Weak approximation of SDEs for tempered distributions and applications

Abstract

The paper shows a new weak approximation for generalized expectation of composition of a Schwartz tempered distribution and a solution to stochastic differential equation. Any order discretization is provided by using stochastic weights which do not depend on the Schwartz distribution. The error bound is obtained through stochastic analysis, which is consistent with the results of numerical experiments. It can also be confirmed that the proposed approximation gives high numerical accuracy.

Appendices

Appendix A. Proof of Proposition 1

In order to show the assertion, we first consider the expansion of $P_{t} f (x) = \langle f , {p^{X}_{t}} (x, \cdot ) \rangle$ around $P_{t}^{0,z} f (x) |_{z=x} = \langle f, p^{\bar {X}^{z}}_{t} (x, \cdot )\rangle |_{z=x}$based on the following perturbation formula:

$$ \begin{array}{@{}rcl@{}} P_{t} f(x) = P_{t}^{0,z} f(x) |_{z=x} + {{\int}_{0}^{t}} P_{t-s} (\mathscr{L} - \mathscr{L}_{0}^{z} ) P_{t}^{0,z} f (x) ds |_{z=x}. \end{array} $$

(A.1)

Furthermore, we introduce

$$ \begin{array}{@{}rcl@{}} P_{s}^{i,z} f(x) = \sum\limits_{k=0}^{i-1} {{\int}_{0}^{t}} P_{t-s}^{0,z} \mathscr{L}_{i-k}^{z} P_{s}^{k,z} f (x) ds, i =1,2, \ldots, 2 m + 2 \ell + 1, \end{array} $$

(A.2)

which approximates the target P_tf(x) and is naturally defined in the process of iterative expansion of the equation (A.1). In particular, the definition of $P_{s}^{i,z} f(x)$ is relies on the expansion of ${\mathscr{L}}$ around ${\mathscr{L}}_{0}^{z}$ as ${\mathscr{L}} - {\mathscr{L}}_{0}^{z} = \textstyle {{\sum }_{k=1}^{2m + 2\ell +1}} {\mathscr{L}}_{k}^{z} + \widetilde {{\mathscr{L}}}^{z}$ where $\widetilde {{\mathscr{L}}}^{z}$ is defined as

$$ \begin{array}{@{}rcl@{}} \widetilde{\mathscr{L}}^{z} \varphi (x)&=&\sum\limits_{\beta_{1},\ldots,\beta_{2m + 2\ell+2}=1}^{N} \prod\limits_{k=1}^{2m+ 2\ell + 2} (x_{\beta_{k}}-z_{\beta_{k}}) \left\{ \sum\limits_{r_{1}=1}^{N} h^{\beta_{1},\ldots,\beta_{2m+ 2\ell+ 2}}_{r_{1}}(x,z) \frac{\partial}{\partial x_{r_{1}} } \varphi (x)\right.\\ && \left.+ \sum\limits_{r_{1},r_{2}=1}^{N} h^{\beta_{1},\ldots,\beta_{2m+ 2\ell + 2}}_{r_{1},r_{2}}(x,z) \frac{\partial^{2}}{\partial x_{r_{1}} \partial x_{r_{2}}} \varphi (x) \right\}, \ \ \ \varphi \!\in\! C_{b}^{\infty}({\mathbb R}^{N}), \ x \in {\mathbb R}^{N}, \end{array} $$

for some bounded functions $h^{\beta _{1},\ldots ,\beta _{2m+2\ell +2}}_{r_{1},\ldots ,r_{k}}(\cdot ,z)$, β₁,…,β_{2m+ 2ℓ+ 2} = 1,⋯ ,N, k = 1, 2.

Then the strategy of the proof of assertion is as follows.

• Step 1. For $f \in \mathcal {S}(\mathbb {R}^{N})$ and integers m and ℓ, we define

  $$ \mathcal{E}_{1,t}^{m,\ell} f(x) := \langle f, {p^{X}_{t}} (x, \cdot) \rangle - \sum\limits_{i=0}^{2m + 2\ell +1 } P_{t}^{i,z} f(x), \ \ t >0, x \in \mathbb{R}^{N}. $$

  (A.3)

  We show that the term $\mathcal {E}_{1,t}^{m,\ell } f(x)$ is given in the form of $\mathcal {E}_{1,t}^{m,\ell } f(x) = \langle f , {e}_{1,t}^{m,\ell } (x, \cdot ) \rangle$ where ${e}_{1,t}^{m,\ell } : \mathbb {R}^{N} \times \mathbb {R}^{N} \to \mathbb {R}$ satisfies $| {e}_{1,t}^{m,\ell } (x,y )| \leq C t^{m+\ell +1} \frac {1}{t^{N/2}} e^{-c |y-x|^{2} / t }$ for some positive constants C and c.

• Step 2. We decompose the expansion formula $\textstyle {{\sum }_{i=0}^{2m +2 \ell +1 }} P_{t}^{i,z} f(x)$ into computation term $\langle f, \vartheta _{t}^{(m)}(x, \cdot ) \rangle$ and error term $\mathcal {E}_{2,t}^{m,\ell } f(x)$, i.e.

  $$ \begin{array}{@{}rcl@{}} \sum\limits_{i=0}^{2m + 2 \ell + 1} P_{t}^{i,z} f(x) = \langle f, \vartheta_{t}^{(m)} (x, \cdot) \rangle + \mathcal{E}_{2,t}^{m,\ell} f(x) \end{array} $$

  (A.4)

  where $\mathcal {E}^{m,\ell }_{2,t}f: \mathbb {R}^{N} \rightarrow \mathbb {R}$ has the form $\textstyle {\mathcal {E}^{m,\ell }_{2, t}f(x)= t^{m+1}{\sum }_{i=1}^{{j}(m,\ell )} C_{\alpha ^{i}}(t,x)}$ $\langle \partial ^{\alpha ^{i}} f, p^{\bar {X}}_{t}(x,\cdot ) \rangle$ with some integer ${j}(m,\ell ) \in \mathbb {N}$, multi-indices αⁱ, i = 1,⋯ ,j(m,ℓ) and bounded functions $C_{\alpha ^{i}}:[0,1] \times \mathbb {R}^{N} \rightarrow \mathbb {R}$.

From Step 1 and 2, we immediately have

$$ \langle f, {p^{X}_{t}} (x, \cdot) \rangle = \sum\limits_{i=0}^{2m + 2\ell +1 } P_{t}^{i,z} f(x) + \mathcal{E}_{1,t}^{m,\ell} f(x) = \langle f, \vartheta_{t}^{(m)} (x, \cdot) \rangle + \mathcal{E}_{1,t}^{m,\ell} f(x) + \mathcal{E}_{2,t}^{m,\ell} f(x). $$

(A.5)

1.1 A.1 Step 1: On the error term $\mathcal {E}_{1,t}^{m,\ell } f(x)$

According to (B.14) in [13], it follows that

$$ \begin{array}{@{}rcl@{}} && \mathcal{E}_{1,t}^{m,\ell} f(x) = \sum\limits_{k=0}^{2m + 2 \ell + 1} {{\int}_{0}^{t}} P_{t-s} \left( \mathscr{L} - \sum\limits_{i=0}^{2m + 2 \ell + 1 -k} \mathscr{L}_{i}^{z} \right)P_{s}^{k,z} f (x) ds |_{z=x} \\ & = & {{\int}_{0}^{t}} P_{t-s} \widetilde{\mathscr{L}^{z}} P_{s}^{0,z} f (x) ds |_{z=x} + \sum\limits_{i=1}^{2m + 2 \ell + 1 }{{\int}_{0}^{t}} P_{t-s} \left( \mathscr{L}^{z}_{2m + 2 \ell +1 } + {\cdots} + \mathscr{L}^{z}_{2m + 2 \ell - (i-1) } + \widetilde{\mathscr{L}}^{z} \right) P_{s}^{i,z} f (x) ds |_{z=x} . \end{array} $$

Hence, due to the definition of the semigroup {P_t}_t≥ 0 and the differential operators $\widetilde {{\mathscr{L}}}^{z}$ and ${\mathscr{L}}^{z}_{2m + 2 \ell + 1 -q }, q=0, \ldots , i$, $\mathcal {E}_{1,t}^{m,\ell } f(x)$ is given as the sum of the following terms: for i = 0, 1,…, 2m + 2ℓ + 1,

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}} E \left[ \partial^{\alpha} (P_{s}^{i,z} f) (X(t-s,x) ) g^{\alpha, q} (X(t-s, x)) \prod\limits_{j=1}^{2m + 2 \ell + 2 - q } (X^{\beta_{j}} (t-s,x) - z_{\beta_{j}} ) \right]ds |_{z=x}, \end{array} $$

