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A higher order weak approximation of McKean–Vlasov type SDEs

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Abstract

The paper introduces a new weak approximation algorithm for stochastic differential equations (SDEs) of McKean–Vlasov type. The arbitrary order discretization scheme is available and is given using Malliavin weights, certain polynomial weights of Brownian motion, which play a role as correction of the approximation. The new weak approximation scheme works even if the test function is not smooth. In other words, the expectation of irregular functionals of McKean–Vlasov SDEs such as probability distribution functions are approximated through the proposed scheme. The effectiveness of the higher order scheme is confirmed by numerical examples for McKean–Vlasov SDEs including the Kuramoto model.

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Acknowledgements

We thank an associate editor and anonymous two referees for valuable comments and suggestions. We thank Prof. Jiro Akahori (Ritsumeikan University) and Prof. Arturo Kohatsu-Higa (Ritsumeikan University) for advices on the proposed scheme at Math-Finance seminar. We also thank Prof. Hirofumi Yamada (Kyoto University) for comments on models in physics.

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Correspondence to Toshihiro Yamada.

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Communicated by David Cohen.

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This work is supported by JSPS KAKENHI (Grant Number 19K13736), MEXT, Japan and JST PRESTO (Grant Number JPMJPR2029), Japan.

Appendices

Appendix

Proof of Lemma 3.1

For \(k \in \mathbb {N}\) and for multi-indices \(\alpha ^{e} \in \{ 0,1,\ldots ,d \}^{r(e)}\), \(r(e) \in \mathbb {N}\), \(e=1,\ldots ,k\) such that \(\Vert \alpha ^{e}\Vert \ge 2\), \(e=1,\ldots ,k\), we have

$$\begin{aligned} \prod _{e=1}^k I_{\alpha ^e,t,s}^{\mathrm {Strat}} = \sum _{ \begin{array}{c} \sigma \in \mathrm {Shuffles}(r_1,\ldots ,r_k) \end{array} } I_{\sigma ^{-1}\cdot (\alpha ^1*\ldots \alpha ^k),t,s}^{\mathrm {Strat}}. \end{aligned}$$
(A.1)

By Proposition 3.2, it holds that

$$\begin{aligned}&{}_{\mathbb {D}^{-\infty }} \langle \delta _\xi (W_{t,s}), \prod _{e=1}^k I_{\alpha ^e,t,s} ^{\mathrm {Strat}} \rangle _{\mathbb {D}^{\infty }} \nonumber \\&\quad = \sum _{ \begin{array}{c} \beta =\sigma ^{-1} \cdot (\alpha ^1*\ldots *\alpha ^k) \sim _p \gamma , \ p\in \mathbb {N},\\ \sigma \in \mathrm {Shuffles}(r_1,\ldots ,r_k) \end{array} } \frac{1}{2^{\eta (\beta ,\gamma )}} {}_{\mathbb {D}^{-\infty }} \langle \delta _\xi (W_{t,s}), I_{\gamma ,t,s}^{\mathrm {It}{\hat{\mathrm{o}}}} \rangle _{\mathbb {D}^{\infty }}. \end{aligned}$$

By integration by parts, we have

$$\begin{aligned} {}_{\mathbb {D}^{-\infty }} \langle \delta _\xi (W_{t,s}), I_{\gamma ,t,s}^{{\mathrm {It}}{\hat{o}}} \rangle _{\mathbb {D}^{\infty }}= & {} \frac{1}{|\gamma |!} (s-t)^{|\gamma |} {} _{\mathbb {D}^{-\infty }} \langle \partial ^{\gamma ^{\star }} \delta _\xi (W_{t,s}), 1 \rangle _{\mathbb {D}^{\infty }}\nonumber \\= & {} \frac{1}{|\gamma |!}(s-t)^{|\gamma |-|\gamma ^\star |} {}_{\mathbb {D}^{-\infty }} \langle \delta _\xi (W_{t,s}), \mathbb {W}_{\gamma ^\star ,t,s}^{\mathrm {Skor}} \rangle _{\mathbb {D} ^{\infty }}. \end{aligned}$$

Then we obtain the assertion. \(\square \)

Proof of Lemma 3.2

By (2.2), we have for \(G \in \mathbb {D}^\infty \),

$$\begin{aligned} E[\partial _i \varphi ({\bar{X}}_s^{t,x,z})G]=&E[ \varphi ({\bar{X}}_s^{t,x,z}) \sum _{j=1}^N \delta ( G \gamma _{ij}^{{\bar{X}}_s^{t,x,z}} D{\bar{X}}_s^{t,x,z,j} ) ] \end{aligned}$$
(B.1)

and note that the following holds:

$$\begin{aligned} \delta (Gh \mathbf{1}_A)=G \sum _{k=1}^d \int _{A} h^k(u)dW_u^k-\sum _{k=1}^d \int _{A} D_{k,u}G h^k(u)du, \end{aligned}$$
(B.2)

for \(h \in L^2([0,T];\mathbb {R}^d)\) and \(A \in \mathscr {B}([0,T])\) where \( \mathscr {B}([0,T])\) is the Borel \(\sigma \)-field over [0, T], by the property of Skorohod integral (see Proposition 1.3.5 of Nualart [20]). Since it holds that \(D_{k,u}{\bar{X}} _s^{t,x,z,j}=V_{k}^j(x,z) \mathbf{1}_{[t,s]}(u)\) for \(k=1,\ldots ,d\), \(j=1,\ldots ,N\), and

$$\begin{aligned} \sigma _{ij}^{{\bar{X}}_s^{t,x,z,j}}&=\langle D{\bar{X}}_s^{t,x,z,i},D{\bar{X}}_s^{t,x,z,j} \rangle _H=(s-t) \sum _{k=1}^d V_{k}^i(x,z)V_{k}^j(x,z) =(s-t)A_{i,j}(x,z), \nonumber \\& 1\le i,j \le N, \end{aligned}$$
(B.3)

we have

$$\begin{aligned}&E[\partial _i \varphi ({\bar{X}}_s^{t,x,z})G]\nonumber \\&\quad =E[\varphi ({\bar{X}}_s^{t,x,z}) (s-t)^{-1}\sum _{j=1}^N\sum _{k=1}^d A_{i,j}^{-1}(x,z) \{ G\delta _k( V^j_{k}(x,z) \mathbf{1}_{[t,s]}(\cdot ) )\nonumber \\&\qquad -\int _t^s D_{k,u}G V^j_{k}(x,z) du \} ] \nonumber \\&\quad =E[\varphi ({\bar{X}}_s^{t,x,z}) (s-t)^{-1}\sum _{j=1}^N\sum _{k=1}^d A_{i,j}^{-1} (x,z)V^j_{k}(x,z) \{GW_{t,s}^k-\int _t^s D_{k,u}Gdu \} ]. \end{aligned}$$
(B.4)

If \(G=\mathbb {W}_{\gamma ,t,s}^{\mathrm {Skor}}\), \(\gamma \in \{1,\ldots ,d \}^\ell \), \(\ell \in \mathbb {N}\), we have

$$\begin{aligned} GW_{t,s}^k-\int _t^s D_{k,u}Gdu=\mathbb {W}_{\gamma *(k),t,s}^{\mathrm {Skor}}. \end{aligned}$$
(B.5)

\(\square \)

Proof of Lemma 5.1

We recall the following estimate on Malliavin covariance matrix from Theorem 3.5 of Kusuoka and Stroock [15], for \(p\ge 1\), there exists \(C>0\) such that for all \(t<s \), \(x,y\in \mathbb {R}^N\),

$$\begin{aligned} \Big \Vert \det ( \sigma ^{{X}_{s}^{t,x,y}} )^{-1} \Big \Vert _{p} \le C \frac{1}{(s-t)^N}. \end{aligned}$$
(C.1)
  1. 1.

