Multilevel particle filters for the non-linear filtering problem in continuous time

Abstract

In the following article we consider the numerical approximation of the non-linear filter in continuous-time, where the observations and signal follow diffusion processes. Given access to high-frequency, but discrete-time observations, we resort to a first order time discretization of the non-linear filter, followed by an Euler discretization of the signal dynamics. In order to approximate the associated discretized non-linear filter, one can use a particle filter. Under assumptions, this can achieve a mean square error of $\mathcal {O}(\epsilon ^2)$, for $\epsilon >0$ arbitrary, such that the associated cost is $\mathcal {O}(\epsilon ^{-4})$. We prove, under assumptions, that the multilevel particle filter of Jasra et al. (SIAM J Numer Anal 55:3068–3096, 2017) can achieve a mean square error of $\mathcal {O}(\epsilon ^2)$, for cost $\mathcal {O}(\epsilon ^{-3})$. This is supported by numerical simulations in several examples.

Proofs

1.1 Objective and structure

The main objective of this appendix is to provide a meaningful bound, in terms of l and N, on

$$\begin{aligned} \overline{\mathbb {E}}[\{[\eta _{t}^{l}-\eta _{t}^{l-1}]^{N}-[\eta _{t}^{l}-\eta _{t}^{l-1}]\}(\varphi )^2]. \end{aligned}$$

This is the main result which we will need to prove Theorem 4.1. The typical way that this can be achieved is to consider the predictor $\check{\pi }_t^{l,N}$ in (14). Our strategy to consider this latter object first in Lemma A.1, which provides an upper-bound in terms of N and the expectation of some operators. The remainder of the proof is then concerned with the control of these expectations in terms of l; this result is given in Lemma A.4. Finally these results are used to prove the bound of interest in Proposition A.1. The remaining technical developments are all used to achieve these results. At a first reading, one can then proceed as just looking at Lemmata A.1 and A.4 followed by Proposition A.1.

The structure of this appendix is as follows. In Sect. A.2 we provide some additional notations and results which will be used in the appendix. In Sect. A.3, we give results on the coupled particle filter and in particular Lemma A.1. Within this section are two additional subsections; Sects. A.3.1 and A.3.2. In Sect. A.3.1, we consider the ‘coupling ability’ of the coupled particle filter. That is, how close particle pairs are, in terms of $\Delta _l$, which is the critical property of CPFs and indeed multilevel methods. Our results in this section are simply specializations of the results already proved in Jasra et al. (2017). In Sect. A.3.2, we give the important results Lemma A.4 and Proposition A.1. The results in the afore-mentioned appendices depend themselves on some important properties of non-linear filtering problems in continuous time and particle filters. The non-linear filtering properties are given in Sect. A.4 and this can be read more-or-less linearly. The results are fairly well understood in the literature, but are provided for completeness of the article. The particle filter is considered in Sect. A.5 and the results are essentially the standard ones, for instance in Del Moral (2013), given in the context of this paper. One way to read the appendix is, begin with Sect. A.2 and then to read, linearly, Sect. A.3 in its entirety, accepting the results in Sects. A.4 and A.5 and those latter sections can then be read (which can both be read linearly).

1.2 Some notations

Some operators are now defined. For $(l,p,n)\in \mathbb {N}_0^3$, $n>p$, $(u_p,\varphi )\in E_l\times \mathcal {B}_b(E_l)$, let

$$\begin{aligned}&\mathbf {Q}_{p,n}^l(\varphi )(u_p) := \int \varphi (u_n)\\&\quad \Big (\prod _{q=p}^{n-1} \mathbf {G}_q^l(u_q)\Big ) \prod _{q=p+1}^{n}M^l(u_{q-1},du_q), \end{aligned}$$

where we use the convention $\mathbf {Q}_{p,p}^l(\varphi )(u_p)=\varphi (u_p)$. One could interpret this as a type of weighted evolution operator from time p to n when a discretization of $\Delta _l$ is used. In addition, for $(l,p,n)\in \mathbb {N}_0^3$, $n>p$, $(u_p,\varphi )\in E_l\times \mathcal {B}_b(E_l)$, let

$$\begin{aligned} \mathbf {D}_{p,n}^l(\varphi )(u_p) := \frac{\mathbf {Q}_{p,n}^l(\varphi -\pi _n^l(\varphi ))(u_p)}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))} \end{aligned}$$

where $\mathbf {D}_{p,p}^l(\varphi )(u_p)=\varphi (u_p)-\pi _p^l(\varphi )$. The latter operator would facilitate certain martingale decompositions in the following.

Throughout our arguments, C is a finite constant whose value may change from line to line, but does not depend upon l nor N. The particular dependencies of a given constant will be clear from the statement of a given result.

As h is bounded, there exists $-\infty<\underline{C}<\overline{C} <+\infty$, such that for any $(l,n)\in \mathbb {N}\times \mathbb {N}_0$ and $u_{n}\in E_l$, almost surely

$$\begin{aligned} \mathbf {G}_{n}^l(u_n)\le & {} \overline{\mathbf {G}}_{n}^l \\ \mathbf {G}_{n}^l(u_n)\ge & {} \underline{\mathbf {G}}_{n}^l \end{aligned}$$

where

$$\begin{aligned} \overline{\mathbf {G}}_{n}^l= & {} \overline{C}\exp \Big \{\sum _{i=0}^{\Delta _l^{-1}-1}\sum _{k=1}^{d_y} \Big (\overline{C}\mathbb {I}_{[0,\infty )}((Y_{n+(i+1)\Delta _l}^{(k)} -Y_{n+i\Delta _l}^{(k)}))\nonumber \\&\quad (Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)})\nonumber \\&+\underline{C}\mathbb {I}_{(-\infty ,0)}((Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)}))(Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)}) \Big )\Big \}\nonumber \\ \end{aligned}$$