(A.6)

where α ∈{1,…,N}^|α| is a multi-index with length |α| = 1, 2, q = 1,…,i, β_j ∈{1,…,N},j = 1,…, 2m + 2ℓ + 2 − q and the function g^α,q belongs to $C_{b}^{\infty } (\mathbb {R}^{N} )$. In particular, we assume q = 0 when i = 0. Here, applying the Malliavin integration by parts formula, we obtain

$$ \begin{array}{@{}rcl@{}} && E \left[ \partial^{\alpha} (P_{s}^{i,z} f) (X(t-s,x) ) g^{\alpha, q} (X(t-s, x)) \prod\limits_{j=1}^{2m + 2 \ell + 2 - q } (X^{\beta_{j}} (t-s,x) - z_{\beta_{j}} ) \right] |_{z=x} \\ &= & E \left[ (P_{s}^{i,z} f) (X(t-s,x) ) H_{\alpha} \left( X(t-s, x), g^{\alpha, q} (X(t-s, x)) \prod\limits_{j=1}^{2m + 2 \ell + 2 - q } (X^{\beta_{j}} (t-s,x) - z_{\beta_{j}} ) \right) \right] |_{z=x} . \end{array} $$

(A.7)

In particular, applying the lemma below to $(P_{s}^{i,z} f) (X(t-s,x) )$, we find that the above expectation is given as the sum of the following term:

$$ \begin{array}{@{}rcl@{}} &&\mathcal{M}_{f} (t,s,x) := E \left[s^{\iota} \psi(z) \prod\limits_{j=1}^{p} \left( X^{k_{j}} (t-s,x) - z_{k_{j}} \right)\right.\\ &&\quad\left.\times \partial^{\gamma} (P_{s}^{0,z} f ) (X(t-s,x)) H_{\alpha} \left( X(t-s, x), g^{\alpha, q} \left( X(t-s, x) \right) \! \! \prod\limits_{j=1}^{2m + 2 \ell + 2 - q } \! \! (X^{\beta_{j}} (t-s,x) - z_{\beta_{j}} ) \right) \right] |_{z=x} , \end{array} $$

(A.8)

where ι ≥ 1, p ≥ 0 and the multi-index γ ∈{1,…,N}^|γ| satisfies 2ι + p −|γ|≥ i with $\psi \in C_{b}^{\infty } (\mathbb {R}^{N} )$, k_j = 1,…,N, j = 1,…,p.

Lemma 1

For $i \in \mathbb {N}$, each term of $P_{s}^{i,z} f (y), s \in (0,1], y, z \in \mathbb {R}^{N}$ is given in the form

$$ \begin{array}{@{}rcl@{}} s^{\iota} \psi(z) \prod\limits_{j=1}^{p} \left( y_{k_{j}} - z_{k_{j}} \right) \partial^{\gamma} (P_{s}^{0,z} f ) (y ) \end{array} $$

(A.9)

where ι ≥ 1, p ≥ 0 and the multi-index γ ∈{1,…,N}^|γ| satisfy 2ι + p −|γ|≥ i with $\psi \in C_{b}^{\infty } (\mathbb {R}^{N} )$, k_j = 1,…,N, j = 1,…,p.

Proof Proof of Lemma 4

See the proof of Lemma B.1. in [13]. □

Using the Malliavin integration by parts formula, we have

$$ \begin{array}{@{}rcl@{}} && \mathcal{M}_{f} (t,s,x) \\ &= & s^{\iota} \psi (x) E [ (P_{s}^{0,z} f ) (X(t-s,x) ) H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right)] |_{z=x} \\ &= & s^{\iota} \psi (x) {\int}_{\mathbb{R}^{N}} (P_{s}^{0,z} f ) (y) {}_{\mathbb{D}^{-\infty}}\langle \delta_{y} \left( X(t-s,x) \right), H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right) \rangle {}_{\mathbb{D}^{\infty}} dy |_{z=x} \\ &= & s^{\iota} \psi (x) {\int}_{\mathbb{R}^{N}} \left( {\int}_{\mathbb{R}^{N}} f (\xi) p^{\bar{X}}_{s} (y , \xi )d \xi \right) {}_{\mathbb{D}^{-\infty}} \langle \delta_{y} \left( X(t-s,x) \right), H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right) \rangle {}_{\mathbb{D}^{\infty}}dy \\ &= & s^{\iota} \psi (x) {\int}_{\mathbb{R}^{N}} f (\xi) {\int}_{\mathbb{R}^{N}} p^{\bar{X}}_{s} (y , \xi ) {}_{\mathbb{D}^{-\infty}} \langle \delta_{y} \left( X(t-s,x) \right), H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right) \rangle {}_{\mathbb{D}^{\infty}} dy d \xi, \end{array} $$

(A.10)

where the random variable G^α,p,q(t − s,x) is defined as

$$ \begin{array}{@{}rcl@{}} G^{\alpha, p, q} (t - s,x) \!:=\! \prod\limits_{j=1}^{p} \left( X^{k_{j}} (t - s,x) - z_{k_{j}} \right) H_{\alpha} \left( X(t - s, x), g^{\alpha, q} \left( X(t - s, x) \right) \! \! \prod\limits_{j=1}^{2m + 2 \ell + 2 - q } \! \! (X^{\beta_{j}} (t - s,x) - z_{\beta_{j}} ) \right). \end{array} $$

We note that Lemma 2 yields

$$ \begin{array}{@{}rcl@{}} && \left| {\int}_{\mathbb{R}^{N}} p^{\bar{X}}_{s} (y , \xi ) {}_{\mathbb{D}^{-\infty}} \langle \delta_{y} \left( X(t-s,x) \right), H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right) \rangle {}_{\mathbb{D}^{\infty}} dy \right| \\ &\leq & C \frac{(t-s)^{\frac{2m + 2 \ell +2 -q + p - |\alpha| - |\gamma|}{2}} }{s^{N/2}} \frac{1}{ (t-s)^{N/2}} {\int}_{\mathbb{R}^{N}} \exp \left( -c_{1} \frac{| \xi - y |^{2}}{s} \right) \exp \left( -c_{2} \frac{| y - x |^{2}}{t-s} \right) dy \\ &\leq & C \frac{(t-s)^{\frac{2m + 2 \ell +2 -q + p - |\alpha| - |\gamma|}{2}} }{t^{N/2}} \exp \left( - c \frac{|\xi - x |^{2}}{t} \right) \end{array} $$

(A.11)

for some positive constants C and c, where we used the estimate $\textstyle { \left \| {\prod }_{j=1}^{\kappa } (X^{l_{j}} (t-s,x) - z_{l_{j}}) \right \|_{k,p} \leq C (t-s)^{\kappa /2}}$ for all $\kappa , k \in \mathbb {N}$ and p ≥ 2, and Lemma 3 on the last inequality. Therefore, each term of $\mathcal {E}_{1,t}^{m, \ell } f (x)$ is given in the form of

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}} \mathcal{M}_{f} (t,s,x) ds = \langle f , \varphi_{t}(x, \cdot) \rangle \end{array} $$

(A.12)

with $\varphi _{t} :\mathbb {R}^{N} \times \mathbb {R}^{N} \to \mathbb {R}$ satisfying for $x, \xi \in \mathbb {R}^{N}$,

$$ \begin{array}{@{}rcl@{}} | \varphi_{t} (x , \xi) | &= & \left| {{\int}_{0}^{t}} s^{\iota} \psi (x) {\int}_{\mathbb{R}^{N}} p^{\bar{X}}_{s} (y , \xi ) {}_{\mathbb{D}^{-\infty}} \langle \delta_{y} \left( X(t-s,x) \right), H_{\gamma} \left( X(t-s,x), G^{\alpha, p, q} (t-s,x) \right) \rangle {}_{\mathbb{D}^{\infty}} dy ds \right| \\ &\leq & C_{1} \left| {{\int}_{0}^{t}} s^{\iota} (t-s)^{\frac{2m + 2 \ell +2 -q + p - |\alpha| - |\gamma|}{2}} \frac{1}{t^{N/2}} \exp \left( - c \frac{|\xi - x |^{2}}{t} \right) ds \right| \\ &\leq & C_{2} t^{\frac{2m + 2 \ell +2 -q + p - |\alpha| - |\gamma|}{2} + \iota +1 } \frac{1}{t^{N/2}} \exp \left( - c \frac{|\xi - x |^{2}}{t} \right) \\ &\leq & C_{2} t^{m + \ell + 1} \frac{1}{t^{N/2}} \exp \left( - c \frac{|\xi - x |^{2}}{t} \right) \end{array} $$

(A.13)

for constants C₁,C₂ > 0 and c > 0 where we used |α|≤ 2, 2ι + p −|γ|≥ i and q ≤ i on the last inequality.