    We first show the representation of \(\frac{\partial ^{\alpha }}{\partial x^{\alpha }}E[ f({X}_{T}^{t_{j+1},x,y}) ]\) with a multi-index \(\alpha =(\alpha _1,\ldots ,\alpha _k) \in \{1,\ldots ,N \}^k\) where \( \frac{\partial ^{\alpha }}{\partial x^{\alpha }}=\frac{\partial ^{k}}{\partial x_{\alpha _1} \ldots \partial x_{\alpha _k}}\). First, one gets

    $$\begin{aligned}&\frac{\partial }{\partial x_i}E[ f({X}_{T}^{t_{j+1},x,y}) ] =\int _{\mathbb {R}^N} f(\xi ) \frac{\partial }{\partial x_i} {}_{\mathbb {D}^{-\infty }}\langle \delta _\xi ({X}_{T}^{t_{j+1},x,y}),1 \rangle _{\mathbb {D}^\infty } d\xi \nonumber \\&\quad =\int _{\mathbb {R}^N} f(\xi ) \sum _{l=1}^N{}_{\mathbb {D}^{-\infty }}\langle \partial _\ell \delta _\xi ({X}_{T}^{t_{j+1},x,y}),\frac{\partial }{\partial x_i}{X}_{T}^{t_{j+1},x,y, \ell } \rangle _{\mathbb {D}^\infty } d\xi \nonumber \\&\quad =\int _{\mathbb {R}^N} f(\xi )\sum _{l=1}^N {}_{\mathbb {D}^{-\infty }}\langle \delta _\xi ({X}_{T}^{t_{j+1},x,y}),H_{(\ell )}({X}_{T}^{t_{j+1},x,y},\frac{\partial }{\partial x_i} {X}_{T}^{t_{j+1},x,y,\ell }) \rangle _{\mathbb {D}^\infty } d\xi \nonumber \\&\quad =E \Big [f({X}_{T}^{t_{j+1},x,y})\sum _{l=1}^NH_{(\ell )} \Big ({X}_{T}^{t_{j+1},x,y}, \frac{\partial }{\partial x_i}{X}_{T}^{t_{j+1},x,y,\ell } \Big ) \Big ], \end{aligned}$$
    (C.2)

    for \(i=1,\ldots ,N\). Iterating this procedure, we have

    $$\begin{aligned} \frac{\partial ^{\alpha }}{\partial x^{\alpha }}E[ f({X}_{T}^{t_{j+1},x,y}) ] =\sum _{\gamma _e,e\le k} E[f({X}_{T}^{t_{j+1},x,y})H_{\gamma _e}({X}_{T}^{t_{j+1},x,y},G_{\gamma _e})], \end{aligned}$$
    (C.3)

    where \(\gamma _e\), \(e=1,\ldots ,k\) are some multi-indices with \(|\gamma _e| \le k\), and \(G_{\gamma _e}\), \(e=1,\ldots ,k\) are some polynomials of \(\frac{\partial ^\beta }{\partial x^{\beta }}{X}_{T-t_{j+1}}^{0,x,y}\) with multi-index \(\beta \) of length \(|\beta | \le k\). By Corollary 3.7 of Kusuoka and Stroock [15], we have that for any \(p\ge 1\) and multi-index \(\gamma \), there exists \(C>0\) such that for \(G \in \mathbb {D}^\infty \), \(t<s\) and \(x,y\in \mathbb {R}^N\),

    $$\begin{aligned}&\Vert H_{\gamma } (X_{s}^{t,x,y},G ) \Vert _p \le C (s-t)^{(2N-1/2)|\gamma |-N} \bigl \Vert \det (\sigma ^{X_{s}^{t,x,y}})^{-1} \bigr \Vert ^{(| \gamma |+1)(2| \gamma |-1)}_{(8N+4)| \gamma |(| \gamma |+1)p} \nonumber \\& \times \Vert DX_{s}^{t,x,y} \Vert ^{2N | \gamma |(2| \gamma |-1)}_{| \gamma |,(16N+8)N | \gamma |^2,H} \Vert G \Vert _{| \gamma |,(4N+2)| \gamma |p}. \end{aligned}$$
    (C.4)

    Note that the following bound holds: for \(k\in \mathbb {N}\), \(p>1\), \(\Vert DX_{s}^{t,x,y} \Vert _{k,p,H}=O((s-t)^{1/2})\), for \(x,y \in \mathbb {R}^N\), by Theorem 2.19 of Kusuoka and Stroock [15]. Combining this with the Kusuoka-Stroock’s estimate (C.1), we have that for any \(p\ge 1\), \(\Vert H_{\gamma }({X}_{s}^{t,x,y},G) \Vert _p \le C (s-t)^{-|\gamma |/2} \Vert G \Vert _{|\gamma |,q}\), for some \(q>1\). To show the bounds of \(G_{\gamma _e}\), \(e=1,\ldots ,k\) in (C.3), we use the result on stochastic flow (Theorem 3.4 of Kusuoka [13]): for any \(k\in \mathbb {N}\), \(p>1\) and multi-index \(\alpha \), there is \(C>0\) such that \(\textstyle {\Vert \frac{\partial ^{\alpha }}{\partial x^{\alpha }}{X}_{s}^{t,x,y} \Vert _{k,p}} \le C\) for all \(t<s \le T\) and \(x,y\in \mathbb {R}^N\). Therefore, it holds that for \(e=1,\ldots ,k\) and \(p\ge 1\), there is \(C>0\) such that \(\Vert H_{\gamma _e}({X}_{T}^{t_{j+1},x,y},G_{\gamma _e}) \Vert _{p}\le C (T-t_{j+1})^{-|\gamma _e|/2}\) for all \(n\ge 1\), \(j\le n-1\) and \(x,y\in \mathbb {R}^N\). Then, we see that there exist \(C>0\) and \(Q>0\) such that

    $$\begin{aligned} \Big | \frac{\partial ^{\alpha }}{\partial x^{\alpha }}E[ f({X}_{T}^{t_{j+1},x,y}) ] \Big | \le C \Vert f \Vert _{\infty } \frac{1}{(T-t_{j+1})^Q}, \end{aligned}$$
    (C.5)

    for all \(n\ge 1\), \(j\le n-1\) and \(x,y\in \mathbb {R}^N\), which proves the assertion.

  2. 2.