(20)

$$\begin{aligned} \underline{\mathbf {G}}_{n}^l= & {} \underline{C}\exp \Big \{\sum _{i=0}^{\Delta _l^{-1}-1}\sum _{k=1}^{d_y}\Big (\underline{C}\mathbb {I}_{[0,\infty )}((Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)}))\nonumber \\&\quad (Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)})\nonumber \\&+ \overline{C}\mathbb {I}_{(-\infty ,0)}((Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)}))(Y_{n+(i+1)\Delta _l}^{(k)}-Y_{n+i\Delta _l}^{(k)}) \Big )\Big \}.\nonumber \\ \end{aligned}$$

(21)

Moreover, for any $r\in \mathbb {N}$, it is straightforward to verify that these upper and lower bounds have finite $\mathbb {L}_r$ and $\mathbb {L}_{-r}$ moments that do not depend upon l.

1.3 Results for the coupled particle filter

Set, for $l\in \mathbb {N}$, $(n,p,\varphi )\in \mathbb {N}_0^2\times \mathcal {B}_b(\mathbb {R}^{d_x})$, $p\le n$

$$\begin{aligned} T_{p,n}^{l,1}(\varphi ):= & {} \overline{\mathbb {E}}[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} (\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})\\&-\, \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}))^2] \end{aligned}$$

and for $p<n$

$$\begin{aligned} T_{p,n}^{l,2}(\varphi ):= & {} \overline{\mathbb {E}}[(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})- \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})\\&(\bar{U}_p^{l-1,1}))^2]^{1/2}+ \Vert \varphi \Vert ^2\overline{\mathbb {E}}[(\mathbf {G}_{p}^l(U_p^{l,1})\\&-\,\mathbf {G}_{p}^{l-1} (\bar{U}_p^{l-1,1}))^2]^{1/2}. \end{aligned}$$

Lemma A.1

Assume (D1). Then for any $n\in \mathbb {N}_0$, there exists a $C<+\infty$ such that for any $(l,N,\varphi )\in \mathbb {N}\times \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})$

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \max \{\check{\pi }_n^{l,N}(\mathbf {G}_n^l\otimes 1)^{-2},\check{\pi }_n^{l,N}(1\otimes \mathbf {G}_n^{l-1})^{-2}\} (\check{\pi }_n^{l,N}-\check{\pi }_n^l)\right. \\&\qquad \left. ((\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1 - 1\otimes (\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))^2\right] \\&\quad \le C\left( \frac{1}{N}\sum _{p=0}^nT_{p,n}^{l,1}(\varphi ) + \frac{1}{N^{3/2}}\sum _{p=0}^{n-1}T_{p,n}^{l,2}(\varphi ) \right) . \end{aligned}$$

Proof

We have the following standard Martingale plus remainder decomposition (Del Moral et al. 2012, Lemma 6.3)

$$\begin{aligned}&(\check{\pi }_n^{l,N}-\check{\pi }_n^l)((\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1 - 1\otimes (\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))\\&\quad = \sum _{p=0}^n(\check{\pi }_p^{l,N}-\check{\Phi }_p^l(\check{\pi }_{p-1}^{l,N}))(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1 - 1\\&\qquad \otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})) \\&\qquad + \sum _{p=0}^{n-1}\Big \{\frac{\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1)}{\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)}[\check{\pi }_p^l-\check{\pi }_p^{l,N}](\mathbf {G}_p^l\otimes 1) \\&\qquad - \frac{\check{\pi }_p^{l,N}(1\otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))}{\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})}[\check{\pi }_p^l-\check{\pi }_p^{l,N}](1\otimes \mathbf {G}_p^{l-1}) \Big \}. \end{aligned}$$

Using the $C_2-$inequality multiple times:

$$\begin{aligned}&\overline{\mathbb {E}}[ \max \{\check{\pi }_n^{l,N}(\mathbf {G}_n^l\otimes 1)^{-2},\check{\pi }_n^{l,N}(1\otimes \mathbf {G}_n^{l-1})^{-2}\} (\check{\pi }_n^{l,N}-\check{\pi }_n^l)\nonumber \\&((\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1 - 1\otimes (\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))^2] \nonumber \\&\qquad \le C\Big (\sum _{p=0}^n\overline{\mathbb {E}}[T_1(p)^2] + \sum _{p=0}^{n-1}\overline{\mathbb {E}}[T_2(p)^2]\Big ) \end{aligned}$$

(22)

where

$$\begin{aligned} T_1(p):= & {} \max \{\check{\pi }_pn^{l,N}(\mathbf {G}_n^l\otimes 1)^{-2},\check{\pi }_n^{l,N} (1\otimes \mathbf {G}_n^{l-1})^{-2}\}^{1/2}\\&(\check{\pi }_p^{l,N}-\check{\Phi }_p^l(\check{\pi }_{p-1}^{l,N})) (\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1 - 1\\&\otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})) \\ T_2(p):= & {} \max \{\check{\pi }_n^{l,N}(\mathbf {G}_n^l\otimes 1)^{-2},\check{\pi }_n^{l,N}(1\otimes \mathbf {G}_n^{l-1})^{-2}\}^{1/2}\\&\times \left( \frac{\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1)}{\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)}[\check{\pi }_p^l-\check{\pi }_p^{l,N}](\mathbf {G}_p^l\otimes 1)\right. \\&\left. - \frac{\check{\pi }_p^{l,N}(1\otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))}{\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})}[\check{\pi }_p^l-\check{\pi }_p^{l,N}](1\otimes \mathbf {G}_p^{l-1})\right) . \end{aligned}$$

It thus suffices to control the terms $T_1(p)$, $p\in \{0,1,\dots ,n\}$ and $T_2(p)$, $p\in \{0,1,\dots ,n-1\}$ in an appropriate way.