1.2 A.2 Step 2: On the error term $\mathcal {E}_{2,t}^{m,\ell } f(x)$

From the definition of $\{P_{t}^{i,z} \}_{t \geq 0}$, we have for t > 0, $x \in \mathbb {R}^{N}$ and $f \in \mathcal {S}(\mathbb {R}^{N} )$,

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{i=0}^{2m + 2 \ell + 1 } P_{t}^{i,z} f (x) |_{z=x} = P_{t}^{0,z} f (x) |_{z=x} \\ && + \sum\limits_{i=1}^{2m + 2 \ell + 1 } \sum\limits_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} {{\int}_{0}^{t}} {\int}_{t_{i}}^{t} {\cdots} {\int}_{t_{2}}^{t} P_{t_{i}}^{0,z} \mathscr{L}_{k_{1}}^{z} P_{t_{i-1} - t_{i}}^{0,z} \mathscr{L}_{k_{2}}^{z} {\cdots} \mathscr{L}_{k_{i}}^{z} P_{t-t_{1}}^{0,z} f (x) dt_{1} {\cdots} d t_{i} |_{z=x}. \end{array} $$

(A.14)

Then, we recursively apply the following Baker-Campbell-Hausdorff formula to the integrand of (A.14) so that we split the term into computation part and error part $\mathcal {E}_{2,t}^{m, \ell } f(x)$.

Lemma 2 (Baker-Campbell-Hausdorff formula)

Let 0 < s < t ≤ 1, $i \in \mathbb {N}$ and $\widehat {{\mathscr{L}}}_{i} \in \mathcal {DO}$ be a differential operator of the form $\widehat {{\mathscr{L}}}_{i} =c \psi _{i}(\cdot ) \partial ^{\beta }$ where c is a constant, ψ_i(⋅) is a polynomial of the degree at most i and ∂^β is a partial derivative with a multi-index β ∈{1,⋯ ,N}^ℓ, $\ell \in \mathbb {N}$. Then we have the explicit formula:

$$ \begin{array}{@{}rcl@{}} P_{s}^{0,z} \widehat{\mathscr{L}}_{i} P_{t-s}^{0,z}\varphi(\cdot) =\sum\limits_{k=0}^{i} \frac{s^{k}}{k!} {{\underbrace{[\mathscr{L}_{0}^{z}, [\cdots [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\widehat{\mathscr{L}}_{i}]]\cdots]]}_{k - \text{times} }}} P_{t}^{0,z}\varphi(\cdot), \end{array} $$

(A.15)

for any $\varphi \in \mathcal {S}(\mathbb {R}^{N})$, where ${{\underbrace { [{\mathscr{L}}_{0}^{z}, [{\cdots } [{\mathscr{L}}_{0}^{z},[{\mathscr{L}}_{0}^{z},\widehat {{\mathscr{L}}}_{i}]]\cdots ]]}_{0 - \text {times}}}} \equiv \widehat {{\mathscr{L}}_{i}}$.

Proof Proof of Lemma 5

See the proof of Proposition 2.1 in [13]. □

By Lemma 5, we have

$$ \begin{array}{@{}rcl@{}} && P_{t_{i}}^{0,z} \mathscr{L}_{k_{1}}^{z} P_{t_{i-1} - t_{i}}^{0,z} \mathscr{L}_{k_{2}}^{z} {\cdots} \mathscr{L}_{k_{i}}^{z} P_{t-t_{1}}^{0,z} f (x) \\ &= & \sum\limits_{\alpha = 0}^{k_{1}} \frac{(t_{i})^{\alpha}}{ \alpha !} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{1}}^{z}]]\cdots]]}_{\alpha - \text{times}}}} P_{t_{i-1}}^{0,z} \mathscr{L}_{k_{2}}^{z} P_{t_{i-2} - t_{i-1}}^{0,z} \mathscr{L}_{k_{3}}^{z} {\cdots} \mathscr{L}_{k_{i}}^{z} P_{t-t_{1}}^{0,z} f (x) \\ &= & \\ && {\vdots} \\ &=& \prod\limits_{l=1}^{i} \left( \sum\limits_{\alpha =0}^{k_{l}} \frac{(t_{i+1-l})^{\alpha} }{\alpha !} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{l}}^{z}]]\cdots]]}_{\alpha - \text{times}}}} \right) (P_{t}^{0,z} f )(x) \\ &= & \sum\limits_{\begin{array}{cc}0 {\leq} \alpha_{1} {\leq} k_{1} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \end{array}} \frac{(t_{1})^{\alpha_{1}} {\cdots} (t_{i})^{\alpha_{i}}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} \left( \prod\limits_{l=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [\cdots [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{l}}^{z}]]\cdots]]}_{\alpha_{l} - \text{times}}}} \right) (P_{t}^{0,z} f )(x). \end{array} $$

(A.16)

Thus, it follows that

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{i=0}^{2m + 2 \ell + 1 } P_{t}^{i,z} f (x) |_{z=x} = P_{t}^{0,z} f(x) |_{z=x} \\ && + \sum\limits_{i=1}^{2m + 2 \ell + 1 } \sum\limits_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\begin{array}{cc}0 {\leq} \alpha_{1} {\leq} k_{1} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ &= & P_{t}^{0,z} f(x) |_{z=x} \\ && + \sum\limits_{i=1}^{m -1 } {\sum}_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\substack{1 \leq \alpha_{1} \leq k_{1} \\ 0 \leq \alpha_{2} \leq k_{2} \\ {\cdots} \\ 0 \leq \alpha_{i} \leq k_{i}}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ && + \sum\limits_{i=m}^{2m + 2 \ell + 1 } {\sum}_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\begin{array}{cc}1 {\leq} \alpha_{1} {\leq}k_{1} \\ 0 \leq \alpha_{2} {\leq} k_{2} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ &=: & P_{t}^{0,z} f(x) |_{z=x} + B_{1} + B_{2}, \end{array} $$

(A.17)

where I(α) is defined by (??). Note that the above summation is taken for α₁ = 1,…,k₁ (does not include 0) since

$$ \begin{array}{@{}rcl@{}} {\left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [\cdots [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x}} = 0 \end{array} $$

(A.18)

whenever α₁ = 0. Furthermore, the term B₁ is decomposed as

$$ \begin{array}{@{}rcl@{}} B_{1}& = & \sum\limits_{i=1}^{m -1 } \sum\limits_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\begin{array}{cc}1 {\leq} \alpha_{1} {\leq} k_{1} \\ 0 {\leq} \alpha_{2} {\leq} k_{2} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \\ {\sum}_{j=1}^{i} \alpha_{j} {+} i \leq m \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! \cdots (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \end{array} $$

$$ \begin{array}{@{}rcl@{}} && \!+ \sum\limits_{i=1}^{m -1 } \sum\limits_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\begin{array}{cc}1 {\leq} \alpha_{1} {\leq} k_{1} \\ 0 {\leq} \alpha_{2} {\leq} k_{2} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \\ {\sum}_{j=1}^{i} \alpha_{j} {+} i > m \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ & = & {\sum}_{i=1}^{m -1 } {\sum}_{k_{1} + {\cdots} + k_{i} =i}^{2m + 1} \sum\limits_{\begin{array}{cc}1 {\leq} \alpha_{1} {\leq} k_{1} \\ 0 {\leq} \alpha_{2} {\leq} k_{2} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \\ {\sum}_{j=1}^{i} \alpha_{j} {+} i \leq m \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ & &\!+ \sum\limits_{i=1}^{m -1 } \sum\limits_{k_{1} + {\cdots} + k_{i} =i}^{2m + 2 \ell + 1} \sum\limits_{\begin{array}{cc}1 {\leq} \alpha_{1} {\leq} k_{1} \\ 0 {\leq} \alpha_{2} {\leq} k_{2} \\ {\cdots} \\ 0 {\leq} \alpha_{i} {\leq} k_{i} \\ {\sum}_{j=1}^{i} \alpha_{j} {+} i > m \end{array}} \frac{t^{{\sum}_{j=1}^{i} \alpha_{j} + i}}{(\alpha_{1})! {\cdots} (\alpha_{i})!} I (\alpha ) \left( \prod\limits_{j=1}^{i} {{\underbrace{ [\mathscr{L}_{0}^{z}, [{\cdots} [\mathscr{L}_{0}^{z},[\mathscr{L}_{0}^{z},\mathscr{L}_{k_{j}}^{z}]]\cdots]]}_{\alpha_{j} - \text{times}}}} \right) (P_{t}^{0,z} f )(x) |_{z=x} \\ &\!=:\! & B_{1,1} + B_{1,2}. \end{array} $$

