A generic construction for high order approximation schemes of semigroups using random grids

Abstract

Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\ge 0$. In order to do it, we fix a time horizon T and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in {\mathbb {N}}$ and we suppose that we have at hand some short time approximation operators $Q_{l}$ such that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<\cdots <t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in {\mathbb {N}}$, and we associate the approximation discrete semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}\ldots Q_{l_{1}}$. Our main result is the following: for any approximation order $\nu$, we can construct random grids $\Pi _{i}(\omega )$ and coefficients $c_{i}$, with $i=1,\ldots ,r$ such that

$$\begin{aligned} P_{t}f=\sum _{i=1}^{r}c_{i}{\mathbb {E}}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu }) \end{aligned}$$

with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $Card (\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order n, for any order of approximation $\nu$. The standard example concerns diffusion processes, using the Euler approximation for $Q_l$. In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup $P_{t}$ and approximations. Besides, approximation schemes sharing the same $\alpha$ lead to the same random grids $\Pi _{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions.

Technical results for the variance analysis

We introduce some notation. We consider smooth random fields, that is functions $\varphi :\Omega \times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}$ which are measurable with respect to $(\omega ,x)$ and such that, for each $\omega ,$ the function $x\mapsto \varphi (\omega ,x)$ is of class $C^{\infty }({\mathbb {R}}^{d})$. For such a random field we denote

$$\begin{aligned} \left\| \varphi \right\| _{0,p,\infty }= & {} \sup _{x}\left\| \varphi (x)\right\| _{p}=\sup _{x}(\int \left| \varphi (\omega ,x)\right| ^{p} d{\mathbb {P}}(\omega ))^{1/p}, \end{aligned}$$

(79)

$$\begin{aligned} \left\| \varphi \right\| _{q,p,\infty }= & {} \sum _{\left| \alpha \right| \le q}\left\| \partial ^{\alpha }\varphi \right\| _{0,p,\infty }. \end{aligned}$$

(80)

Moreover, we will say that a sequence of random fields $\varphi _{i},i=1,\ldots ,m$ are independent if there are some independent $\sigma -$ algebras ${\mathcal {G}}_{i},i=1,\ldots ,m$ such that $\varphi _{i}$ is ${\mathcal {G}}_{i}\otimes {\mathcal {B}}({\mathbb {R}}^{d})$ measurable. We will use this property as follows. Suppose that $\Phi$ is ${\mathcal {G}}_{m}\otimes {\mathcal {B}}({\mathbb {R}}^{d})$ measurable and $\Psi$ and $\Theta$ are $\vee _{i=1}^{m-1}{\mathcal {G}}_{i}\otimes {\mathcal {B}}({\mathbb {R}}^{d})$ measurable. Then, for every $x\in {\mathbb {R}}^{d}$ and every $p\ge 1$

$$\begin{aligned} {\mathbb {E}}(\left| \Phi (\omega ,\Psi (\omega ,x))\right| ^{p} |\Theta (\omega ,x)|)= & {} {\mathbb {E}}\left( |\Theta (\omega ,x)| {\mathbb {E}}\left( \left| \Phi (\omega ,\Psi (\omega ,x))\right| ^{p} \bigg | \vee _{i=1}^{m-1}{\mathcal {G}}_{i} \right) \right) \nonumber \\\le & {} \left\| \Phi \right\| _{0,p,\infty }^{p}{\mathbb {E}}(\left| \Theta (\omega ,x)\right| ). \end{aligned}$$

(81)

In the sequel we consider a sequence of smooth random fields $\Phi _{i}:\Omega \times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}$ and $\phi _{i}:\Omega \times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}$ , $i\in {\mathbb {N}}$ and moreover, a vector field $\varphi :\Omega \times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}$. We assume that $\varphi$ and $(\Phi _{j},\phi _{j}),j\in {\mathbb {N}}$ are independent. We fix $r\in {\mathbb {N}}$ and, for a set $\Lambda \subset \{1,\ldots ,r\},$ we define

$$\begin{aligned} \theta _{i}^{\Lambda }= & {} 1_{\Lambda }(i)\phi _{i}+1_{\Lambda ^{c}}(i)\Phi _{i},\quad i=1,\ldots ,r\quad and \\ \theta _{(r)}^{\Lambda }= & {} \theta _{r}^{\Lambda }\circ \cdots \circ \theta _{1}^{\Lambda } \end{aligned}$$