Now, using (21) and applying the conditional Marcinkiewicz-Zygmund inequality

$$\begin{aligned}&\overline{\mathbb {E}}[T_1(p)^2] \le \frac{C}{N}\overline{\mathbb {E}}[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}\nonumber \\&(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})-\mathbf {D}_{p,n}^{l-1} (\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}))^2]. \end{aligned}$$

(23)

For $T_2(p)$ we have

$$\begin{aligned} T_2(p)= & {} \max \{\check{\pi }_n^{l,N}(\mathbf {G}_n^l\otimes 1)^{-2},\check{\pi }_n^{l,N}(1\otimes \mathbf {G}_n^{l-1})^{-2}\}^{1/2}\left( T_3(p)\right. \nonumber \\&\quad +\left. T_4(p) + T_5(p)\right) \end{aligned}$$

where

$$\begin{aligned} T_3(p):= & {} [\check{\pi }_p^l-\check{\pi }_p^{l,N}](\mathbf {G}_p^l\otimes 1)\frac{\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1)}{\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})}\nonumber \\&\Big \{\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})-\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)\Big \} \end{aligned}$$

(24)

$$\begin{aligned} T_4(p):= & {} [\check{\pi }_p^l-\check{\pi }_p^{l,N}](\mathbf {G}_p^l\otimes 1)\frac{1}{\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})}\nonumber \\&\Big \{\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1)-\check{\pi }_p^{l,N}(1\otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))\Big \} \nonumber \\ \end{aligned}$$

(25)

$$\begin{aligned} T_5(p):= & {} \frac{\check{\pi }_p^{l,N}(1\otimes \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1} \pmb {\varphi }^{l-1}))}{\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})} [\check{\pi }_p^l-\check{\pi }_p^{l,N}]\nonumber \\&(\mathbf {G}_p^l\otimes 1-1\otimes \mathbf {G}_p^{l-1}). \end{aligned}$$

(26)

By using (21) and the $C_2-$inequality three times

$$\begin{aligned} \overline{\mathbb {E}}[T_2(p)^2]\le C\sum _{j=3}^5\overline{\mathbb {E}}[\max \{(\underline{\mathbf {G}}_{n}^l)^{-2}, (\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}T_j(p)^2] \end{aligned}$$

so we consider bounding the R.H.S. of this inequality.

For $T_3(p)$, using Cauchy–Schwarz then Hölder gives

$$\begin{aligned}&\overline{\mathbb {E}}[\max \{(\underline{\mathbf {G}}_{n}^l)^{-2}, (\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}T_3(p)^2]\\&\quad \le \overline{\mathbb {E}}\Big [\Big |\frac{\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l (\mathbf {G}_n^l\pmb {\varphi }^l)\otimes 1)\max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}}{\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})}\Big |^{12} \Big ]^{1/6}\\&\qquad \times \overline{\mathbb {E}}[[\check{\pi }_p^l-\check{\pi }_p^{l,N}](\mathbf {G}_p^l\otimes 1)^{12}]^{1/6} \overline{\mathbb {E}}[|\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})\\&\qquad -\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)|^6]^{1/6}\\&\qquad \times \overline{\mathbb {E}}[|\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1})-\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)|^2]^{1/2}. \end{aligned}$$

For the left-most term on the R.H.S. one can apply Hölder, Lemma A.10 and Corollary A.3. For the term after on the R.H.S. one can apply Proposition A.2. For the next term, one can use (20). For the right-most term on the R.H.S. one has

$$\begin{aligned}&\overline{\mathbb {E}}[|\check{\pi }_p^{l,N}(1\otimes \mathbf {G}_p^{l-1}) -\check{\pi }_p^{l,N}(\mathbf {G}_p^l\otimes 1)|^2]^{1/2}\le \overline{\mathbb {E}}[(\mathbf {G}_{p}^l(U_p^{l,1})\\&\quad -\mathbf {G}_{p}^{l-1}(\bar{U}_p^{l-1,1}))^2]^{1/2}. \end{aligned}$$

Hence, we have that

$$\begin{aligned} \overline{\mathbb {E}}[T_3(p)^2] \le \frac{C\Vert \varphi \Vert ^2}{N^{2}}\overline{\mathbb {E}}[(\mathbf {G}_{p}^l(U_p^{l,1})-\mathbf {G}_{p}^{l-1}(\bar{U}_p^{l-1,1}))^2]^{1/2}. \end{aligned}$$

(27)

For $T_4(p), T_5(p)$, using almost the same strategy , except for using Proposition A.2 for terms such as (for any $r\in \mathbb {N}$)

$$\begin{aligned}&\mathbb {E}\left[ \Big |\check{\pi }_p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l \pmb {\varphi }^l)\otimes 1)-\check{\pi }_p^{l,N}(1\otimes \mathbf {D}_{p,n}^{l-1}\right. \\&\quad \left. (\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1}))\Big |^r\right] ^{1/r} \end{aligned}$$

yields

$$\begin{aligned}&\overline{\mathbb {E}}[\max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}T_4(p)^2] \le \frac{C}{N^{3/2}}\overline{\mathbb {E}}[(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})\nonumber \\&\qquad - \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}))^2]^{1/2}. \end{aligned}$$