(A.19)

Since $P_{t}^{0,z} f(x)$ is represented as

$$ \begin{array}{@{}rcl@{}} P_{t}^{0,z} f(x) = {\int}_{\mathbb{R}^{N}} f (y) {}_{\mathbb{D}^{-\infty}} \langle \delta_{y} (\bar{X}^{z} (t,x) ) , 1 \rangle {}_{\mathbb{D}^{\infty}} dy = \langle f, p^{\bar{X}^{z}}_{t} (x, \cdot) \rangle, \end{array} $$

(A.20)

we see that $P_{t}^{0,z} f(x) |_{z=x} + B_{1,1} = \langle f, {\vartheta ^{m}_{t}} (x, \cdot ) \rangle$ from the definition of the weight function $\vartheta _{t}^{(m)} (x, \cdot )$ in (??). Here, we also define $\mathcal {E}_{2,t}^{m, \ell } f(x) := B_{1,2} + B_{2}$. Thus,

$$ \begin{array}{@{}rcl@{}} \mathcal{E}_{2,t}^{m, \ell} f(x) &= & t^{m+1}\sum\limits_{i=1}^{{j}(m,\ell)} C_{\alpha^{i}}(t,x) \partial^{\alpha^{i}} (P_{t}^{0,z} f) (x) |_{z=x} \\ &= & t^{m+1}\sum\limits_{i=1}^{{j}(m,\ell)} C_{\alpha^{i}}(t,x) {\int}_{\mathbb{R}^{N}} f (y ) \partial^{\alpha^{i}} p_{t}^{\bar{X}^{z}} (\cdot, y) (x) dy |_{z=x} \\ &= & t^{m+1}\sum\limits_{i=1}^{{j}(m,\ell)} C_{\alpha^{i}}(t,x) {\int}_{\mathbb{R}^{N}} f (y ) \partial^{\alpha^{i}} p_{t}^{\bar{X}} (x, \cdot ) (y) dy \\ &= & t^{m+1}\sum\limits_{i=1}^{{j}(m,\ell)} C_{\alpha^{i}}(t,x) \langle \partial^{\alpha^{i}} f , p_{t}^{\bar{X}} (x, \cdot) \rangle, \end{array} $$

(A.21)

with some integer ${j}(m,\ell ) \in \mathbb {N}$, multi-indices αⁱ, i = 1,⋯ ,j(m,ℓ) and bounded functions $C_{\alpha ^{i}}:[0,1] \times \mathbb {R}^{N} \rightarrow \mathbb {R}$, where we used $\partial _{x_{i}} p^{\bar {X}^{z}} (x, y) = \partial _{y_{i}} p^{\bar {X}^{z}} (x, y), i=1,\ldots , N$ and integration by parts.

Appendix B. Proof of Lemma 2

2.1 B.1 Proof of (??)

For the simplicity, we denote by B_j− 1,j the Brownian increment B_jT/n − B_{(j− 1)T/n} and define

$$ \begin{array}{@{}rcl@{}} \pi_{T/n}^{m, j} &:= & \pi_{T/n}^{m, \bar{X}^{\text{EM}, (n), x}_{(j-1)T/n}} (B_{jT/n} - B_{(j-1)T/n} ), \end{array} $$

(B.1)

$$ \begin{array}{@{}rcl@{}} \mathscr{W}_{T/n}^{m, k} &:= & \prod\limits_{j=1}^{k} (1 + \pi_{T/n}^{m, \bar{X}^{\text{EM}, (n), x}_{(j-1)T/n}} (B_{jT/n} - B_{(j-1)T/n} ) ) \end{array} $$

(B.2)

for j,k = 1, 2,…,n. Then, we will show that for all $j \in \mathbb {N}$ and p > 1, there exists a constant C(T) such that

$$ \begin{array}{@{}rcl@{}} \| \mathscr{W}_{T/n}^{m, k} \|_{j, p}^{p} := E [| \mathscr{W}_{T/n}^{m, k} |^{p} ] + \sum\limits_{i=1}^{j} E [ \| D^{i} \mathscr{W}_{T/n}^{m, k} \|_{H^{\otimes i}}^{p} ] \leq & C(T), \end{array} $$

(B.3)

for k = 1,…,n where Dⁱ means the i-th order Malliavin derivative. Especially we consider the case p = 2e,e ≥ 1 and j = 1. According to [13], it holds that for all k = 1,…,n

$$ \begin{array}{@{}rcl@{}} E [| \mathscr{W}_{T/n}^{m, k} |^{p} ] \leq \left( 1 + c \frac{T}{n} \right)^{k} \end{array} $$

(B.4)

for some constant c > 0. Then, it suffices to show that

$$ \begin{array}{@{}rcl@{}} E [ \| D \mathscr{W}_{T/n}^{m, k} \|_{H}^{p} ] \leq & C(T). \end{array} $$

(B.5)

First, we have from the triangle inequality of L^p(Ω) norm

$$ \begin{array}{@{}rcl@{}} E [ \| D \mathscr{W}_{T/n}^{m, k} \|_{H}^{p} ] &= & E \left[ \left( \sum\limits_{j=1}^{d} {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, k} \right)^{2} dt \right)^{e} \right] \\ &\leq & \left( \sum\limits_{j=1}^{d} \left\| {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, k} \right)^{2} dt \right\|_{e} \right)^{e} \\ &\leq & T^{e-1} \left( \sum\limits_{j=1}^{d} \left\| \left( {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, k} \right)^{2e} dt \right)^{1/e} \right\|_{e} \right)^{e} \end{array} $$

(B.6)

where we applied the Hölder’s inequality to the time integral on the last inequality. In what follows, using the argument of induction with respect to k = 1,…,n, which is the number of product of Malliavin weights, we show that there exists a constant C > 0 such that for any j = 1,…,d and m ≥ 2,

$$ \begin{array}{@{}rcl@{}} E \left[ {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, k} \right)^{2e} dt \right] \leq \left( 1 + C \frac{T}{n} \right)^{k}. \end{array} $$

(B.7)

Throughout the proof, the following two results play an important role:

• 1. ${B_{t}^{1}}, \ldots , {B_{t}^{d}}$ are independent and for j = 1,…,d and t > 0

  $$ \begin{array}{@{}rcl@{}} E[({B_{t}^{j}})^{r} ] = \begin{cases} 0 & (r : \text{odd}) \\ \frac{r!}{2^{r/2}(r/2)!}t^{r/2} & (r : \text{even}). \end{cases} \end{array} $$

  (B.8)

• 2. The weight $\pi _{T/n}^{m, j}$ is represented as

  $$ \begin{array}{@{}rcl@{}} \pi_{T/n}^{m, j} &= & \sum\limits_{i_{1},i_{2},i_{3} = 1}^{d} V_{(i_{1}, i_{2}, i_{3})} (\bar{X}^{\text{EM}, (n), x}_{(j-1)T/n}) \left( \frac{T}{n} \right)^{-1} \\ &&\times \{ B_{j-1, j}^{i_{1}} B_{j-1, j}^{i_{2}} B_{j-1, j}^{i_{3}} - B_{j-1, j}^{i_{1}} \frac{T}{n} 1_{i_{2} = i_{3}} - B_{j-1, j}^{i_{2}} \frac{T}{n} 1_{i_{3} = i_{1}} \\ &&- B_{j-1,j}^{i_{3}} \frac{T}{n} 1_{i_{2} = i_{1}} \} + r_{T/n}^{m}(\bar{X}^{\text{EM}, (n), x}_{(j-1)T/n}, B_{j-1,j}) , \end{array} $$

  (B.9)

  where $V_{(i_{1}, i_{2}, i_{3})} \in C_{b}^{\infty } (\mathbb {R}^{N})$, the function $r_{T/n}^{m} : \mathbb {R}^{N} \times \mathbb {R}^{d} \to \mathbb {R}$ satisfies $r_{T/n}^{m} (\cdot , \xi ) \in C_{b}^{\infty } (\mathbb {R}^{N})$ for any $\xi \in \mathbb {R}^{d}$ and the random variable $\textstyle {r_{T/n}^{m}(\bar {X}^{\text {EM}, (n), x}_{(j-1)T/n}, B_{j-1,j} )}$ satisfies for any q ≥ 1, $\| r_{T/n}^{m} (\bar {X}^{\text {EM}, (n), x}_{(j-1)T/n}, B_{j-1,j} ) \|_{q} \leq c T/n$ for some c > 0.