Moreover, given a multi-index $\alpha ,$ we define

$$\begin{aligned} \Gamma _{r}^{\alpha }\varphi (x)=\sum _{\Lambda \subset \{1,\ldots ,r\}}(-1)^{Card ( \Lambda ) }\partial _{x}^{\alpha }[\varphi (\theta _{(r)}^{\Lambda })](x) \end{aligned}$$

(82)

Proposition A.1

Suppose that for every $p,q\in {\mathbb {N}}$, there exists $C_{q,p}$ such that

$$\begin{aligned} \forall i\in \{1,\ldots ,r\}, \ \left\| \Phi _{i}\right\| _{q,p,\infty }+\left\| \phi _{i}\right\| _{q,p,\infty }+\left\| \varphi \right\| _{q+1,p,\infty }\le C_{q,p}<\infty . \end{aligned}$$

(83)

Then, for every $p\ge 1$ and every multi-index $\alpha$ we have

$$\begin{aligned} \left\| \Gamma _{r}^{\alpha }\varphi \right\| _{0,p,\infty }\le C\times \prod _{i=1}^{r}\left\| \Phi _{i}-\phi _{i}\right\| _{\left| \alpha \right| +r,2 p,\infty } \end{aligned}$$

(84)

for some C depending on r, $\left| \alpha \right|$ and $C_{\left| \alpha \right| +r,2|\alpha | p}.$

Remark A.2

This proposition says the following: if at each step the error is of order $\delta _{i}=\left\| \Phi _{i}-\phi _{i}\right\| _{q,p^{\prime },\infty }$, then after r steps we have an error of order $\delta _{1}\times \cdots \times \delta _{r}.$ This may seem a little surprising, and one may have expected an error of order $\delta _{1}+\cdots +\delta _{r}$, but this is due to the way how terms are summed with $\sum _{\Lambda \subset \{1,\ldots ,r\}}(-1)^{Card ( \Lambda )}.$

Proof

Step 1. We use the Faá di Bruno formula $\partial ^{\alpha }[f\circ g]=\sum _{\left| \beta \right| \le \left| \alpha \right| }(\partial ^{\beta }f)(g)P_{\alpha ,\beta }(g)$ (see (8)) and the inequality between geometric and arithmetic means to upper bound the terms $\vert \prod _{i=1}^{k}\partial ^{\gamma _{i}} g^{j_i} \vert$ defining $P_{\alpha ,\beta }(g)$. We then obtain for random functions g

$$\begin{aligned} \left\| P_{\alpha ,\beta }(g)\right\| _{0,p,\infty }^p\le C \left\| g\right\| _{\left| \alpha \right| ,|\alpha | p,\infty }^{|\alpha | p}. \end{aligned}$$

Besides, for two random fields $g_{1}$ and $g_{2}$, we write

$$\begin{aligned} \prod _{i=1}^{k}\partial ^{\gamma _{i}}g_1^{j_i}-\prod _{i=1}^{k}\partial ^{\gamma _{i}}g_2^{j_i}=\sum _{i'=1}^k \left( \prod _{i<i'}\partial ^{\gamma _{i}}g_2^{j_i} \right) (\partial ^{\gamma _{i'}}g_1^{j_{i'}}-\partial ^{\gamma _{i'}}g_2^{j_{i'}} ) \left( \prod _{i>i'}\partial ^{\gamma _{i}}g_1^{j_i}\right) . \end{aligned}$$

Using the inequality between geometric and arithmetic means for the product on $i\not =j$ and then the Cauchy–Schwarz inequality, we get

$$\begin{aligned}&\left\| P_{\alpha ,\beta }(g_{1})-P_{\alpha ,\beta }(g_{2})\right\| _{0,p,\infty }^p\le C\left( \left\| g_{1}\right\| _{\left| \alpha \right| ,2(|\alpha |-1)p,\infty }\right. \nonumber \\&\quad \left. +\left\| g_{2}\right\| _{\left| \alpha \right| ,2(|\alpha |-1)p,\infty }\right) ^{(|\alpha |-1)p} \left\| g_{1}-g_{2}\right\| ^p_{\left| \alpha \right| ,2 p,\infty }. \end{aligned}$$