(28)

and

$$\begin{aligned}&\overline{\mathbb {E}}[\max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}T_5(p)^2]\nonumber \\&\quad \le \frac{C\Vert \varphi \Vert ^2}{N^{3/2}}\overline{\mathbb {E}}[(\mathbf {G}_{p}^l(U_p^{l,1})-\mathbf {G}_{p}^{l-1}(\bar{U}_p^{l-1,1}))^2]^{1/2}. \end{aligned}$$

(29)

Combining (27)-(29) gives

$$\begin{aligned} \overline{\mathbb {E}}[T_2(p)^2] \le \frac{C}{N^{3/2}}T_{p,n}^{l,2}(\varphi ). \end{aligned}$$

(30)

The proof is easily completed by noting the bounds in (22), (23) and (30). $\square$

1.3.1 Additional technical results for coupled particle filters

The following section is essentially an adaptation of Jasra et al. (2017, Lemmata D.3-D.4). Many of the arguments are very similar to that article, with a modification to the context here. The entire proofs are included for completeness of this paper.

For $(i,l,n)\in \{1,\dots ,N\}\times \mathbb {N}\times \mathbb {N}_0$:

• $\widehat{U}_{n}^{l,i},\widehat{\bar{U}}_{n}^{l-1,i}$ denote the particles immediately after resampling

• $(I_{n}^{l,i},\bar{I}_{n}^{l-1,i})\in \{1,\dots ,N\}$ represent the resampled indices of $(u_n^{l,i},\bar{u}_n^{l-1,i})$ and let $I_{n}^l(i):=I_{n}^{l,i}$ and $\bar{I}_{n}^{l-1}(i):=\bar{I}_{n}^{l-1,i}$.

For $(l,n)\in \mathbb {N}\times \mathbb {N}_0$, let $\textsf {S} _n^l$ be the particle indices that choose the same ancestor at each resampling stage:

$$\begin{aligned} \textsf {S} _n^l=\{i\in \{1,\ldots , N\}:&I_{n}^l(i)=\bar{I}_{n}^{l-1}(i),I_{n-1}^l\circ I_{n}^l(i)=\bar{I}_{n-1}^{l-1}\circ \bar{I}_{n}^{l-1}(i),\dots , I_{0}^l\circ \cdots \circ I_{n}^l(i)=\bar{I}_{0}^{l-1}\circ \cdots \circ \bar{I}_{n}^{l-1}(i)\}. \end{aligned}$$

For $n=-1$, set $\textsf {S} _n^l=\{1,\ldots , N\}$. Let, for $(l,n)\in \mathbb {N}\times \mathbb {N}_0$

$$\begin{aligned} \mathcal {G}_n^l=&\sigma \left( \left\{ U_{p}^{l,i},\bar{U}_{p}^{l-1,i}, \widehat{U}_{p}^{l,i}, \widehat{\bar{U}}_{p}^{l-1,i}, I_{p}^l, \bar{I}_{p}^{l-1};0\le p<n,1\le i\le N\right\} \cup \left\{ U_{n}^{l,i},\bar{U}_{n}^{l-1,i},1\le i\le N\right\} \right) \vee \mathcal {Y}_{n+1},\\ \widehat{\mathcal {G}}_n^l=&\sigma \left( \left\{ U_{p}^{l,i},\bar{U}_{p}^{l-1,i}, \widehat{U}_{p}^{l,i}, \widehat{\bar{U}}_{p}^{l,i}, I_{p}^l, \bar{I}_{p}^{l-1};0\le p<n,1\le i\le N\right\} \cup \left\{ U_{n}^{l,i},\bar{U}_{n}^{l-1,i},\widehat{U}_{n}^{l,i},\widehat{\bar{U}}_{n}^{l-1,i},1\le i\le N\right\} \right) \vee \mathcal {Y}_{n+1}. \end{aligned}$$

To avoid ambiguity in the subsequent notations, we set for $(i,l,n)\in \{1,\dots ,N\}\times \mathbb {N}\times \mathbb {N}_0$

$$\begin{aligned} u_{n}^{l,i}= & {} (x_{n,n}^{l,i},x_{n,n+\Delta _l}^{l,i},\dots ,x_{n,n+1}^{l,i}) \in E_l \\ \bar{u}_{n}^{l,i}= & {} (\bar{x}_{n,n}^{l-1,i},\bar{x}_{n,n+\Delta _{l-1}}^{l-1,i},\dots ,\bar{x}_{n,n+1}^{l-1,i}) \in E_{l-1}. \end{aligned}$$

Lemma A.2

Assume (D1). Then for any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,N)\in \mathbb {N}\times \mathbb {N}$

$$\begin{aligned} \overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert X_{n,n+1}^{l,i}-\bar{X}_{n,n+1}^{l-1,i}\Vert _2^r\right] ^{1/r}\le C\Delta _{l}^{1/2}. \end{aligned}$$

Proof

The proof is by induction on n. The case $n=0$ follows immediately, for instance by Jasra et al. (2017, Proposition D.1). For a general n; following the first four lines of the proof of Jasra et al. (2017, Lemma D.3), one has

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert X_{n,n+1}^{l,i}-\bar{X}_{n,n+1}^{l-1,i}\Vert _2^r\right] ^{1/r}\\&\quad \le C\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert X_{n-1,n}^{l,I_{n-1}^{l,i}}-\bar{X}_{n-1,n}^{l-1,\bar{I}_{n-1}^{l,i}}\Vert _2^r\right] ^{1/r}+C\Delta _{l}^{1/2}. \end{aligned}$$