Then, for k = 1 it is easy to see that there exists a positive constant C such that

$$ \begin{array}{@{}rcl@{}} E \left[ {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, 1} \right)^{2e} dt \right] = E \left[ {{\int}_{0}^{T}} \left( D_{j, t} \pi_{T/n}^{m, 1} \right)^{2e} dt \right] \leq 1 + C T/n. \end{array} $$

(B.10)

Now, we assume that for $k=1,2, \ldots , l, l \in \mathbb {N}$ the bound (B.7) holds. By the chain rule of Malliavin derivative, one has

$$ \begin{array}{@{}rcl@{}} E \left[ {{\int}_{0}^{T}} \left( D_{j, t} \mathscr{W}_{T/n}^{m, l+1} \right)^{2e} dt \right]& = & E \left[ {{\int}_{0}^{T}} \left( (1 + \pi_{T/n}^{m, l+1}) D_{j, t} \mathscr{W}_{T/n}^{m, l} + \mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \right] \\ &= & E \left[ {\int}_{0}^{l T/n} \left( (1 + \pi_{T/n}^{m, l+1}) D_{j, t} \mathscr{W}_{T/n}^{m, l} + \mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \right] \\ && + E \left[ {\int}_{lT/n}^{(l+1) T/n} \left( \mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \right] =:F_{1} + F_{2} \end{array} $$

(B.11)

where we used $D_{j, t} {\mathscr{W}}_{T/n}^{m, l} = 0$ for t ≥ lT/n on the second equality.

For the second term of (B.11), we have

$$ \begin{array}{@{}rcl@{}} F_{2} &=& E \left[ (\mathscr{W}_{T/n}^{m, l} )^{2e} {\int}_{lT/n}^{(l+1) T/n} \left( D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \right] \\ &=& E \left[ (\mathscr{W}_{T/n}^{m, l} )^{2e} E \left[ {\int}_{lT/n}^{(l+1) T/n} \left( D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \left. \right| \mathcal{F}_{lT/n} \right] \right] \leq C \frac{T}{n} E \left[ (\mathscr{W}_{T/n}^{m, l} )^{2e} \right] \leq C \frac{T}{n} \left( 1 + c \frac{T}{n} \right)^{l}\\ \end{array} $$

(B.12)

for some constants C > 0 and c > 0, where we used (B.4) and

$$ \begin{array}{@{}rcl@{}} E \left[ {\int}_{lT/n}^{(l+1) T/n} \left( D_{j, t} \pi_{T/n}^{m, l+1} \right)^{2e} dt \left. \right| \mathcal{F}_{lT/n} \right] \leq C \frac{T}{n}. \end{array} $$

(B.13)

Moreover, the binomial expansion of integrand of time integral in F₁ gives

$$ \begin{array}{@{}rcl@{}} F_{1} &= & E \left[ {\int}_{0}^{l T/n} \left\{ (1 + \pi_{T/n}^{m, l+1}) D_{j, t} \mathscr{W}_{T/n}^{m, l} \right\}^{2e} dt \right] \\ && + \sum\limits_{i=1}^{2e-1} \begin{pmatrix} 2e \\ i \end{pmatrix} E \left[ {\int}_{0}^{l T/n} \left\{ (1 + \pi_{T/n}^{m, l+1}) D_{j, t} \mathscr{W}_{T/n}^{m, l} \right\}^{2e - i} (\mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} )^{i} dt \right] \\ & &+ E \left[ {\int}_{0}^{l T/n} (\mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} )^{2e} dt \right] =: F_{1,1} + F_{1,2} + F_{1,3}. \end{array} $$

(B.14)

From now on, let us estimate the terms F_1,1,F_1,2 and F_1,3. The upper bound of F_1,1 and F_1,3 are given as follows.

$$ \begin{array}{@{}rcl@{}} F_{1,1} &= & E \left[ {\int}_{0}^{l T/n} (D_{j, t} \mathscr{W}_{T/n}^{m, l} )^{2e} dt E \left[ (1 + \pi_{T/n}^{m, l+1})^{2e} \left. \right| \mathcal{F}_{l T/n} \right] \right] \\ &\leq & \left( 1 + C_{1} \frac{T}{n} \right) E \left[ {\int}_{0}^{l T/n} (D_{j, t} \mathscr{W}_{T/n}^{m, l} )^{2e} dt \right] \leq \left( 1 + C_{1} \frac{T}{n} \right)^{l+1}, \end{array} $$

(B.15)

and

$$ \begin{array}{@{}rcl@{}} F_{1,3}& = & E \left[ \left( \mathscr{W}_{T/n}^{m, l} \right)^{2e} E \left[ {\int}_{0}^{l T/n} (D_{j, t} \pi_{T/n}^{m, l+1} )^{2e} dt \left. \right| \mathcal{F}_{lT/n} \right] \right] \\ &\leq & C_{2} \frac{T}{n} E[ \left( \mathscr{W}_{T/n}^{m, l} \right)^{2e} ] \leq \left( 1 + C_{2} \frac{T}{n} \right)^{l+1}, \end{array} $$

(B.16)

for some positive constants C₁ and C₂. Finally, let us consider the term F_1,2. We have

$$ \begin{array}{@{}rcl@{}} |F_{1,2} | &\leq & \sum\limits_{i=1}^{2e-1} \begin{pmatrix} 2e \\ i \end{pmatrix} \left| E \left[ {\int}_{0}^{l T/n} \left\{ (1 + \pi_{T/n}^{m, l+1}) D_{j, t} \mathscr{W}_{T/n}^{m, l} \right\}^{2e - i} (\mathscr{W}_{T/n}^{m, l} D_{j, t} \pi_{T/n}^{m, l+1} )^{i} dt \right] \right| \\ &\leq & \sum\limits_{i=1}^{2e-1} \begin{pmatrix} 2e \\ i \end{pmatrix} E \left[ \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{m, l} \right|^{2e-i} | E \left[ (1 + \pi_{T/n}^{m, l+1} )^{2e - i} (D_{j, t} \pi_{T/n}^{m, l+1} )^{i} \left. \right| \mathcal{F}_{lT/n} \right] | dt \right] \\ &\leq & C \left( \frac{T}{n} \right)\sum\limits_{i=1}^{2e-1} E \left[ \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{m, l} \right|^{2e-i} \left| G_{j,t}^{(l)} \right| dt \right], \end{array} $$

(B.17)

where $(G_{j,t}^{(l)})_{t \geq 0}$ is a stochastic process containing $D_{j,t} \bar {X}^{\text {EM}, (n), x, i}_{l T/n}, i=1,\ldots , N$, partial derivatives of $V_{(i_{1}, i_{2}, i_{3})}(\bar {X}^{\text {EM}, (n), x}_{lT/n} )$, i₁,i₂,i₃ = 1,…,d and $r_{T/n}^{m} (\bar {X}^{\text {EM}, (n), x}_{l T/n}, B_{l,l+1} )$. Here, applying Hölder’s inequality to the time integral in (B.17) with Hölder’s conjugates $p^{\prime }= 2e/(2e-i) \geq 1$ and $q^{\prime } = 2e/i \geq 1$, i = 1,…, 2e − 1, we obtain

$$ \begin{array}{@{}rcl@{}} && E \left[ \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{(m), l} \right|^{2e-i} \left| G_{j,t}^{(l)} \right| dt \right] \\ &\leq & E \left[ \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} \left( {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{m, l} \right|^{2e} dt \right)^{1/p^{\prime}} \left( {\int}_{0}^{lT/n} \left| G_{j,t}^{(l)} \right|^{q^{\prime}} dt \right)^{1/q^{\prime}} \right] \\ &\leq & \left\| \left( {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{m, l} \right|^{2e} dt \right)^{1/p^{\prime}} \right\|_{p^{\prime}} \left\| \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} \left( {\int}_{0}^{lT/n} \left| G_{j,t}^{(l)} \right|^{q^{\prime}} dt \right)^{1/q^{\prime}} \right\|_{q^{\prime}} \\ &\leq & \left\| \left( {\int}_{0}^{l T/n} \left| D_{j, t} \mathscr{W}_{T/n}^{m, l} \right|^{2e} dt \right)^{1/p^{\prime}} \right\|_{p^{\prime}} \left\| \left| \mathscr{W}_{T/n}^{m, l} \right|^{i} \right\|_{2q^{\prime}} \left\| \left( {\int}_{0}^{lT/n} \left| G_{j,t}^{(l)} \right|^{q^{\prime}} dt \right)^{1/q^{\prime}} \right\|_{2q^{\prime}} \\ &\leq & \left( 1 + C_{1} \frac{T}{n} \right)^{l/ p^{\prime}} \times \left( 1 + C_{2} \frac{T}{n} \right)^{l/(2q^{\prime})} \times C \\ &\leq & C \left( 1 + c \frac{T}{n} \right)^{l(1/p^{\prime} + 1/q^{\prime})} = C \left( 1 + c \frac{T}{n} \right)^{l},\ \end{array} $$