(85)

Step 2. We prove (84) for $r=1.$ In this case $\Lambda =\varnothing$ or $\Lambda =\{1\}$ so that

$$\begin{aligned} \Gamma _{1}^{\alpha }\varphi (x)= & {} \partial ^{\alpha }[\varphi (\Phi _{1})](x)-\partial ^{\alpha }[\varphi (\phi _{1})](x) \\= & {} \sum _{\left| \beta \right| \le \left| \alpha \right| }(\partial ^{\beta }\varphi )(\Phi _{1})P_{\alpha ,\beta }(\Phi _{1})-(\partial ^{\beta }\varphi )(\phi _{1})P_{\alpha ,\beta }(\phi _{1}) \\= & {} \sum _{\left| \beta \right| \le \left| \alpha \right| }A_{\beta }+B_{\beta } \end{aligned}$$

with

$$\begin{aligned} A_{\beta }= & {} ((\partial ^{\beta }\varphi )(\Phi _{1})-(\partial ^{\beta }\varphi )(\phi _{1}))P_{\alpha ,\beta }(\Phi _{1}) \\= & {} P_{\alpha ,\beta }(\Phi _{1})\int _{0}^{1}\left\langle \nabla (\partial ^{\beta }\varphi )(\lambda \Phi _{1}+(1-\lambda )\phi _{1}),\Phi _{1}-\phi _{1}\right\rangle d\lambda \end{aligned}$$

and

$$\begin{aligned} B_{\beta }=(\partial _{\beta }\varphi )(\phi _{1})(P_{\alpha ,\beta }(\Phi _{1})-P_{\alpha ,\beta }(\phi _{1})). \end{aligned}$$

Using (83) and the fact that $\varphi$ is independent of $\lambda \Phi _{1}+(1-\lambda )\phi _{1}$ we get (see (81))

$$\begin{aligned} \left\| \left\langle \nabla (\partial ^{\beta }\varphi )(\lambda \Phi _{1}+(1-\lambda )\phi _{1}),\Phi _{1}-\phi _{1}\right\rangle \right\| _{p}\le & {} \left\| \varphi \right\| _{\left| \beta \right| +1,p,\infty }\left\| \Phi _{1}-\phi _{1}\right\| _{0,p,\infty } \\\le & {} C_{\left| \alpha \right| ,p}\left\| \Phi _{1}-\phi _{1}\right\| _{0,p,\infty } \end{aligned}$$

so that $\left\| A_{\beta }\right\| _{0,p,\infty }\le C\left\| \Phi _{1}-\phi _{1}\right\| _{0,p,\infty }.$ Moreover, using again (81 ) first and then (85), we get

$$\begin{aligned} \left\| B_{\beta }\right\| _{0,p,\infty }\le \left\| \varphi \right\| _{\left| \beta \right| ,p,\infty }\left\| P_{\alpha ,\beta }(\Phi _{1})-P_{\alpha ,\beta }(\phi _{1})\right\| _{0,p,\infty }\le C\left\| \Phi _{1}-\phi _{1}\right\| _{|\alpha |,2p,\infty } \end{aligned}$$

so (84) is proved for $r=1$.

Step 3. Suppose (84) is true for $r-1,$ for every $\alpha$ and every $p\ge 1.$ We prove it for r. We do it first for $\alpha =\varnothing$ (without derivatives) because it is simpler. We write

$$\begin{aligned} \Gamma _{r}^{\varnothing }\varphi= & {} \sum _{\Lambda \subset \{1,\ldots ,r\}}(-1)^{Card ( \Lambda ) }\varphi (\theta _{(r)}^{\Lambda }) \\= & {} \sum _{\Lambda ^{\prime }\subset \{2,\ldots ,r\}}(-1)^{Card (\Lambda ^{\prime }) }(\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})(\Phi _{1})-\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})(\phi _{1})) \\= & {} \sum _{i=1}^{d}(\Phi _{1}^{i}-\phi _{1}^{i})\int _{0}^{1}\sum _{\Lambda ^{\prime }\subset \{2,\ldots ,r\}}(-1)^{Card ( \Lambda ^{\prime }) }\partial ^{i}[\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})](\lambda \Phi _{1}+(1-\lambda )\phi _{1})d\lambda \\= & {} \sum _{i=1}^{d}(\Phi _{1}^{i}-\phi _{1}^{i})\int _{0}^{1}\Gamma _{r-1}^{(i)}\varphi (\lambda \Phi _{1}+(1-\lambda )\phi _{1})d\lambda . \end{aligned}$$