    For the process \({\bar{X}}_{t_{i+1}}^{(T/n),x,y}\) \(={\bar{X}}_{t_i}^{(T/n),x,y}+\sum _{\ell =0}^d V_\ell ({\bar{X}}_{t_i}^{(T/n),x,y},Q^{(m),y}_{t_i}\varphi (y) )W^\ell _{t_i,t_{i+1}}\), \(i=0,1,\ldots ,n-1\), \({\bar{X}}_{0}^{(T/n),x,y}=x \in \mathbb {R}^N\), \(y\in \mathbb {R}^N\), we see that for all \(p\ge 1\), at least

    $$\begin{aligned} \Big \Vert \det ( \sigma ^{{\bar{X}}_{t_{j+1}}^{(T/n),x,y}} )^{-1} \Big \Vert _{p} = O(n^{\gamma }) \end{aligned}$$
    (C.6)

    for some \(\gamma >0\), by the elliptic condition on \(V=(V_1,\ldots ,V_d)\). But for all \(j\le n-1\), one has that for all \(k \in \mathbb {N}\), \(p\ge 1\),

    $$\begin{aligned} \Vert {X}_{t_{j+1}}^{t_j,x,y}-{\bar{X}}_{t_{j+1}}^{(T/n),x,y} \Vert _{k,p}=O(n^{-1/2}) \end{aligned}$$
    (C.7)

    (which will be proved after (C.8)). Then, by (C.1) (with \(t=0\) and \(s=t_{j+1}\)), (C.6) and (C.7), we are able to apply Lemma 2.1 of Hu and Watanabe [9] and obtain that there exists \(C>0\) such that

    $$\begin{aligned} \Big \Vert \det ( \sigma ^{{\bar{X}}_{t_{j+1}}^{(T/n),x,y}} )^{-1} \Big \Vert _{q} \le C\frac{1}{t_{j+1}^N}, \end{aligned}$$
    (C.8)

    for all \(x,y\in \mathbb {R}^N\) and \(n\ge 1\). Here, (C.7) is obtained by introducing a process \({\hat{X}}_{t_{0}}^{(T/n),x,y}=x\), \({\hat{X}}_{t_{i+1}}^{(T/n),x,y}\) \(={\hat{X}}_{t_i}^{(T/n),x,y}+\sum _{\ell =0}^d V_\ell ({\bar{X}}_{t_i}^{(T/n),x,y},P^{y}_{0,t_i}\varphi (y) )W^\ell _{t_i,t_{i+1}}\), \(i=0,1,\ldots ,n-1\), or a Itô process \({\hat{X}}_{s}^{(T/n),x,y}=x+\sum _{i=0}^d \int _0^s V_i({\hat{X}}_{\psi (u)}^{(T/n),x,y},P^y_{0,\psi (u)}\varphi (y)) dW_u^i\), \(s\ge 0\) with \(\psi (s):=\max \{ kT/n; s\ge kT/n\}\), as follows: for all \(k \in \mathbb {N}\) and \(p\ge 1\),

    $$\begin{aligned}&\Vert {X}_{t_{j+1}}^{t_j,x,y}-{\bar{X}}_{t_{j+1}}^{(T/n),x,y}\Vert _{k,p} \le \Vert {X}_{t_{j+1}}^{t_j,x,y}-{\hat{X}}_{t_{j+1}}^{(T/n),x,y}\Vert _{k,p}+\Vert {\hat{X}}_{t_{j+1}} ^{(T/n),x,y}-{\bar{X}}_{t_{j+1}}^{(T/n),x,y}\Vert _{k,p} \\&= O(n^{-1/2})+O(n^{-m+\frac{1}{2}})=O(n^{-1/2}) \end{aligned}$$

    where we used \(m\ge 2\), Lemma 5.1 of Bally and Talay [2] or Lemma 4.3 of Gobet and Munos [7] for the estimate \(\Vert {X}_{t_{j+1}}^{t_j,x,y}-{\hat{X}} _{t_{j+1}}^{(T/ n),x,y} \Vert _{k,p}=O(n^{-1/2})\), and for the estimate \(\Vert {\hat{X}}_{t_{j+1}}^{(T/n),x,y}-{\bar{X}} _{t_{j+1}}^{(T/n),x,y}\Vert _{k,p} =O(n^{-m+\frac{1}{2}})\), we used

    $$\begin{aligned} \Vert {\hat{X}}_{t_{j+1}}^{(T/n),x,y}-{\bar{X}}_{t_{j+1}}^{(T/n),x,y}\Vert _{k,p} \le C \Vert {\hat{X}}_{t_{j}}^{(T/n),x,y}-{\bar{X}}_{t_{j}}^{(T/n),x,y}\Vert _{k,p}+C n^{-m-\frac{1}{2}} \le C n^{-m+\frac{1}{2}} \ \ \ \end{aligned}$$

    with \(\Vert {X}_{t_{1}}^{(T/n),x,y}-{\hat{X}}_{t_{1}}^{(T/n),x,y}\Vert _{k,p}=0\), on which the estimate \(|P_{0,t_j}^y \varphi (y)-Q_{0,t_j}^{(m),y} \varphi (y)|= O(n^{-m})\) by Proposition 5.2 (with the Lipschitz continuity of \(V_i\), \(i=0,1, \ldots ,d\)) and the property of Brownian motion \(\Vert W_{t_j,t_{j+1}} \Vert _{k,p}=O(n^{-1/2}) \) are applied.

\(\square \)

Proof of Lemma 5.3

In the proof, we use a generic constant C which does not depend on f, xy and n whose value will change from line to line. For \(j=0,1,\ldots ,n-1\), it holds that

$$\begin{aligned}&Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,k}P^y_{t_{j+1},T}f(x) =E \Big [\mathcal {E}_{t_j,t_{j+1}}^{y,k}P_{t_{j+1},T}^yf( {\bar{X}}_{t_{j}}^{(T/n),x,y} ) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big ] \nonumber \\&\quad =\sum _{p \le \nu _k} E \Big [ \partial ^{\gamma ^p} P^y_{t_{j+1},T}f( {\bar{X}}_{t_{j+1}}^{(T/n),x,y} ) C^{y,k,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big ], \end{aligned}$$
(D.1)

where \(\prod _{i=1}^{0} \cdot \) is understood as \(\prod _{i=1}^{0}\cdot \equiv 1\) for the case \(j=0\). By Lemma 2.11 of Iguchi and Yamada [10], it holds that for \(r \in \{ 0 \} \cup \mathbb {N}\) and \(s \in [1,\infty )\),

$$\begin{aligned} \Big \Vert \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _{r,s} \le C, \end{aligned}$$
(D.2)

for all \(j\le n-1\). Then, for \(p=1,\ldots ,\nu _3\) and for \(r \in \{ 0 \} \cup \mathbb {N}\) and \(s \in [1,\infty )\),

$$\begin{aligned} \Big \Vert C^{y,3,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _{r,s} \le C\frac{1}{n^{m+1}}, \end{aligned}$$
(D.3)

for all \(j\le n-1\). Also, for \(p=1,\ldots ,\nu _4\) and for \(r \in \{ 0 \} \cup \mathbb {N}\) and \(s \in [1,\infty )\),

$$\begin{aligned} \Big \Vert C^{y,4,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _{r,s} \le C\frac{1}{n} | (P^y_{t_j}-Q^{(m),y}_{t_j}) \chi ^m_{\varphi ,V}(y)|, \end{aligned}$$
(D.4)

for all \(j\le n-1\), where the result of Proposition 5.1 is applied.