(B.18)

for some positive constants C₁,C₂,C and c where on the fourth inequality we used the assumption of induction (B.7), the bound (B.4) and

$$ \begin{array}{@{}rcl@{}} E \left[ \left( {\int}_{0}^{lT/n} \left| G_{j,t}^{(l)} \right|^{q^{\prime}} dt \right)^{2} \right] \leq C \end{array} $$

(B.19)

which comes from the boundedness of functions $V_{i_{1},i_{2}, i_{3}}$ and $r_{T/n}^{m}(\cdot , \xi )$ for any $\xi \in \mathbb {R}^{d}$, and for any $k \in \mathbb {N}$ and p ≥ 1, $\textstyle { \left \| D \bar {X}^{\text {EM}, (n), x, i}_{l T/n} \right \|_{k,p}} \leq c$ with some constant c > 0. Hence, we see that the bound (B.7) holds when k = l + 1. For the higher order Malliavin derivative Dⁱ,i ≥ 2, we are able to apply the above discussion in the same way and reach to the conclusion. $\Box$

2.2 B.2 Proof of (??)

Using Hölder’s inequality, we have

$$ \begin{array}{@{}rcl@{}} && \left| E[ H^{y} (\bar{X}_{s}^{\text{EM},(n),x}) H_{\alpha}(\bar{X}_{s}^{\text{EM},(n),x}, {\prod}_{i=1}^{k} (1+{\pi}_{T/n}^{m,\bar{X}_{(i-1)T/n}^{\text{EM},(n),x}}(B_{iT/n}-B_{(i-1)T/n})) ) \right| \\ &\!\leq\! & \left\| H^{y} (\bar{X}_{s}^{\text{EM},(n),x}) \right\|_{2} \left\| H_{\alpha}(\bar{X}_{s}^{\text{EM},(n),x}, {\prod}_{i=1}^{k} (1+{\pi}_{T/n}^{m,\bar{X}_{(i-1)T/n}^{\text{EM},(n),x}}(B_{iT/n}-B_{(i-1)T/n})) ) \right\|_{2} \\ &\!\leq\! & C (T) \left\| H^{y} (\bar{X}_{s}^{\text{EM},(n),x}) \right\|_{2} s^{- \frac{|\alpha|}{2}}, \end{array} $$

(B.20)

where we applied the bound (??), $\| \bar {X}_{s}^{\text {EM},(n),x} \|_{j,q}, j \in \mathbb {N}, q \geq 1$ and the Kusuoka-Stroock estimate [17] for the Malliavin covariance matrix. In order to complete the proof, let us show that

$$ \begin{array}{@{}rcl@{}} \left\| H^{y} (\bar{X}_{s}^{\text{EM},(n),x}) \right\|_{2} \leq C (T ) e^{-c | y -x |^{2} / s}. \end{array} $$

(B.21)

Using the upper bound of the density $p^{\bar {X}_{s}^{\text {EM},(n),x}}(x, \xi ), \xi \in \mathbb {R}^{N}$, we have

$$ \begin{array}{@{}rcl@{}} E \left[ \left| H^{y} (\bar{X}_{s}^{\text{EM},(n),x} ) \right|^{2} \right] = E \left[ \left| H^{y} (\bar{X}_{s}^{\text{EM},(n),x} ) \right| \right] &\leq & \frac{K(T)}{s^{N/2}} {\int}_{\mathbb{R}^{N}} H^{y} (\xi) e^{-c | \xi -x |^{2} /s } d \xi \\ &\leq & K(T) \prod\limits_{i=1}^{N} \frac{1}{\sqrt{s}} {\int}_{\mathbb{R}^{N}} H^{y_{i}} (\xi_{i}) e^{-c | \xi_{i} -x_{i} |^{2} /s } d \xi_{i} \end{array} $$

(B.22)

for some non-decreasing function K(⋅). For the case $y_{i} -x_{i} \leq \sqrt {s}$, we have

$$ \frac{1}{\sqrt{s}} {\int}_{\mathbb{R}^{N}} H^{y_{i}} (\xi_{i}) e^{-c | \xi_{i} -x_{i} |^{2} /s } d \xi_{i} \leq C = C e^{- |y_{i} -x_{i} |^{2}/s} e^{|y_{i} -x_{i} |^{2}/s} \leq C e^{- |y_{i} -x_{i} |^{2}/s}, $$

(B.23)

for some positive constant C. Though we are not able to apply the above discussion in the case $y_{i} -x_{i} \geq \sqrt {s}$, we still have

$$ \begin{array}{@{}rcl@{}} \frac{1}{\sqrt{s}} {\int}_{\mathbb{R}^{N}} H^{y_{i}} (\xi_{i}) e^{-c | \xi_{i} -x_{i} |^{2} /s } d \xi_{i} &= & \frac{1}{\sqrt{s}} {\int}_{y_{i} - x_{i}}^{\infty} e^{-c z^{2} /s } d z \\ &\leq & \frac{1}{\sqrt{s}} {\int}_{y_{i} - x_{i}}^{\infty} \frac{z}{y_{i} - x_{i}} e^{-c z^{2} /s } d z \leq C e^{- c |y_{i} -x_{i} |^{2}/s} \end{array} $$

(B.24)

for some constant C > 0. Hence, the inequality (B.21) holds. $\Box$

Appendix C. Proof of Lemma 1

Substituting m = 2 into (??), we have

$$ \begin{array}{@{}rcl@{}} E[ \delta_{y}(\bar{X}(t,x) ) (1+ \pi_{t}^{2,x}(B_{t}))] &= & E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] |_{z=x} + \sum\limits_{i=1}^{5} \frac{t^{2}}{2} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] |_{z=x} \\ &= & E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] |_{z=x} + \sum\limits_{i=1}^{2} \frac{t^{2}}{2} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] |_{z=x}, \end{array} $$

since $[ {\mathscr{L}}_{0}^{z}, {\mathscr{L}}_{i}^{z}] \varphi (x), \varphi \in C_{b}^{\infty }(\mathbb {R}^{N})$ equals to zero for i ≥ 3 when we substitute z = x. We have

$$ \begin{array}{@{}rcl@{}} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{1}^{z}] \Big|_{z=x} &= & \sum\limits_{j_{1}, j_{2} =1}^{N} b^{j_{2}} (x) \partial_{j_{2}} b^{j_{1}} (x) \frac{ \partial }{ \partial x_{j_{1}}} + \sum\limits_{i_{1}=1}^{d} \sum\limits_{j_{1} ,j_{2}, j_{3} =1}^{N} b^{j_{3}} (x) \partial_{j_{3}} \sigma_{i_{1}}^{j_{1}}(x) \sigma_{i_{1}}^{j_{2}}(x) \frac{ \partial^{2} }{ \partial x_{j_{1}} \partial x_{j_{2}}} \\ &&+ \sum\limits_{i_{1}=1}^{d} \sum\limits_{j_{1},j_{2},j_{3} =1}^{N} \sigma_{i_{1}}^{j_{2}} (x) \sigma_{i_{1}}^{j_{3}} (x) \partial_{j_{3}} b^{j_{1}} (x) \frac{ \partial^{2} }{ \partial x_{j_{1}} \partial x_{j_{2}}} \\ && + \sum\limits_{i_{1}, i_{2}=1}^{d} \sum\limits_{j_{1},j_{2}, j_{3}, j_{4} =1}^{N} \sigma_{i_{1}}^{j_{3}}(x) \sigma_{i_{1}}^{j_{4}} (x) \partial_{j_{4}} \sigma_{i_{2}}^{j_{1}}(x) \sigma_{i_{2}}^{j_{2}}(x) \frac{ \partial^{3} }{ \partial x_{j_{1}} \partial x_{j_{2}} \partial x_{j_{3}} }, \end{array} $$

(C.1)

and

$$ \begin{array}{@{}rcl@{}} && [ \mathscr{L}_{0}^{z}, \mathscr{L}_{2}^{z}] \Big|_{z=x} = \frac{1}{2} \sum\limits_{i_{1}=1}^{d} \sum\limits_{j_{1}, j_{2}, j_{3} =1}^{N} \sigma_{i_{1}}^{j_{1}}(x) \sigma_{i_{1}}^{j_{2}}(x) \partial_{j_{1}} \partial_{j_{2}} b^{j_{3}} (x) \frac{ \partial }{ \partial x_{j_{3}}} \\ &&+ \frac{1}{2} \sum\limits_{i_{1}, i_{2}=1}^{d} \sum\limits_{j_{1}, j_{2}, j_{3},j_{4} = 1}^{N} \sigma_{i_{1}}^{j_{1}}(x) \sigma_{i_{1}}^{j_{2}}(x) \{ \partial_{j_{1}} \partial_{j_{2}} \sigma_{i_{2}}^{j_{3}} (x) \sigma_{i_{2}}^{j_{4}}(x) + \partial_{j_{1}} \sigma_{i_{2}}^{j_{3}}(x) \partial_{j_{2}} \sigma_{i_{2}}^{j_{4}}(x) \} \frac{ \partial^{2} } {\partial x_{j_{3}} \partial x_{j_{4}} }. \end{array} $$