Note that we have made a slight abuse of notation here: the notation $\theta _{(r-1)}^{\Lambda ^{\prime }}$ is used in fact for $\theta ^{\Lambda '}_r \circ \dots \circ \theta ^{\Lambda '}_2$, not for $\theta ^{\Lambda '}_{r-1} \circ \dots \circ \theta ^{\Lambda '}_1$. Since $(\lambda \Phi _{1}+(1-\lambda )\phi _{1})(x)$ is independent of $\Gamma _{r-1}^{(i)}\varphi ,$ we have from (81)

$$\begin{aligned} \left\| \Gamma _{r}^{\varnothing }\varphi (x)\right\| _{p}\le & {} d \left\| \Phi _{1}-\phi _{1}\right\| _{0,p,\infty }\times \left\| \Gamma _{r-1}^{(i)}\varphi \right\| _{0,p,\infty }. \end{aligned}$$

Then, by using the induction hypothesis, we get

$$\begin{aligned} \left\| \Gamma _{r}^{\varnothing }\varphi (x)\right\| _{p}\le & {} C\left\| \Phi _{1}-\phi _{1}\right\| _{0,p,\infty }\times \prod _{j=2}^{r}\left\| \Phi _{j}-\phi _{j}\right\| _{1+r-1,2 p,\infty } \\\le & {} C\prod _{j=1}^{r}\left\| \Phi _{j}-\phi _{j}\right\| _{r,2p,\infty }. \end{aligned}$$

We prove now (84) for a general multi-index $\alpha$ and make the same abuse of notation for $\theta ^{\Lambda '}$. Using (8) and (9) for $f=\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})$ and $g_{1}=\Phi _{1},g_{2}=\phi _{1}$ we obtain

$$\begin{aligned} \Gamma _{r}^{\alpha }\varphi= & {} \sum _{\Lambda \subset \{1,\ldots ,r\}}(-1)^{Card (\Lambda ) }\partial _{x}^{\alpha }[\varphi (\theta _{(r)}^{\Lambda })] \\= & {} \sum _{\Lambda ^{\prime }\subset \{2,\ldots ,r\}}(-1)^{Card ( \Lambda ^{\prime }) }\sum _{\left| \beta \right| \le \left| \alpha \right| }\partial ^{\beta }[\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})](\Phi _{1})P_{\alpha ,\beta }(\Phi _{1}) \\&-\sum _{\Lambda ^{\prime }\subset \{2,\ldots ,r\}}(-1)^{Card ( \Lambda ^{\prime }) }\sum _{\left| \beta \right| \le \left| \alpha \right| }\partial ^{\beta }[\varphi (\theta _{(r-1)}^{\Lambda ^{\prime }})](\phi _{1})P_{\alpha ,\beta }(\phi _{1}) \\= & {} \sum _{\left| \beta \right| \le \left| \alpha \right| }\Gamma _{r-1}^{\beta }\varphi (\Phi _{1})P_{\alpha ,\beta }(\Phi _{1})-\Gamma _{r-1}^{\beta }\varphi (\phi _{1})P_{\alpha ,\beta }(\phi _{1}) \\= & {} \sum _{\left| \beta \right| \le \left| \alpha \right| }A_{\beta }+B_{\beta } \end{aligned}$$

with

$$\begin{aligned} A_{\beta }=(\Gamma _{r-1}^{\beta }\varphi (\Phi _{1})-\Gamma _{r-1}^{\beta }\varphi (\phi _{1}))P_{\alpha ,\beta }(\Phi _{1}) \end{aligned}$$

and

$$\begin{aligned} B_{\beta }=\Gamma _{r-1}^{\beta }\varphi (\Phi _{1})(P_{\alpha ,\beta }(\Phi _{1})-P_{\alpha ,\beta }(\phi _{1})). \end{aligned}$$