Now, $I_{n-1}^{l,i}=\bar{I}_{n-1}^{l-1,i}$ for $i\in \textsf {S} _{n-1}^l$. The conditional distribution of $(X_{n-1,n}^{l,I_{n-1}^{l,i}},\bar{X}_{n-1,n}^{l,\bar{I}_{n-1,n}^{l,i}})\ (i\in \textsf {S} _{n-1}^l)$ given $\textsf {S} _{n-1}^l$ and $\mathcal {G}_{n-1}^l$ is

$$\begin{aligned} \frac{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}\delta _{(X_{n-1,n}^{l,i},\bar{X}_{n-1,n}^{l,i})}}{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}}. \end{aligned}$$

Now, we have, almost surely:

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \left. \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N}\right| \mathcal {G}_{n-1}^l\right] \nonumber \\&\quad = \sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}\nonumber \\&\quad \le \frac{\overline{\mathbf {G}}_{n-1}^l}{\underline{\mathbf {G}}_{n-1}^l}\frac{\text {Card}(\textsf {S} _{n-2}^l)}{N}. \end{aligned}$$

(31)

Therefore

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert X_{n-1,n}^{l,I_{n-1}^{l,i}}-\bar{X}_{n-1,n}^{l-1,\bar{I}_{n-1}^{l,i}}\Vert _2^r\right] \\&\quad =\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \overline{\mathbb {E}}\left[ \left. \Vert X_{n-1,n}^{l,I_{n-1}^{l,i}}-\bar{X}_{n-1,n}^{l-1,\bar{I}_{n-1}^{l-1,i}}\Vert _2^r\right| \textsf {S} _{n-1}^l,\mathcal {G}_{n-1}^l\right] \right] \\&\quad =\overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N}\left\{ \frac{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}\Vert X_{n-1,n}^{l,i}-\bar{X}_{n-1,n}^{l,i}\Vert _2^r}{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}} \right\} \right] \\&\quad =\overline{\mathbb {E}}\left[ \overline{\mathbb {E}}\left[ \left. \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N}\right| \mathcal {G}_{n-1}^l\right] \left\{ \frac{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}\Vert X_{n-1,n}^{l,i}-\bar{X}_{n-1,n}^{l,i}\Vert _2^r}{\sum _{i\in \textsf {S} _{n-2}^l} \frac{\mathbf {G}_{n-1}^l(U_{n-1}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^l(U_{n-1}^{l,k})}\wedge \frac{\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n-1}^{l-1}(\bar{U}_{n-1}^{l-1,k})}} \right\} \right] \\&\quad \le \overline{\mathbb {E}}\left[ \overline{\mathbb {E}}\left[ \left. \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N}\right| \mathcal {G}_{n-1}^l\right] \frac{1}{\text {Card}(\textsf {S} _{n-2}^l)}\sum _{i\in \textsf {S} _{n-2}^l} \Vert X_{n-1,n}^{l,i}-\bar{X}_{n-1,n}^{l-1,i}\Vert ^r \Bigg (\frac{\overline{\mathbf {G}}_{n-1}^l}{\underline{\mathbf {G}}_{n-1}^l}\wedge \frac{\overline{\mathbf {G}}_{n-1}^{l-1}}{\underline{\mathbf {G}}_{n-1}^{l-1}}\Big / \frac{\underline{\mathbf {G}}_{n-1}^l}{\overline{\mathbf {G}}_{n-1}^l}\wedge \frac{\underline{\mathbf {G}}_{n-1}^{l-1}}{\overline{\mathbf {G}}_{n-1}^{l-1}} \Bigg ) \right] . \end{aligned}$$

Then noting (31) and then taking expectations w.r.t. the data on the time interval $[n-1,n]$ yields:

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert X_{n-1,n}^{l,I_{n-1}^{l,i}}-\bar{X}_{n-1,n}^{l-1,\bar{I}_{n-1}^{l,i}}\Vert _2^r\right] \\&\quad \le C\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-2}^l} \Vert X_{n-1,n}^{l,i}-\bar{X}_{n-1,n}^{l-1,i}\Vert ^r\right] . \end{aligned}$$

The result hence follows by induction.

Corollary A.1

Assume (D1). Then for any $(n,r)\in \mathbb {N}\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,N)\in \mathbb {N}\times \mathbb {N}$

$$\begin{aligned} \overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \Vert \widehat{X}_{n-1,n}^{l,i}-\widehat{\bar{X}}_{n-1,n}^{l-1,i}\Vert _2^r\right] ^{1/r}\le C\Delta _{l}^{1/2}. \end{aligned}$$

Proof

Easily follows from the proof of Lemma A.2. $\square$

Lemma A.3

Assume (D1). Then for any $n\in \{-1,0,1,\dots \}$, there exists a $C<+\infty$ such that for any $(l,N)\in \mathbb {N}\times \mathbb {N}$

$$\begin{aligned} 1-\overline{\mathbb {E}}\left[ \frac{{\text {Card}}(\textsf {S} _n^l)}{N}\right] \le C\Delta _{l}^{1/2}. \end{aligned}$$