We now estimate the upper bound of (D.1).

  1. 1.

    (The case \(j=0,\ldots ,[n/2]-1\)) By applying (D.3) and (D.4) with the gradient estimate (5.11) in Proposition 5.1, we have

    $$\begin{aligned}&\Big |Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,3}P^y_{t_{j+1},T}f(x)\Big |\nonumber \\&\quad \le \sum _{p \le \nu _3} \Vert \nabla ^{|\gamma ^p|} P^y_{t_{j+1},T}f \Vert _\infty \Big \Vert C^{y,3,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _1 \\&\quad \le C \Vert f \Vert _{\infty } \frac{1}{n^{m+1}}, \end{aligned}$$

    and

    $$\begin{aligned}&\Big |Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,4}P^y_{t_{j+1},T}f(x) \Big |\nonumber \\&\quad \le \sum _{p \le \nu _4} \Vert \nabla ^{|\gamma ^p|} P^y_{t_{j+1},T}f \Vert _\infty \Big \Vert C^{y,4,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _1 \nonumber \\&\quad \le C \Vert f \Vert _{\infty }\frac{1}{n} | (P_{t_j}^y-Q^{(m),y}_{t_j}) \chi ^m_{\varphi ,V} (y)| \le C \Vert f \Vert _{\infty } \frac{1}{n^{m+1}}, \end{aligned}$$

    for all \(j\le n-1\).

  2. 2.

    (The case \(j=[n/2],\ldots ,n-1\)) We apply integration by parts and get

    $$\begin{aligned}&Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,k}P^y_{t_{j+1},T}f(y)\\&\quad =\sum _{p \le \nu _k} E \Big [ P^y_{t_{j+1},T}f( {\bar{X}}_{t_{j+1}}^{(T/n),x,y} ) H_{\gamma ^p} \Big ({\bar{X}}_{t_{j+1}}^{(T/n),y}, C^{y,k,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y})\\&\qquad \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big )\Big ]. \end{aligned}$$

    It holds that

    $$\begin{aligned}&\Big |E \Big [ P^y_{t_{j+1},T}f( {\bar{X}}_{t_{j+1}}^{(T/n),x,y} ) H_{\gamma ^p} \Big ({\bar{X}}_{t_{j+1}}^{(T/n),x,y}, C^{y,k,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/ n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y} _{t_{i-1},t_{i}} \Big ) \Big ] \Big |\\&\quad \le C \Vert P^y_{t_{j+1},T}f \Vert _{\infty } \Vert \det ( \sigma ^{{\bar{X}}_{t_{j+1}}^{(T/n),x,y}} ) \Vert ^{\kappa _1}_{q_1} \nonumber \\&\qquad \Vert D{\bar{X}}_{t_{j+1}}^{(T/n),x,y} \Vert ^{\kappa _2}_{r_1,q_2,H} \Big \Vert C^{y,k,p}_{t_j,t_{j+1}}({\bar{X}}_{t_{j}}^{(T/n),x,y}) \prod _{i=1}^{j} \vartheta ^{(m),{\bar{X}}_{t_{i-1}}^{(T/n),x,y},y}_{t_{i-1},t_{i}} \Big \Vert _{r_2,q_3}, \end{aligned}$$

    for some \(q_1,q_2,q_3,r_1,r_2,\kappa _1,\kappa _2>1\). By applying (D.3) and (D.4) with the estimate on the inverse of the determinant of the Malliavin covariance matrix (5.12) in Proposition 5.1 and the estimate \(\sup _j \Vert D{\bar{X}}_{t_{j+1}}^{(T/n),y} \Vert _{k,p,H}\le C\), we have

    $$\begin{aligned} |Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,3}P^y_{t_{j+1},T}f(y)|\le & {} C \Vert f \Vert _{\infty } \frac{1}{n^{m+1}} \end{aligned}$$

    and

    $$\begin{aligned} |Q^{(m),y}_{0,t_{j}}\mathcal {E}_{t_j,t_{j+1}}^{y,4}P^y_{t_{j+1},T}f(y)|\le & {} C \Vert f \Vert _{\infty }\frac{1}{n} | (P^y_{t_j}-Q^{(m),y}_{t_j}) \chi ^m_{\varphi ,V}(y)| \le C \Vert f \Vert _{\infty } \frac{1}{n^{m+1}}. \end{aligned}$$

Then we have the assertion. \(\square \)

Proof of Lemma 6.1

By Ito’s formula and Stratonovich Taylor expansion, one has

$$\begin{aligned}&X_{t_{i+1}}^{t_i,x,y} =x+\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}} [\psi ^y_{j_1}(X_{\cdot } ^{t_i,x,y},\cdot )]\\&\quad =x+\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}[\widetilde{\psi }^y_{j_1} (X_{\cdot }^{t_i,x,y},\cdot )] +\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}[(\psi ^y_{j_1}-\widetilde{\psi } ^y_{j_1})(X_{\cdot }^{t_i,x,y},\cdot )] \\&\quad =x+\sum _{j_1=0}^d\widetilde{\psi }^y_{j_1}(x,t_i)I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}} +\sum _{k=2}^r \sum _{j_1,\ldots ,j_k=0}^d\mathcal {V}_{t_i}^{j_1,y} \ldots \mathcal {V} _{t_i}^{j_{k-1},y} \widetilde{\psi }^y_{j_{k}}(x,t_i) I_{(j_1,\ldots ,j_k),t_i,t_{i+1}} ^{\mathrm {Strat}}\\&\qquad +\sum _{j_1,\ldots ,j_{r+1}=0}^d I_{(j_1,\ldots ,j_{r+1}),t_i,t_{i+1}}^{\mathrm {Strat}} [\mathcal {V}_{\cdot }^{j_1,y}\ldots \mathcal {V}_{\cdot }^{j_r,y}\widetilde{\psi }^y_{j_{r +1}}(X_{\cdot }^{t_i,x,y},\cdot )]\\&\qquad +\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}[(\psi ^y_{j_1}-\widetilde{\psi } ^y_{j_1})(X_{\cdot }^{t_i,x,y},\cdot )]. \end{aligned}$$