By assumption $\theta _2,\dots ,\theta _r$ are independent of $(\Phi _{1},\phi _{1})$. Therefore, $\Gamma _{r-1}^{\beta }\varphi (x)$ is independent of $(\Phi _{1},\phi _{1})$. We use (81) first and then the induction hypothesis and (85) to obtain

$$\begin{aligned} \left\| B_{\beta }\right\| _{0,p,\infty }\le & {} \left\| \Gamma _{r-1}^{\beta }\varphi \right\| _{0,p,\infty }\left\| P_{\alpha ,\beta }(\Phi _{1})-P_{\alpha ,\beta }(\phi _{1})\right\| _{0,p,\infty } \\\le & {} C \left\| \Phi _{1}-\phi _{1}\right\| _{|\alpha |,2 p,\infty } \prod _{i=2}^{r}\left\| \Phi _{i}-\phi _{i}\right\| _{\left| \beta \right| +r-1,2 p,\infty } \\\le & {} C \prod _{i=1}^{r}\left\| \Phi _{i}-\phi _{i}\right\| _{\left| \alpha \right| +r,2 p,\infty } . \end{aligned}$$

Moreover

$$\begin{aligned} A_{\beta }=P_{\alpha ,\beta }(\Phi _{1})\int _{0}^{1}\left\langle (\nabla \Gamma _{r-1}^{\beta }\varphi )(\lambda \Phi _{1}+(1-\lambda )\phi _{1}),\Phi _{1}-\phi _{1}\right\rangle d\lambda . \end{aligned}$$

Notice that $\partial ^{i}\Gamma _{r-1}^{\beta }\varphi =\Gamma _{r-1}^{(\beta ,i)}\varphi$. Using again (81) and the recurrence hypothesis, we get

$$\begin{aligned} \left\| A_{\beta }\right\| _{0,p,\infty }\le C\prod _{i=1}^{r}\left\| \Phi _{i}-\phi _{i}\right\| _{\left| \beta \right| +1+r-1,2p,\infty }\le C \prod _{i=1}^{r}\left\| \Phi _{i}-\phi _{i}\right\| _{\left| \alpha \right| +r,2 p,\infty } . \end{aligned}$$

$\square$

Lemma A.3

Let $(X_t(x))_{t\ge 0}$ denote the flow of the SDE (1) and ${\hat{X}}_t(x)=x+b(x)t+\sigma (x)W_t$ the flow of the Euler scheme. We assume that b and $\sigma$ are $C^\infty$, bounded and with bounded derivatives. Then, we have

$$\begin{aligned} \forall p,q\in {\mathbb {N}}, \exists C_{p,q}, \Vert {\hat{X}}_t-X_t\Vert _{q,p,\infty }\le C_{p,q} t^a, \end{aligned}$$

with $a=2$ if $\sigma =0$, $a=3/2$ if $\sigma (x)$ is a constant function and $a=1$ in the general case.

Proof

We show this result by induction on q. We only focus on the general case, the cases $\sigma =0$ or $\sigma (x)$ constant can be then easily deduced. For $q=0$, this result is stated for example in Proposition 1.2 [2]. For simplicity of notation, we do the proof in dimension $d=1$ with $b=0$. We note $\sigma ^{(q)}$ the qth derivative of $\sigma$. For $q=1$, we have ${\hat{X}}_t^{(1)}(x)=1+\sigma ^{(1)}(x)W_t$ and

$$\begin{aligned} X_t^{(1)}(x)=1+\int _0^tX_s^{(1)}(x) \sigma ^{(1)} (X_s(x)) dW_s . \end{aligned}$$

(86)

Since $\sigma ^{(1)}$ is bounded, we have $\forall t>0, \sup _{x}{\mathbb {E}}[\sup _{s\in [0,t]} |X_s^{(1)}(x)|^p]<\infty$. We write

$$\begin{aligned} {\hat{X}}_t^{(1)}(x)- X_t^{(1)}(x)=\int _0^t (X_s^{(1)}(x)-1) \sigma ^{(1)} (X_s(x)) dW_s +\int _0^t \sigma ^{(1)} (X_s(x))-\sigma ^{(1)}(x) dW_s. \end{aligned}$$