Proof

The proof is by induction, with the initialization clear. We have

$$\begin{aligned}&1-\overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _n^l)}{N}\right] \nonumber \\&\quad =\left\{ 1-\overline{\mathbb {E}}\left[ \sum _{i=1}^N \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}\wedge \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N} \mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k})}\right] \right\} \nonumber \\&\qquad +\overline{\mathbb {E}}\left[ \sum _{i\notin \textsf {S} _{n-1}^l} \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^{l-1} (\bar{U}_{n}^{l-1,k})}\right] \nonumber \\&\quad \le C\overline{\mathbb {E}}\left[ \sum _{i\in \textsf {S} _{n-1}^l}\left| \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^{l-1} (\bar{U}_{n}^{l-1,k})}\right| \right] \nonumber \\&\qquad +C\overline{\mathbb {E}}\left[ \sum _{i\notin \textsf {S} _{n-1}^l}\left| \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N} \mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k})}\right| \right] . \end{aligned}$$

(32)

Now, we note that by using the bounds from (20) and (21)

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \sum _{i\notin \textsf {S} _{n-1}^l} \left| \frac{\mathbf {G}_{n}^l (U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N} \mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k})}\right| \right] \nonumber \\&\quad \le \overline{\mathbb {E}} \left[ \frac{\text {Card}((\textsf {S} _{n-1}^l)^c)}{N} \Big (\frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l}+ \frac{\overline{\mathbf {G}}_{n}^{l-1}}{\underline{\mathbf {G}}_{n}^{l-1}}\Big )\right] \nonumber \\&\quad \le C\Big (1-\overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \right] \Big ). \end{aligned}$$

(33)

To conclude the result, we must appropriately deal with the left-most term on the R.H.S. of (32). We have

$$\begin{aligned} \overline{\mathbb {E}}\left[ \sum _{i\in \textsf {S} _{n-1}^l}\left| \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N} \mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k})}\right| \right] \le T_1 + T_2 \end{aligned}$$

where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\left[ \frac{1}{\sum _{k=1}^{N}\mathbf {G}_{n}^l (U_{n}^{l,k})}\sum _{i\in \textsf {S} _{n-1}^l}\left| \mathbf {G}_{n}^l(U_{n}^{l,i}) -\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\right| \right] \\ T_2:= & {} \overline{\mathbb {E}}\left[ \sum _{i\in \textsf {S} _{n-1}^l}\mathbf {G}_{n}^l (\bar{U}_{n}^{l-1,i})\left( \frac{\sum _{k=1}^{N}\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k})-\sum _{k=1}^{N} \mathbf {G}_{n}^l(U_{n}^{l,k})}{\sum _{k=1}^{N}\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,k}) \sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})} \right) \right] . \end{aligned}$$

For $T_1$, applying Cauchy–Schwarz and recalling the bounds from (20) and (21),

$$\begin{aligned} T_1 \le \overline{\mathbb {E}}\left[ \frac{1}{(\underline{\mathbf {G}}_{n}^l)^2}\right] ^{1/2} \overline{\mathbb {E}}\left[ \left( \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \left| \mathbf {G}_{n}^l(U_{n}^{l,i})-\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\right| \right) ^2\right] ^{1/2}. \end{aligned}$$

Then applying Jensen and noting that the left-most term on the R.H.S. is upper-bounded by a constant that does not depend upon l nor N we have

$$\begin{aligned} T_1 \le C\overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l}\left| \mathbf {G}_{n}^l(U_{n}^{l,i}) -\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\right| ^2\right] ^{1/2}. \end{aligned}$$

Conditioning upon $\widehat{\mathcal {G}}_{n-1}^l$ and applying Lemma A.8 followed by Corollary A.1 yields the upper-bound

$$\begin{aligned} T_1 \le C\Delta _l^{1/2}. \end{aligned}$$

For $T_2$, using the bounds (20) and (21), one has

$$\begin{aligned} T_2\le & {} \overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \left( \frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l \underline{\mathbf {G}}_{n}^{l-1}}\right) \right. \\&\left. \left| \frac{1}{N}\sum _{i=1}^N \{\mathbf {G}_{n}^l(U_{n}^{l,i})-\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\}\right| \right] . \end{aligned}$$

Then it easily follows

$$\begin{aligned} T_2\le & {} \overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \left( \frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l \underline{\mathbf {G}}_{n}^{l-1}}\right) \left( \left| \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l} \{\mathbf {G}_{n}^l(U_{n}^{l,i}) -\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\}\right| \right. \right. \\&\left. \left. + \frac{\text {Card} ((\textsf {S} _{n-1}^l)^c)}{N}(\overline{\mathbf {G}}_{n}^l+\overline{\mathbf {G}}_{n}^{l-1}) \right) \right] . \end{aligned}$$

Now we set

$$\begin{aligned} T_3:= & {} \overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \left( \frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l \underline{\mathbf {G}}_{n}^{l-1}}\right) \right. \\&\left. \left| \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l}\{\mathbf {G}_{n}^l(U_{n}^{l,i}) -\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\}\right| \right] \\ T_4:= & {} \overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{n-1}^l)}{N} \left( \frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l \underline{\mathbf {G}}_{n}^{l-1}}\right) \frac{\text {Card}((\textsf {S} _{n-1}^l)^c)}{N}\right. \\&\left. (\overline{\mathbf {G}}_{n}^l +\overline{\mathbf {G}}_{n}^{l-1})\right] . \end{aligned}$$

For $T_3$, applying Cauchy Schwarz and Jensen

$$\begin{aligned} T_3\le & {} \overline{\mathbb {E}}\left[ \left( \frac{\text {Card} (\textsf {S} _{n-1}^l)}{N}\left( \frac{\overline{\mathbf {G}}_{n}^l}{\underline{\mathbf {G}}_{n}^l\underline{\mathbf {G}}_{n}^{l-1}}\right) \right) ^2\right] ^{1/2}\\&\overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{n-1}^l}\{\mathbf {G}_{n}^l(U_{n}^{l,i})-\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\}^2\right] ^{1/2}. \end{aligned}$$