Furthermore,

$$\begin{aligned}&X_{t_{i+1}}^{t_i,x,y} =x+\sum _{j_1=0}^d\bar{\psi }^y_{j_1}(x,t_i) I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}+ \sum _{j_1=0}^d (\widetilde{\psi }^y_{j_1}-\bar{\psi }^y_{j_1})(x,t_i)I_{(j_1),t_i,t_{i +1}}^{\mathrm {Strat}} \\&\ \ +\sum _{k=2}^r \sum _{j_1,\ldots ,j_k=0}^d\bar{\mathcal {V}}_{t_i}^{j_1,y} \ldots \bar{\mathcal {V}}_{t_i}^{j_{k-1},y} \bar{\psi }^y_{j_{k}}(x,t_i)I_{(j_1, \ldots ,j_k),t_i,t_{i+1}}^{\mathrm {Strat}}\\&\ \ \ \ \ + \sum _{k=2}^r \sum _{j_1,\ldots ,j_k=0}^d(\mathcal {V}_{t_i}^{j_1,y} \ldots \mathcal {V}_{t_i}^{j_{k-1},y} \widetilde{\psi }^y_{j_{k}}-\bar{\mathcal {V}}_{t_i} ^{j_1,y} \ldots \bar{\mathcal {V}}_{t_i}^{j_{k-1},y} \bar{\psi }^y_{j_{k}}) (x,t_i)I_{(j_1,\ldots ,j_k),t_i,t_{i+1}}^{\mathrm {Strat}}\\&\ \ +\sum _{j_1,\ldots ,j_{r+1}=0}^d I_{(j_1,\ldots ,j_{r+1}),t_i,t_{i+1}} ^{\mathrm {Strat}}[\mathcal {V}_{\cdot }^{j_1,y} \ldots \mathcal {V}_{\cdot }^{j_r,y} \widetilde{\psi }^y_{j_{r+1}}(X_{\cdot }^{t_i,x,y},\cdot )]\nonumber \\&+\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}[(\psi ^y_{j_1}-\widetilde{\psi } ^y_{j_1})(X_{\cdot }^{t_i,x,y},\cdot )]. \end{aligned}$$

\(\square \)

Proof of Lemma 6.2

In the proof, we use a generic constant C which does not depend on t, \(t_i\) and y whose value will change from line to line. Using Itô formula iteratively, we obtain

$$\begin{aligned} \Big |P^y_t\varphi (y)-P^y_{t_i}\varphi (y)+\sum _{k=1}^{m-1}\frac{(t-t_i)^k}{k!}P^y_{t_i}(\mathcal{L}_{t_i}^y)^k\varphi (y) \Big | \le C (t-t_i)^{m}, \end{aligned}$$
(F.1)

where C depends on \(\varphi \), and

$$\begin{aligned} |(\psi ^{l,y}_{j_1}-\widetilde{\psi }^{l,y}_{j_1})(x,t)|\le C (t-t_i)^{m}, \end{aligned}$$
(F.2)

by the Lipschitz continuity of \(V_{j_1}^l\), \(j_1=0,1,\ldots ,d\), \(l=1,\ldots ,N\).

Let us prove (6.5). For the case \(k=1\), we obtain the assertion from the Lipschitz continuity of \(V_{j_1}^l\), \(j_1=0,1,\ldots ,d\), \(l=1,\ldots ,N\):

$$\begin{aligned} |(\widetilde{\psi }_{j_1}^{l,y}-\bar{\psi }_{j_1}^{l,y})(x,t_i)|= & {} |V_{j_1}^l(x,P^y_{t_i}\varphi (y))-V_{j_1}^l(x,Q^{(m),y}_{t_i}\varphi (y))| \nonumber \\\le & {} C|(P^y_{t_i}-Q^{(m),y}_{t_i})\varphi (y)|. \ \ \ \ \end{aligned}$$
(F.3)

We consider the case \(k=2\). Note that \((\mathcal {V}_{t_i}^{j_1,y}\widetilde{\psi }_{j_2}^y- \bar{\mathcal {V}}_{t_i}^{j_1,y}\bar{\psi }_{j_2}^y)(x,t_i) =(\mathcal {V}_{t_i}^{j_1,y}-\bar{\mathcal {V}}_{t_i}^{j_1,y})\widetilde{\psi }_{j_2}^y(x,t_i) +\bar{\mathcal {V}}_{t_i}^{j_1,y}(\widetilde{\psi }_{j_2}^y-\bar{\psi }_{j_2}^y)(x,t_i)\), \(j_1,j_2=0,1,\ldots ,d\). Since \(\mathcal {V}\) and \(\bar{\mathcal {V}}\) are given by (3.7), (3.8), (3.20) and (3.21), the first term above is estimated by

$$\begin{aligned} |(\mathcal {V}_{t_i}^{j_1,y}-\bar{\mathcal {V}}_{t_i}^{j_1,y})\widetilde{\psi }_{j_2}^{l_1,y} (x,t_i)|\le & {} \Big |\sum _{l_2=1}^N(\psi _{j_1}^{l_2,y}(x,t_i)-\bar{\psi }_{j_1}^{l_2,y}(x,t_i)) \partial _{l_2} \widetilde{\psi }_{j_2}^{l_1,y}(x,t_i) \Big |\nonumber \\\le & {} C|(P^y_{t_i}-Q^{(m),y}_{t_i})\varphi (y)| \ \ \ \ \ \ \ \ \end{aligned}$$
(F.4)

using (F.3) and the Lipschitz continuity of \(V_{j_1}^{l_1}\), \(l_1=1,\ldots ,N\). Also, we have that if \(j_1=1,\ldots ,d\),

$$\begin{aligned} |\bar{\mathcal {V}}_{t_i}^{j_1,y}(\widetilde{\psi }_{j_2}^{l_1,y}-\bar{\psi }_{j_2} ^{l_1,y})(x,t_i)|\le & {} \Big |\sum _{l_2=1}^{N}\bar{\psi }^{l_2,y}_{j_1}(x,t_i)\partial _{l_2} (\widetilde{\psi }_{j_2}^{l_1,y}-\bar{\psi }_{j_2}^{l_1,y})(x,t_i) \Big |\nonumber \\\le & {} C|(P^y_{t_i}-Q^{(m),y}_{t_i})\varphi (y)| \end{aligned}$$
(F.5)

and if \(j_1=0\),

$$\begin{aligned}&|\bar{\mathcal {V}}_{t_i}^{j_1,y}(\widetilde{\psi }^{l_1,y}_{j_2}-\bar{\psi }^{l_1,y} _{j_2})(x,t_i)|\nonumber \\&\le \Big |\sum _{l_2=1}^{N}\bar{\psi }^{l_2,y}_{j_1}(x,t_i)\partial _{l_2} (\widetilde{\psi }^{l_1,y}_{j_2}-\bar{\psi }_{j_2}^{l_1,y})(x,t_i)+ \partial _t(\widetilde{\psi }_{j_2}^{l_1,y}- \bar{\psi }_{j_2}^{l_1,y})(x,t_i) \Big | \nonumber \\&\le C|(P^y_{t_i}-Q^{(m),y}_{t_i})\chi _{\varphi ,V}(y)|, \end{aligned}$$
(F.6)

for \(l_1=1,\ldots ,N\) with a smooth bounded function \(\chi _{\varphi ,V}\) which has the form \(c_1\varphi +c_2\bar{\mathcal {L}}_{t_i}^y \varphi \) for some constants \(c_1\) and \(c_2\) (\( \bar{\mathcal {L}}_{t_i}^y\) depends on \(V_i\), \(i=0,1,\ldots ,d\)), where we used the bounded smoothness of \(V_i\), \(i=0,1,\ldots ,d\) and the property of linear operators \(P^y_{t_i}\) and \(Q^{(m),y}_{t_i}\). We proceed in the similar argument for \(k>2\) and get the estimate