(C.2)

Hence, we obtain

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{2} \frac{t^{2}}{2} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} & = & \frac{t^{2}}{2} \sum\limits_{j_{1}=1}^{N} \mathscr{L} b^{j_{1}} (x) E [ \partial_{j_{1}} \delta_{y} (\bar{X}(t,x) ) ] \\ &&+ \frac{t^{2}}{2} \sum\limits_{i_{1} =1}^{d} \sum\limits_{j_{1}, j_{2} =1}^{N} \left\{ \mathcal{L} \sigma_{i_{1}}^{j_{1}}(x) + \mathcal{V}_{i_{1}} b^{j_{1}}(x) \right\} \sigma_{i_{1}}^{j_{2}}(x) E [ \partial_{j_{1}} \partial_{j_{2}} \delta_{y} (\bar{X} (t,x) )] \\ &&+ \frac{t^{2}}{2} \sum\limits_{i_{1}, i_{2} =1}^{d} \sum\limits_{j_{1}, j_{2}, j_{3}=1}^{N} \mathcal{V}_{i_{1}} \sigma_{i_{2}}^{j_{1}} (x) \sigma_{i_{1}}^{j_{2}}(x) \sigma_{i_{2}}^{j_{3}}(x) E [ \partial_{j_{1}} \partial_{j_{2}} \partial_{j_{3}} \delta_{y} (\bar{X} (t,x) )] \\ &&+ \frac{t^{2}}{4} \sum\limits_{i_{1}, i_{2} =1}^{d} \sum\limits_{j_{1}, j_{2} =1}^{N} \mathcal{V}_{i_{1}} \sigma_{i_{2}}^{j_{1}} (x) \mathcal{V}_{i_{1}} \sigma_{i_{2}}^{j_{2}} (x) E [ \partial_{j_{1}} \partial_{j_{2}} \delta_{y} (\bar{X} (t,x) )]. \end{array} $$

(C.3)

Furthermore, it follows that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j_{2} =1}^{N} \sigma_{i_{1}}^{j_{2}}(x) E [ \partial_{j_{1}} \partial_{j_{2}} \delta_{y} (\bar{X} (t,x) )] & = & E \left[ \frac{1}{t} {\int}_{0}^{\infty} D_{i_{1},s} \partial_{j_{1}} \delta_{y} (\bar{X} (t,x) ) \mathbf{1}_{s \leq t} ds \right] \\ & = & \frac{1}{t} E \left[ \partial_{j_{1}} \delta_{y} (\bar{X} (t,x) ) {\int}_{0}^{\infty} \mathbf{1}_{s \leq t} dB_{s}^{i_{1}} \right] \\ & = & \frac{1}{t} E \left[ \delta_{y} (\bar{X} (t,x) ) H_{(j_{1})} (\bar{X}(t,x), B_{t}^{i_{1}}) \right], \end{array} $$

(C.4)

where we used the duality formula (??) on the second equality. Similarly, we have

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j_{2}, j_{3}=1}^{N} \sigma_{i_{1}}^{j_{2}}(x) \sigma_{i_{2}}^{j_{3}}(x) E [ \partial_{j_{1}} \partial_{j_{2}} \partial_{j_{3}} \delta_{y} (\bar{X} (t,x) )] &= & \frac{1}{t} \sum\limits_{j_{2} =1 }^{N} \sigma_{i_{1}}^{j_{2}}(x) E [ \partial_{j_{1}} \partial_{j_{2}} \delta_{y} (\bar{X} (t,x) ) B_{t}^{i_{2}} ] \\ &= & \frac{1}{t^{2}} E [ {\int}_{0}^{\infty} D_{i_{1},s} \partial_{j_{1}} \delta_{y} (\bar{X} (t,x) ) B_{t}^{i_{2}} \mathbf{1}_{s \leq t} ds ] \\ &= & \frac{1}{t^{2}} E [ \partial_{j_{1}} \delta_{y} (\bar{X} (t,x) ) (B_{t}^{i_{1}} B_{t}^{i_{2}} - t \mathbf{1}_{i_{1} = i_{2}})] \\ &= & \frac{1}{t^{2}} E [ \delta_{y} (\bar{X} (t,x) ) H_{(j_{1})} (\bar{X}(t,x), B_{t}^{i_{1}} B_{t}^{i_{2}} - t \mathbf{1}_{i_{1} = i_{2}})], \end{array} $$

(C.5)

where on the third equality we used the formula (??). Then, we reach to the assertion of Lemma 1. $\Box$

Appendix D. The stochastic weights used in the numerical study

4.1 D.1 Second and third order weight used in Section ??

Based on the second order stochastic weight formula (??), the second order weight $\pi _{t}^{2,x, \text {BS}}(B_{t})$ is given by

$$ \begin{array}{@{}rcl@{}} \pi_{t}^{2, x, \text{BS} } (B_{t}) = \frac{\sigma^{2}}{4}\left( (B_{t})^{2} - t \right) + \frac{\sigma}{2 t} \left( (B_{t})^{3} - 3 B_{t} t \right) \end{array} $$

(D.1)

for $(t,x) \in (0, \infty ) \times \mathbb {R}$.

From now on, let us see the representation of the third order stochastic weight $\pi _{t}^{3, x, \text {BS} } (B_{t})$. Firstly, substituting m = 3 into (??), we have

$$ \begin{array}{@{}rcl@{}} && E[ \delta_{y}(\bar{X}(t,x) ) (1+ \pi_{t}^{3,x}(B_{t}))] = E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \sum\limits_{i=1}^{2} \frac{t^{2}}{2} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} \\ && + \frac{t^{3}}{6} \sum\limits_{i=2}^{4} [\mathscr{L}_{0}^{z}, [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}]] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \frac{t^{3}}{6} \sum\limits_{i_{1}=1}^{2} \sum\limits_{i_{1} + i_{2} \leq 4} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i_{1}}^{z}] \mathscr{L}_{i_{2}}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x}. \end{array} $$

(D.2)

Note that the second term of (D.2) corresponds to the second order weight $\pi _{t}^{2,x,\text {BS}}$. Furthermore, since we have

$$ \begin{array}{@{}rcl@{}} \mathscr{L}_{0}^{z} = & \frac{\sigma^{2} z^{2}}{2} \frac{\partial^{2}}{ (\partial x)^{2}}, \ \ \mathscr{L}_{1}^{z} = \sigma^{2} z (x - z) \frac{\partial^{2}}{ (\partial x)^{2}}, \ \ \mathscr{L}_{2}^{z} = \frac{1}{2} \sigma^{2} (x - z)^{2} \frac{\partial^{2}}{ (\partial x)^{2}}, \ \ \mathscr{L}_{i}^{z} = 0, i \geq 3, \end{array} $$

it holds that

$$ \begin{array}{@{}rcl@{}} && \frac{t^{3}}{6} \sum\limits_{i=2}^{4} [\mathscr{L}_{0}^{z}, [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i}^{z}]] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \frac{t^{3}}{6} \sum\limits_{i_{1}=1}^{2} \sum\limits_{i_{1} + i_{2} \leq 4} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{i_{1}}^{z}] \mathscr{L}_{i_{2}}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} \\ &= & \frac{t^{3}}{6} [\mathscr{L}_{0}^{z}, [ \mathscr{L}_{0}^{z}, \mathscr{L}_{2}^{z}]] E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \frac{t^{3}}{6} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{1}^{z}] \mathscr{L}_{1}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x}\\ &&+ \frac{t^{3}}{6} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{1}^{z}] \mathscr{L}_{2}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} \\ && + \frac{t^{3}}{6} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{2}^{z}] \mathscr{L}_{1}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \frac{t^{3}}{6} [ \mathscr{L}_{0}^{z}, \mathscr{L}_{2}^{z}] \mathscr{L}_{2}^{z} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} \\ &= & \frac{2}{3} t^{3} \sigma^{6} x^{4} \frac{\partial^{4}}{ (\partial x)^{4}} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x} + \frac{2}{3} t^{3} \sigma^{6} x^{3} \frac{\partial^{3}}{ (\partial x)^{3}} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x}\\ &&+ \frac{1}{12} t^{3} \sigma^{6} x^{2} \frac{\partial^{2}}{ (\partial x)^{2}} E[ \delta_{y}(\bar{X}^{z}(t,x) ) ] \Big|_{z=x}. \end{array} $$