Again, noting that the left-most term on the R.H.S. is upper-bounded by a constant that does not depend upon l nor N

$$\begin{aligned} T_3 \le C \overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in S_{n-1}^l}\{\mathbf {G}_{n}^l(U_{n}^{l,i})-\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})\}^2\right] ^{1/2}. \end{aligned}$$

Then, applying the above arguments, we have

$$\begin{aligned} T_3 \le C\Delta _l^{1/2}. \end{aligned}$$

For $T_4$, taking expectations w.r.t. the data on the time interval $[n,n+1]$ yields:

$$\begin{aligned} T_4 \le C\overline{\mathbb {E}}\left[ \frac{\text {Card}((\textsf {S} _{n-1}^l)^c)}{N}\right] . \end{aligned}$$

Thus, we have

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \sum _{i\in \textsf {S} _{n-1}^l}\left| \frac{\mathbf {G}_{n}^l(U_{n}^{l,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^l(U_{n}^{l,k})}- \frac{\mathbf {G}_{n}^{l-1}(\bar{U}_{n}^{l-1,i})}{\sum _{k=1}^{N}\mathbf {G}_{n}^{l-1} (\bar{U}_{n}^{l-1,k})}\right| \right] \nonumber \\&\quad \le C\Big (\Delta _l^{1/2}+1-\overline{\mathbb {E}}\left[ \frac{\text {Card} (\textsf {S} _{n-1}^l)}{N}\right] \Big ). \end{aligned}$$

(34)

Combining (32)–(34), one can conclude the result via induction. $\square$

1.3.2 Rate proofs for the coupled particle filter

Lemma A.4

Assume (D1). Then for any $n\in \mathbb {N}_0$, there exists a $C<+\infty$ such that for any $(l,N,\varphi )\in \mathbb {N}\times \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$

$$\begin{aligned} \sum _{p=0}^nT_{p,n}^{l,1}(\varphi )\le & {} C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})^2\Delta _l^{1/2} \\ \sum _{p=0}^{n-1}T_{p,n}^{l,2}(\varphi )\le & {} C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})^2\Delta _l^{1/4}. \end{aligned}$$

Proof

We consider $T_{p,n}^{l,1}(\varphi )$ only, as the case of $T_{p,n}^{l,2}(\varphi )$ is very similar.

We have the upper-bound

$$\begin{aligned}&\overline{\mathbb {E}}[\max \{(\underline{\mathbf {G}}_{n}^l)^{-2}, (\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}(\mathbf {D}_{p,n}^l (\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})\\&\quad - \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}))^2] \le C\sum _{j=1}^5T_j \end{aligned}$$

where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\left[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} \left( \frac{1}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))}\right. \right. \\&\left. \left. \quad \{\mathbf {Q}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})- \mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1})\}\right) ^2\right] \\ T_2:= & {} \overline{\mathbb {E}}\left[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} \left( \frac{\mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1})}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))} \right. \right. \\&\left. \left. \{\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))-\pi _p^l(\mathbf {Q}_{p,n}^l(1))\}\right) ^2\right] \\ T_3:= & {} \overline{\mathbb {E}}\left[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} \left( \frac{\mathbf {Q}_{p,n}^{l-1}(1)(\bar{U}_p^{l-1,1})}{\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))} \right. \right. \\&\left. \left. \{\pi _n^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})-\pi _n^l(\mathbf {G}_n^{l}\pmb {\varphi }^{l})\}\right) ^2\right] \\ T_4:= & {} \overline{\mathbb {E}}\left[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} \left( \frac{\pi _n^l(\mathbf {G}_n^{l}\pmb {\varphi }^{l})}{\pi _p^{l-1} (\mathbf {Q}_{p,n}^{l-1}(1))}\right. \right. \\&\left. \left. \{\mathbf {Q}_{p,n}^{l-1}(1)(\bar{U}_p^{l-1,1})-\mathbf {Q}_{p,n}^l(1)(U_p^{l,1})\}\right) ^2\right] \\ T_5:= & {} \overline{\mathbb {E}}\left[ \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} \left( \frac{\mathbf {Q}_{p,n}^{l}(1)(U_p^{l,1})}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))} \right. \right. \\&\left. \left. \{\pi _p^l(\mathbf {Q}_{p,n}^l(1))-\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))\}\right) ^2\right] . \end{aligned}$$

We will give bounds on the terms $T_1$ and $T_2$ only. The proofs for appropriate bounds on $T_3, T_4, T_5$ are very similar and hence omitted.