$$\begin{aligned}&\sup _{x,l,j_1,\ldots ,j_k} |(\mathcal {V}^{j_k,y}_{t_i}\ldots \mathcal {V}^{j_2,y}_{t_i} \widetilde{\psi }^{l,y}_{j_1}-\bar{\mathcal {V}}^{j_k,y}_{t_i}\ldots \bar{\mathcal {V}} ^{j_2,y}_{t_i}\bar{\psi }^{l,y}_{j_1})(x,t_i)| \le C |(P_{t_i}^y-Q^{(m),y}_{t_i}) \chi ^k_{\varphi ,V}(y)|, \end{aligned}$$
(F.7)

where \(\chi ^k_{\varphi ,V}\) is a smooth bounded function which is given by a linear combination of \(\varphi \) and their derivatives \(\partial ^{\alpha }\varphi \) with the coefficients depending on \(V_i\), \(i=0,1,\ldots ,d\). \(\square \)

Proof of Lemma 6.3

By Taylor expansion,

$$\begin{aligned}&E[ \Phi ( X^{t_i,x,y}_{t_{i+1}} )] \nonumber \\&\quad =E[ \Phi ({\bar{X}}^{t_i,x,y}_{t_{i+1}})]+\sum _{\ell =1}^{2m+1} \frac{1}{\ell !} \sum _{|I|= \ell } E[ \Phi ({\bar{X}}^{t_i,x,y}_{t_{i+1}}) H_I ({\bar{X}}^{t_i,x,y}_{t_{i+1}}, \prod _{k=1}^{\ell } [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k} ) ] \nonumber \\&+\frac{1}{(2m+1)!}\int _0^1 (1-\lambda )^{2m+1} \sum _{|I|= 2m+2} \int _{\mathbb {R}^N} \Phi (\xi ) E[ \delta _\xi ({\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}}) \nonumber \\& \ H_I({\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}}, \prod _{k=1}^{2m+2} [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k} ) ] d\lambda +O(\Vert \Phi \Vert _{\infty } n^{-(m+1)} ), \end{aligned}$$
(G.1)

where \({\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}}=\lambda {X}^{t_i,x,y}_{t_{i+1}}+(1-\lambda ){\bar{X}}^{t_i,x,y}_{t_{i+1}}+n^{-\delta } {\bar{W}}_1\) with an sufficiently large real number \(\delta >0\) and an additional N-dimensional Brownian motion \(\{{\bar{W}}_t\}_t\) on a Wiener space (cf. [9]). Let \(\textstyle {\Xi _{2m+2,t_{i+1}}^{t_i,x,y}=\prod _{k=1}^{2m+2} [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k}}\). To estimate the error term with respect to the interval \((t_{i+1}-t_i)=T/n\), we introduce a stochastic matrix \(M_{t_{i+1}}^{\lambda ,t_i,x,y}\) given by \([M_{t_{i+1}}^{\lambda ,t_i,x,y}]_{j_1,j_2}= \langle (t_{i+1}-t_i)^{-1/2}D{\widehat{X}}^{\lambda ,t_i,x,y,j_1}_ {t_{i+1}},(t_{i+1}-t_i)^{-1/2}D{\widehat{X}}^{\lambda ,t_i,x,y,j_2}_ {t_{i+1}}\rangle _H\), \(1 \le j_1,j_2 \le N\). By using the property of the map \(\delta (\cdot )\) and the Sobolev norm \(\Vert \cdot \Vert _{k,p}\), the weight in the error term is estimated as follows: for \(k \in \mathbb {N}\), \(p>1\),

$$\begin{aligned}&\bigl \Vert H_I({\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}}, \prod _{k=1}^{2m+2} [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k} ) \bigr \Vert _{k,p} \le C (t_{i+1}-t_i)^{-|I|/2} \bigl \Vert \det (M_{t_{i+1}}^{\lambda ,t_i,x,y})^{-1} \bigr \Vert ^{r_1}_{q_1} \nonumber \\& \times \Vert (t_{i+1}-t_i)^{-1/2} D{\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}} \Vert ^{r_2}_{s,q_2,H} \Vert \Xi _{2m+2,t_{i+1}}^{t_i,x,y} \Vert _{s,q_3}, \end{aligned}$$

for some \(C>0\) and \(q_1,q_2,q_3,r_1,r_2,s >1\) which are independent of n. The factor \(\Vert (t_{i+1}-t_i)^{-1/2} D{\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}} \Vert _{k,p,H}\) is bounded as follows;

$$\begin{aligned} \sup _{\lambda \in [0,1],x,y \in \mathbb {R}^N, i \le n} \Vert (t_{i+1}-t_i)^{-1/2} D{\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}} \Vert _{k,p,H} \le C, \end{aligned}$$

for some \(C>0\) independent of n. Furthermore, \(\Vert \Xi _{2m+2,t_{i+1}}^{t_i,x,y} \Vert _{k,p}\) is estimated as

$$\begin{aligned}&\Vert \Xi _{2m+2,t_{i+1}}^{t_i,x,y} \Vert _{k,p} \\&\quad \le C n^{-(2m+2)} + C \sum _{\ell =0,1,\ldots ,2m+1} n^{-(\ell +m+1)/2} | (P^y_{t_i}- Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y) |^{2m+2-\ell }, \end{aligned}$$

for some \(C>0\) independent of n. We give the upper bound of \(\Vert \det (M_{t_{i+1}}^{\lambda ,t_i,x,y})^{-1} \bigr \Vert _p\). We decompose \(E[ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} ]\) into two parts as follows:

$$\begin{aligned}&E[ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} ]\nonumber \\&\quad = E \Big [ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} \mathbf{1}_{\{ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 | \le 1/2 \} } \Big ] \nonumber \\&\qquad + E \Big [ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} \mathbf{1}_{ \{ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 | > 1/2 \} } \Big ] . \end{aligned}$$
(G.2)