Hence, we have the third order stochastic weight as follows:

$$ \begin{array}{@{}rcl@{}} \pi_{t}^{3, x, \text{BS}} (B_{t})&=& \pi_{t}^{2, x, \text{BS}} (B_{t}) + \frac{2}{3} t^{3} \sigma^{6} x^{4} H_{(1,1,1,1)}(\bar{X}(t,x), 1)\\ &&+ \frac{2}{3} t^{3} \sigma^{6} x^{3} H_{(1,1,1)}(\bar{X}(t,x), 1) + \frac{1}{12}t^{3} \sigma^{6} x^{2} H_{(1,1)}(\bar{X}(t,x), 1)\\ &=& \pi_{t}^{2, x, \text{BS}} (B_{t}) + \frac{2 \sigma^{2}}{3 t} \left( (B_{t})^{4} - 6 (B_{t})^{2} t + 3 t^{2} \right)\\ &&+ \frac{2 \sigma^{3}}{3} \left( (B_{t})^{3} - 3 B_{t} t \right) + \frac{ \sigma^{4} t }{12} \left( (B_{t})^{2} - t \right). \end{array} $$

(D.3)

4.2 D.2 Second order weight used in Section ??

We set d = 2, N = 2 and the coefficients of the SDE are given by

$$ b^{1} (x) = b^{2} (x) = 0, {\sigma_{1}^{1}}(x) = (x_{1})^{\beta} x_{2}, {\sigma_{2}^{1}}(x) = 0, {\sigma_{1}^{2}}(x) = \alpha \rho x_{2}, {\sigma_{2}^{2}}(x) = \alpha \sqrt{1- \rho^{2}} x_{2}, $$

(D.4)

for $x \in \mathbb {R}^{2}$. Then, from the formula (??), the second order weight for stochastic volatility model $\pi _{t}^{2, x, \text {SV}}(B_{t})$ is given by

$$ \begin{array}{@{}rcl@{}} && \pi_{t}^{2, x, \text{SV}}(B_{t}) \\ &= & \frac{1}{2} \mathscr{V}_{1} {\sigma_{1}^{1}}(x) H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}}{B_{t}^{1}} - t \right) + \frac{1}{2} \mathscr{V}_{1} {\sigma_{1}^{2}}(x) H_{(2)} \left( \bar{X}(t,x), {B_{t}^{1}}{B_{t}^{1}} - t \right)\\ &&+ \frac{1}{2} \mathscr{V}_{1} {\sigma_{2}^{2}}(x) H_{(2)} \left( \bar{X}(t,x), {B_{t}^{1}}{B_{t}^{2}} \right) \\ && + \frac{1}{2} \mathscr{V}_{2} {\sigma_{1}^{1}}(x) H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}}{B_{t}^{2}} \right) + \frac{1}{2} \mathscr{V}_{2} {\sigma_{1}^{2}} (x) H_{(2)} \left( \bar{X}(t,x), {B_{t}^{1}}{B_{t}^{2}} \right)\\ &&+ \frac{1}{2} \mathscr{V}_{2} {\sigma_{2}^{2}} (x) H_{(2)} \left( \bar{X}(t,x), {B_{t}^{2}}{B_{t}^{2}} - t \right) \end{array} $$

$$ \begin{array}{@{}rcl@{}} && + \frac{t}{2} \mathscr{L} {\sigma_{1}^{1}}(x) H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}}\right) + \frac{t^{2}}{4} \left( \mathscr{V}_{1} {\sigma_{1}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right)\\ &&+ \frac{t^{2}}{2} \mathscr{V}_{1} {\sigma_{1}^{1}}(x) \mathscr{V}_{1} {\sigma_{1}^{2}}(x) H_{(1,2)} \left( \bar{X}(t,x),1 \right) \\ && + \frac{t^{2}}{4} \left\{ \left( \mathscr{V}_{1} {\sigma_{1}^{2}}(x) \right)^{2} + \left( \mathscr{V}_{1} {\sigma_{2}^{2}}(x) \right)^{2} \right\} H_{(2,2)} \left( \bar{X}(t,x),1 \right)\\ &&+ \frac{t^{2}}{4} \left( \mathscr{V}_{2} {\sigma_{1}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right) \\ && + \frac{t^{2}}{2} \mathscr{V}_{2} {\sigma_{1}^{1}}(x) \mathscr{V}_{2} {\sigma_{1}^{2}}(x) H_{(1,2)} \left( \bar{X}(t,x),1 \right)\\ &&+ \frac{t^{2}}{4} \left\{ \left( \mathscr{V}_{2} {\sigma_{1}^{2}}(x) \right)^{2} + \left( \mathscr{V}_{2} {\sigma_{2}^{2}}(x) \right)^{2} \right\} H_{(2,2)} \left( \bar{X}(t,x),1 \right). \end{array} $$

(D.5)

The inverse of Malliaivin covariance $\gamma ^{\bar {X}(t,x)}$ appearing in H_α is given as follows.

$$ \begin{array}{@{}rcl@{}} \gamma^{\bar{X}(t,x)} = \begin{pmatrix} \frac{\left( {\sigma_{1}^{2}}(x)\right)^{2} + \left( {\sigma_{2}^{2}}(x)\right)^{2}}{\left( {\sigma_{1}^{1}}(x) {\sigma_{2}^{2}}(x) \right)^{2} t } & - \frac{{\sigma_{1}^{2}}(x)}{{\sigma_{1}^{1}}(x) \left( {\sigma_{2}^{2}}(x) \right)^{2} t} \\ - \frac{{\sigma_{1}^{2}}(x)}{{\sigma_{1}^{1}}(x) \left( {\sigma_{2}^{2}}(x) \right)^{2} t} & \frac{1}{\left( {\sigma_{2}^{2}}(x) \right)^{2} t}\\ \end{pmatrix}. \end{array} $$

(D.6)

4.3 D.3 Second order weight used in Section ??

In this case d = 2 and N = 1. The coefficients of the SDE are given by

$$ \begin{array}{@{}rcl@{}} b^{1}(x) = (\mu_{2} - \sigma_{2} \rho \lambda)x, {\sigma_{1}^{1}}(x) = \sigma_{2} \rho x, {\sigma_{2}^{1}}(x) = \sigma_{2} \sqrt{1 - \rho^{2}} x, \end{array} $$

for $x \in \mathbb {R}$. Then, the second order stochastic weight $\pi _{t}^{2, x, \text {UIP}}(B_{t})$ is given as follows.

$$ \begin{array}{@{}rcl@{}} \pi_{t}^{2, x, \text{UIP}}(B_{t}) &= & \frac{1}{2} \mathscr{V}_{1} {\sigma_{1}^{1}}(x) H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}} {B_{t}^{1}} -t \right) + \frac{1}{2} \left\{ \mathscr{V}_{1} {\sigma_{2}^{1}}(x) + \mathscr{V}_{2} {\sigma_{1}^{1}}(x) \right\} H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}} {B_{t}^{2}} \right) \\ && + \frac{1}{2} \mathscr{V}_{2} {\sigma_{2}^{1}}(x) H_{(1)} \left( \bar{X}(t,x), {B_{t}^{2}} {B_{t}^{2}} -t \right) + \frac{t}{2} \left\{ \mathscr{L} {\sigma_{1}^{1}}(x) + \mathscr{V}_{1} b^{1}(x) \right\} H_{(1)} \left( \bar{X}(t,x), {B_{t}^{1}} \right) \\ && + \frac{t}{2} \left\{ \mathscr{L} {\sigma_{2}^{1}}(x) + \mathscr{V}_{2} b^{1}(x) \right\} H_{(1)} \left( \bar{X}(t,x), {B_{t}^{2}} \right) + \frac{t^{2}}{4} \left( \mathscr{V}_{1} {\sigma_{1}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right) \\ && + \frac{t^{2}}{4} \left( \mathscr{V}_{2} {\sigma_{1}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right) + \frac{t^{2}}{4} \left( \mathscr{V}_{1} {\sigma_{2}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right) \\ && + \frac{t^{2}}{4} \left( \mathscr{V}_{2} {\sigma_{2}^{1}}(x) \right)^{2} H_{(1,1)} \left( \bar{X}(t,x),1 \right) + \frac{t^{2}}{2} \mathscr{L} b^{1}(x) H_{(1)} \left( \bar{X}(t,x),1 \right). \end{array} $$

(D.7)