | \sigma (s^{-1}_n ( \bar{x} ) )  - \sigma      (s^{-1}    ( \bar{x} ) )  |^2 d\bar{x} 
\leq \int_{S_n} | f_l (s^{-1}_n (\bar{x}) )  -f_l (s^{-1}    (\bar{x}) )  | dx 
    \\&=\int_{S_n} |s^{-1}_n (\bar{x})  - s^{-1} (\bar{x}) | \left| \int_0^1  f'_l (\theta s^{-1}_n (\bar{x})  +(1-\theta) (s^{-1}    (\bar{x}) )  ) d \theta  \right| d\bar{x} 
    \\&\leq c'_{b,\sigma}  n^{-1} 
         \int_0^1 \int_{S_n}
           c_{s2} g'_\theta (\bar{x}) 
          f'_l ( g_\theta (\bar{x})  ) d\bar{x} d \theta . 
    \\&\leq c'_{b,\sigma}  n^{-1}  \| f \|_{\infty}.
$$

Therefore we obtain $$\begin{aligned}
\label{scale invariant}
&\int_{S_n}\left| \bar{\sigma}_n (\bar{x}) - \bar{\sigma} (\bar{x}) \right| d \bar{x}
    \\ & \leq
    \left( \|\sigma_n  \|_{S_n, \infty} 
    \left( c_{s2} c_{b,\sigma}  +2  c'_{b,\sigma}\| {b}{\sigma^{-2}}  \|_{L^1(I)} \right)
    +
    c_{s2} 
    (  c_{s2} C_0 + c'_{b,\sigma}   \| f \|_{S_n, \infty } ) \right) 
     n^{-1},\end{aligned}$$ and hence we obtain the invariant property.

In short, we obtain the extended results of Theorem \[Thm:main1\] with the drifts.

\[Thm:main3\] Let $T>0$ and $p \geq 1$ and let $X_n$ for $n \in {\mathbb{N}}$ be solutions of $(\ref{sde:stability})$ and $X$ be a solution of $(\ref{sde0})$ such that ${\mathbb{E}}\left| X (0) \right|^p < \infty $ and $X_n (0) \equiv X (0)$. Suppose that $\sigma$ and $\{ \sigma_n \}_{n \in {\mathbb{N}}}$ satisfy the Nakao-Le Gall condition $\sigma, \sigma_n \in \mathcal{C}_{NL} (\epsilon, \| f\|_{\infty} )$.

If the stability rate conditions holds, i.e. there exists a positive constant $C_0$ such that $$\begin{aligned}
\int_{x \in {\mathbb{R}}} | b_n (x)- b(x)  | + | \sigma_n (x) - \sigma (x)  | dx \leq C_0 n^{-1}, \end{aligned}$$ Then there exist positive constants $C_p \ (p \geq 1)$ such that $$\begin{aligned}
{\mathbb{E}}\left[ \left| X (t) - X_n (t) \right| \right] \leq C_1 (\log n)^{-\frac{1}{2}} \ ( 0 \leq t \leq T )\end{aligned}$$ and $$\begin{aligned}
{\mathbb{E}}\left[ \sup_{0 \leq t \leq T} \left| X (t) - X_n (t) \right|^p \right] \leq C_p (\log n)^{-\frac{p}{4(p+1)}} \end{aligned}$$ holds for $p > 1$ and $n>2$.

Assume ${\mathbb{E}}[\sup_{ 0\leq t \leq T}|X (t)|^2] < \infty$. This implies that ${\mathbb{E}}[ \int_{{\mathbb{R}}} L_T^a (X) da  ] = {\mathbb{E}}[\langle X \rangle]_T < \infty$, and then the same convergence rate is obtained under the stability rate of $(\ref{con:L1})$.

[10]{}

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