We note that it obviously holds that \(\sup _{x,y}E \Big [ |\det (M_{t_{i+1}}^{0,t_i,x,y})|^{-p} \Big ] \le C\) for some \(C>0\) independent of n. If \(|\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 | \le 1/2\), then \(\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})^{-1} \le 2 \det ({M_{t_{i+1}}^{0,t_i,x,y}})^{-1}\). Thus, we obtain

$$\begin{aligned}&E \Big [ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} \mathbf{1}_{\{ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 | \le 1/2 \} } \Big ]\nonumber \\&\quad \le 2^p E \Big [ |\det (M_{t_{i+1}}^{0,t_i,x,y})|^{-p} \Big ] \le C \end{aligned}$$

for some \(C>0\) uniformly in xy and n. Next, we give the upper bound of the second term of (G.2). Note that the followings hold: \(\textstyle {\sup _{x,y}} (t_{i+1}-t_i)^{-1/2}\Vert {\widehat{X}}^{\lambda ,t_i,x,y}_{t_{i+1}}-{\widehat{X}}^{0,t_i,x,y}_{t_{i+1}} \Vert _{1,p}=O(\lambda n^{-1/2})=O( n^{-1/2})\) and \(\textstyle {\sup _{x,y}} E[|\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} ] \le C n^{2p\delta N}\). Then we have

$$\begin{aligned}&E \Big [ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-p} \mathbf{1}_{ \{ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 |> 1/2 \} } \Big ] \nonumber \\&\quad \le E \Big [ |\det (M_{t_{i+1}}^{\lambda ,t_i,x,y})|^{-2p} \Big ]^{1/2} P( |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 | > 1/2 )^{1/2} \nonumber \\&\quad \le c_1 n^{2p\delta N} 2^{2r} E[ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )^{-1}\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -1 |^{2r} ] \nonumber \\&\quad \le c_1 n^{2p\delta N} 2^{2r} E[ |\det ( M_{t_{i+1}}^{\lambda ,t_i,x,y} ) -\det ( M_{t_{i+1}}^{0,t_i,x,y} ) |^{4r} ]^{1/2} E[ |\det ( M_{t_{i+1}}^{0,t_i,x,y} )|^{-4r} ]^{1/2} \nonumber \\&\quad \le c_2 n^{2p\delta N} 2^{2r} n^{-rN}= c_3 n^{(2p\delta -r)N}, \end{aligned}$$

for arbitrary \(r>1\), where \(c_1,c_2,c_3>0\) are constants independent of \(\lambda \), xy and n. It is enough to take r so that \(r=2p\delta \), and we have

$$\begin{aligned} \sup _{\lambda \in [0,1], x,y \in \mathbb {R}^N}\bigl \Vert \det (M_{t_{i+1}}^{\lambda ,t_i,x,y})^{-1} \bigr \Vert _p \le C, \end{aligned}$$

for some \(C>0\) independent of n. Therefore, we obtain

$$\begin{aligned}&\sup _{\lambda \in [0,1], x,y \in \mathbb {R}^N} \bigl \Vert H_I({\widehat{X}}^{\lambda ,t_i,x,y}_ {t_{i+1}}, \prod _{k=1}^{2m+2} [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k} ) \bigr \Vert _{k,p} \nonumber \\& \le C n^{-(m+1)} + C \sum _{\ell =0,1,\ldots ,2m+1} n^{-\ell /2} | (P^y_{t_i}-Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y) |^{2m+2-\ell }, \end{aligned}$$

for some \(C>0\) independent of n. Thus, it holds that

$$\begin{aligned}&\sup _{x,y \in \mathbb {R}^N} \Big | E[ \Phi ( X^{t_i,x,y}_{t_{i+1}} )]-\{E[ \Phi ({\bar{X}}^{t_i,x,y}_{t_{i+1}})]\nonumber \\&\qquad +\sum _{\ell =1}^{2m+1} \frac{1}{\ell !} \sum _{|I|= \ell } E[\Phi ({\bar{X}}^{t_i,x,y}_{t_{i+1}}) H_I({\bar{X}}^{t_i,x,y}_{t_{i+1}}, \prod _{k=1}^{\ell } [ X^{t_i,x,y}_{t_{i+1}}-{\bar{X}}^{t_i,x,y}_{t_{i+1}} ]^{I_k} ) ] \} \Big | \nonumber \\&\quad \le C \Vert \Phi \Vert _{\infty } n^{-(m+1)}+C \Vert \Phi \Vert _{\infty } \sum _{k=0,1,\ldots ,2m+1} n^{-k/2} | (P^y_{t_i}-Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y) |^{2m+2-k}. \end{aligned}$$

\(\square \)

Proof of Lemma 6.5

We use a generic constant \(C>0\) which does not depend on n. Using Lemma 6.4 and Lemma 6.2, we prove the assertion. We define \(G_1^{I_k},G_2^{I_k},G_3^{I_k},G_4^{I_k}\), \(I_k=1,\ldots ,N\), \(k=1,\ldots ,\ell \) as

$$\begin{aligned} G_{1}^{I_k}&=\sum _{ \Vert (j_1,\ldots ,j_{p}) \Vert \ge r(k), p\le r(k)-1} \bar{\mathcal {V}}_{t_i}^{j_1,y} \ldots \bar{\mathcal {V}}_{t_i}^{j_{p-1},y} \bar{\psi }^y_{j_{p}}(x,t_i)I_{(j_1,\ldots ,j_p),t_i,t_{i+1}}^{\mathrm {Strat}},\\ G_{2}^{I_k}&= \sum _{j_1,\ldots ,j_{r(k)}=0}^d I_{(j_1,\ldots ,j_{r(k)}),t_i,t_{i+1}}^{\mathrm {Strat}}[\mathcal {V}_{\cdot }^{j_1,y} \ldots \mathcal {V}_{\cdot }^{j_{r(k)-1},y}\widetilde{\psi }^{I_k,y}_{j_{r(k)}}(X_{\cdot }^{t_i,x,y},\cdot )], \\ G_{3}^{I_k}&=\sum _{p=1}^{r(k)-1} \sum _{j_1,\ldots ,j_p=0}^d(\mathcal {V}_{t_i}^{j_1,y} \ldots \mathcal {V}_{t_i}^{j_{p-1},y} \widetilde{\psi }^{I_k,y}_{j_{p}}-\bar{\mathcal {V}}_{t_i}^{j_1,y} \ldots \bar{\mathcal {V}}_{t_i}^{j_{p-1},y} \bar{\psi }^{I_k,y}_{j_{p}})(x,t_i)\\& \times I_{(j_1,\ldots ,j_p),t_i,t_{i+1}}^{\mathrm {Strat}} \end{aligned}$$

with \(\sum _{k=1}^{\ell } r(k)=2(m+1)+\ell \), and

$$\begin{aligned} G_{4}^{I_k}=\sum _{j_1=0}^d I_{(j_1),t_i,t_{i+1}}^{\mathrm {Strat}}[(\psi ^{I_k,y}_{j_1}-\widetilde{\psi }^{I_k,y}_{j_1})(X_{\cdot }^{t_i,x,y},\cdot )]. \end{aligned}$$