Since $\sigma ^{(1)}$ is bounded and Lipschitz, we get by using the Burkholder–Davis–Gundy inequality and then Jensen inequality

$$\begin{aligned} {\mathbb {E}}[|{\hat{X}}_t^{(1)}(x)- X_t^{(1)}(x)|^p\le C t^{p/2-1}\int _0^t {\mathbb {E}}[|X_s^{(1)}(x)-1|^p]+{\mathbb {E}}[|X_s(x)-x|^p]ds, \end{aligned}$$

with a constant C that does not depend on x. We check then again with the BDG inequality that ${\mathbb {E}}[|X_s(x)-x|^p]\le Cs^{p/2}$ since $\sigma$ is bounded and ${\mathbb {E}}[|X_s^{(1)}(x)-1|^p]\le Cs^{p/2}$ since $\sigma ^{(1)}$ is bounded and (86). Thus, we have ${\mathbb {E}}[|{\hat{X}}_t^{(1)}(x)- X_t^{(1)}(x)|^p]\le C t^p$.

We suppose now the result true for $q-1\in {\mathbb {N}}^*$ and that we have shown that

$$\begin{aligned} {\mathbb {E}}[|X_s^{(r)}(x)|^p]\le Cs^{p/2},\text { for }2\le r\le q-1, \end{aligned}$$

(87)

for each p, with a constant C that does not depend on x. We have ${\hat{X}}_t^{(q)}(x)=\sigma ^{(q)}(x)W_t,$ and by the Faá di Bruno formula

$$\begin{aligned} d X^{(q)}_t(x)&=\sum _{m_1+\dots + q m_q=q}c_{m_1,\dots ,m_q} \prod _{k=1}^q( X^{(k)}_t(x))^{m_k} \sigma ^{(m_1+\dots +m_q)}(X_t(x))dW_t \\&= X^{(q)}_t(x) \sigma ^{(1)}(X_t(x))dW_t + (X^{(1)}_t(x))^q \sigma ^{(q)}(X_t(x))dW_t +A_tdW_t, \end{aligned}$$

with $A_t=\sum _{m_1+\dots + q m_q=q, m_1\not = q,m_q\not =0 }c_{m_1,\dots ,m_q} \prod _{k=1}^q( X^{(k)}_t(x))^{m_k} \sigma ^{(m_1+\dots +m_q)}(X_t(x))$. Note that in this sum is equal to 0 for $q=2$ and otherwise there is at least one $k\in \{2,\dots ,q-1\}$, such that $m_k\ge 1$. This gives ${\mathbb {E}}[|A_t|^p]\le Ct^{p/2}$ by using the induction hypothesis (87) and Hölder type inequalities. Since $X^{(q)}_0(x)=0$, $\sigma ^{(1)}$ and $\sigma ^{(q)}$ are bounded and $\sup _{x}{\mathbb {E}}[\sup _{s\in [0,t]} |X_s^{(1)}(x)|^p]<\infty$ for any p, we get ${\mathbb {E}}[|X_t^{(q)}(x)|^p]\le Ct^{p/2}$ by using BDG and Gronwall inequalities. Therefore, ${\tilde{A}}_t=A_t+ X^{(q)}_t(x) \sigma ^{(1)}(X_t(x))$ also satisfies ${\mathbb {E}}[|{\tilde{A}}_t|^p]\le Ct^{p/2}$.

We now repeat the same arguments as for $q=1$: from

$$\begin{aligned} {\hat{X}}_t^{(q)}(x)- X_t^{(q)}(x)=&\int _0^t ((X_s^{(1)}(x))^q-1) \sigma ^{(q)} (X_s(x)) dW_s +\int _0^t \sigma ^{(q)} (X_s(x))-\sigma ^{(q)}(x) dW_s \\&+ \int _0^t {\tilde{A}}_tdW_t, \end{aligned}$$

we get ${\mathbb {E}}[|{\hat{X}}_t^{(q)}(x)- X_t^{(q)}(x)|^p]\le C t^p$.

$\square$