First, for the case \( | I |=\ell \in \{ 1,\ldots , 2m+1\}\), we estimate \(\textstyle {E[\partial ^{I} \Phi ( {\bar{X}}_{t_{i+1}}^{t_i,x,y} ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ]}\) whose upper bound depends on whether \(G_{3}^{I_k}\) is included or not, due to the result of Lemma 6.2. If \(\#\{ k ; \ \alpha _k=3 \}=0\), we have

$$\begin{aligned} \Big |E[\partial ^{I} \Phi ( {\bar{X}}_{t_{i+1}}^{t_i,x,y} ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ] \Big | \le C \Vert \Phi \Vert _{\infty } n^{-(m+1)}, \end{aligned}$$

by the estimate \(\textstyle {\Vert H_{I}({\bar{X}}_{t_{i+1}}^{t_i,x,y},\prod _{k=1}^{\ell } G_{\alpha _k}^{I_k}) \Vert _p \le C n^{-(m+1)}}\) since it holds that for all \(s \in \mathbb {N}\), \(p\ge 1\), \(\textstyle {\Vert \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} \Vert _{s,p}=O(n^{-(m+1+\ell )})}\). If \(\#\{ k ; \ \alpha _k=3 \}\ge 1\), we have

$$\begin{aligned} E[\partial ^{I} \Phi ( {\bar{X}}_{t_{i+1}}^{t_i,x,y} ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ]=\sum _{k\le e} C^k_{t_i,t_{i+1}}(x) E[\partial ^{\alpha (k)} \Phi ( {\bar{X}}_{t_{i+1}}^{t_i,x,y} )] \end{aligned}$$

by using Lemma 6.4, where for some \(e\in \mathbb {N}\) and multi-indices \(\alpha (k)\), \(k=1,\ldots ,e\), and \(C^{y,k}_{t_i,t_{i+1}}(x)\), \(k=1,\ldots ,e\), satisfy \( |C^{y,k}_{t_i,t_{i+1}}(x)| \le C n^{-1} | (P^y_{t_i} -Q^{(m),y}_{t_i})\chi (y)|\) where \(\chi \) is a smooth bounded function.

For the case \( | I |=\ell =2m+2\), we estimate \(\textstyle {E[\partial ^{I} \Phi ( {\widehat{X}}_{t_{i+1}}^{t_i,x,y,\lambda } ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ]}\) without using the derivatives of \(\Phi \) whose upper bound again depends on whether \(G_{3}^{I_k}\) is included or not. If \(\#\{ k ; \ \alpha _k=3 \}=0\), we have

$$\begin{aligned} \sup _{\lambda \in [0,1]} \Big |E[\partial ^{I} \Phi ( {\widehat{X}}_{t_{i+1}}^{t_i,x,y,\lambda } ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ] \Big |\le C \Vert \Phi \Vert _{\infty } n^{-(m+1)}, \end{aligned}$$

since for all \(s \in \mathbb {N}\), \(p\ge 1\), at least \(\Vert G_{\alpha _k}^{I_k} \Vert _{s,p}=O(n^{-1})\). If \(\#\{ k ; \ \alpha _k=3 \}=q \ge 1\), we have

$$\begin{aligned}&\sup _{\lambda \in [0,1]} \Big |E[\partial ^{I} \Phi ( {\widehat{X}}_{t_{i+1}}^{t_i,x,y,\lambda } ) \prod _{k=1}^{\ell } G_{\alpha _k}^{I_k} ] \Big |\nonumber \\&\quad \le C \Vert \Phi \Vert _{\infty } \left( \frac{1}{\sqrt{n}}\right) ^{-(2m+2)} \Big ( \left( \frac{1}{\sqrt{n}}\right) | (P_{t_i}^y-Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y)| \Big )^{q}\left( \frac{1}{n}\right) ^{2m+2-q}, \end{aligned}$$

by the integration by parts argument, since it holds that for all \(s \in \mathbb {N}\), \(p\ge 1\), \(\Vert G_{3}^{I_k} \Vert _{s,p}\le C n^{-1/2} | (P_{t_i}^y-Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y)| \) and at least \(\Vert G_{\alpha _k}^{I_k} \Vert _{s,p}=O(n^{-1})\) when \( \alpha _k\ne 3\).

Using these estimates with the formula in Lemma 3.1, we have the decomposition:

$$\begin{aligned} E[\Phi ( X_{t_{i+1}}^{t_i,x,y} )]=\mathcal{T}^y_{t_i,t_{i+1}} \Phi (x)+e^{1,y}_{t_i,t_{i+1}} \Phi (x)+e^{2,y}_{t_i,t_{i+1}} \Phi (x)+e^{3,y}_{t_i,t_{i+1}} \Phi (x) \end{aligned}$$

where

$$\begin{aligned}&\mathcal{T}^y_{t_i,t_{i+1}} \Phi (x)=E[\Phi ({\bar{X}}_{t_{i+1}}^{t_i,x,y})] +\sum _{j=1}^{2m+1} \sum _{k=1}^{j} \ \sum _{ \begin{array}{c} r_i \ge 2, \ i=1,\ldots ,k, \\ \sum _{i=1}^k r_i =j+k \end{array} } \sum _{I=(I_1,\ldots ,I_k) \in \{1,\ldots ,N \}^k} \frac{1}{k!} \nonumber \\&\quad \sum _{ \begin{array}{c} \alpha ^{e} \in \{0,1,\ldots ,d \}^{r_e}, \\ e=1,\ldots ,k \end{array} } \prod _{e=1}^k \bar{\mathcal {V}}_{t_i}^{\alpha ^e_1,y} \ldots \bar{\mathcal {V}}_{t_i}^{\alpha ^e_{r_e-1},y} \bar{\psi }^{I_k,y}_{\alpha ^e_{r_e}}(x,t_i) \nonumber \\&\quad \sum _{ \begin{array}{c} \beta =\sigma ^{-1} \cdot (\alpha ^1*\ldots *\alpha ^k) \sim _p \gamma , \ p\in \mathbb {N},\\ \sigma \in \mathrm {Shuffles}(r_1,\ldots ,r_k), \\ \Vert \gamma \Vert \le 2m+1+k \end{array} } \frac{1}{2^{\eta (\beta ,\gamma )}} E[ \partial ^I \Phi ({\bar{X}}_{t_{i+1}}^{t_i,x,y}) I_{\gamma ,t_i,t_{i+1}}^{\mathrm {It}{\hat{\mathrm{o}}}} ] \end{aligned}$$

and \(e^{p,y}_{t_i,t_{i+1}} \Phi (x)\), \(p=1,2,3\) satisfy \(\sup _{x,y} |e^{1,y}_{t_i,t_{i+1}} \Phi (x)| \le C \Vert \Phi \Vert _{\infty } n^{-(m+1)} \),

$$\begin{aligned} \sup _{x} |e^{2,y}_{t_i,t_{i+1}} \Phi (x)| \le C \Vert \Phi \Vert _{\infty } \sum _{k=0,1, \ldots ,2m+1} n^{-k/2} | (P_{t_i}^y-Q^{(m),y}_{t_i}) \chi ^m_{\varphi ,V}(y)|^{2m+2-k}. \end{aligned}$$

and \(e^{3,y}_{t_i,t_{i+1}} \Phi (x)=\sum _{k\le e} C^{y,k}_{t_i,t_{i+1}}(x) E[\partial ^{\alpha (k)} \Phi ( {\bar{X}}_{t_{i+1}}^{t_i,x,y} )]\) with \(\sup _x |C^{y,k}_{t_i,t_{i+1}}(x)| \le C \frac{1}{n} | (P_{t_i}^y-Q^{(m),y} _{t_i}) \chi ^m_{\varphi ,V}(y)|\). \(\square \)

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Naito, R., Yamada, T. A higher order weak approximation of McKean–Vlasov type SDEs. Bit Numer Math 62, 521–559 (2022). https://doi.org/10.1007/s10543-021-00880-1

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