For $T_1$, we have $T_1=T_6+T_7$, where

$$\begin{aligned} T_6:= & {} \overline{\mathbb {E}}\left[ \frac{\max \{(\underline{\mathbf {G}}_{n}^l)^{-2}, (\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))^2} \frac{1}{N}\right. \\&\left. \sum _{i\in \textsf {S} _{p-1}^l}\{\mathbf {Q}_{p,n}^l(\mathbf {G}_n^l \pmb {\varphi }^l)(U_p^{l,i})- \mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1} \pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,i})\}^2\right] \\ T_7:= & {} \overline{\mathbb {E}}\left[ \frac{\max \{(\underline{\mathbf {G}}_{n}^l)^{-2}, (\underline{\mathbf {G}}_{n}^{l-1})^{-2}\}}{\pi _p^l(\mathbf {Q}_{p,n}^l(1))^2} \frac{1}{N}\right. \\&\left. \sum _{i\in (\textsf {S} _{p-1}^l)^c}\{\mathbf {Q}_{p,n}^l(\mathbf {G}_n^l \pmb {\varphi }^l)(U_p^{l,i})- \mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1} \pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,i})\}^2\right] . \end{aligned}$$

For $T_6$, applying Cauchy–Schwarz (twice) with Lemma A.10 and conditional Jensen yields

$$\begin{aligned} T_6\le & {} C \overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{p-1}^l}\{\mathbf {Q}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,i})\right. \\&\left. - \mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,i})\}^4\right] ^{1/2}. \end{aligned}$$

Then conditioning upon the entire data trajectory and the information up-to resampling at time $p-1$, followed by Lemma A.8 gives the upper-bound

$$\begin{aligned}&T_6 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\\&\quad \left( \overline{\mathbb {E}}\left[ \frac{1}{N}\sum _{i\in \textsf {S} _{p-1}^l} \Vert \widehat{X}_{p-1,p}^{l,i}-\widehat{\bar{X}}_{p-1,p}^{l-1,i}\Vert _2^4\right] +\Delta _l^{2}\right) ^{1/2}. \end{aligned}$$

Applying Corollary A.1 gives

$$\begin{aligned} T_6 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\Delta _l. \end{aligned}$$

(35)

For $T_7$, noting (20) and (21) and taking expectations w.r.t. $\{Y_t\}$ on the time interval $[p,n+1]$ one has the upper-bound

$$\begin{aligned} T_7 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\left( 1-\overline{\mathbb {E}}\left[ \frac{\text {Card}(\textsf {S} _{p-1}^l)}{N}\right] \right) . \end{aligned}$$

Then applying Lemma A.3 one has

$$\begin{aligned} T_7 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\Delta _l^{1/2}. \end{aligned}$$

Thus, using (35)

$$\begin{aligned} T_1 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\Delta _l^{1/2}. \end{aligned}$$

For $T_2$ applying Cauchy–Schwarz and Corollary A.2

$$\begin{aligned} T_2 \le C\overline{\mathbb {E}}\left[ \left( \frac{\mathbf {Q}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}) \max \{(\underline{\mathbf {G}}_{n}^l)^{-2},(\underline{\mathbf {G}}_{n}^{l-1})^{-2}\} }{\pi _p^l(\mathbf {Q}_{p,n}^l(1))\pi _p^{l-1}(\mathbf {Q}_{p,n}^{l-1}(1))}\right) ^4\right] ^{1/2}\Delta _l. \end{aligned}$$

Applying Hölder (thrice) and Lemma A.10 one has

$$\begin{aligned} T_2 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\Delta _l. \end{aligned}$$

Hence we have shown that

$$\begin{aligned}&\overline{\mathbb {E}}[(\mathbf {D}_{p,n}^l(\mathbf {G}_n^l\pmb {\varphi }^l)(U_p^{l,1})- \mathbf {D}_{p,n}^{l-1}(\mathbf {G}_n^{l-1}\pmb {\varphi }^{l-1})(\bar{U}_p^{l-1,1}))^2]\\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^2\Delta _l^{1/2}. \end{aligned}$$

$\square$

Proposition A.1

Assume (D1). Then for any $n\in \mathbb {N}_0$, there exists a $C<+\infty$ such that for any $(l,N,\varphi )\in \mathbb {N}\times \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$

$$\begin{aligned}&\overline{\mathbb {E}}\left[ \left( [\eta _{n}^{l}-\eta _{n}^{l-1}]^{N}(\varphi ) -[\eta _{n}^{l}-\eta _{n}^{l-1}](\varphi )\right) ^2\right] \\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})^2\left( \frac{\Delta _l^{1/2}}{N}+\frac{\Delta _l^{1/4}}{N^{3/2}}\right) . \end{aligned}$$

Proof

The result follows, essentially, by using Jasra et al. (2017, Lemma C.5) along with Lemmata A.1, A.4, along with Corollary A.2 (see also Remark A.2).

1.4 Results for the non-linear filter

In this section, we consider the case of the non-linear filter, with a probability space $(\Omega ,\mathcal {F})$, with $\mathcal {F}_t$ the filtration, that includes $\{Y_t\}_{t\ge 0}$ as standard Brownian motion independent of a diffusion process $\{X_t^{x}\}_{t\ge 0}$ which obeys (2) with initial condition $x\in \mathbb {R}^{d_x}$ and associated Euler discretization (with the same Brownian increments) at level l $(\widetilde{X}_{\Delta _l}^x,\widetilde{X}_{2\Delta _l}^{x},\dots )$. We will also consider another diffusion process $\{X_t^{x_{*}}\}_{t\ge 0}$ which obeys (2), initial condition ${x_{*}}\in \mathbb {R}^{d_x}$ and the same Brownian motion as $\{X_t^x\}_{t\ge 0}$ and associated Euler discretization (with the same Brownian increments) at level l $(\widetilde{X}_{\Delta _l}^{x_{*}},\widetilde{X}_{2\Delta _l}^{x_{*}},\dots )$. Expectations are written $\overline{\mathbb {E}}$. We set for $(p,n)\in \mathbb {N}_0^2$, $n+1>p$

$$\begin{aligned} Z_{p,n+1}^x = \exp \Big \{\int _{p}^{n+1}h(X_s^x)^*dY_s -\frac{1}{2}\int _{p}^{n+1}h(X_s^x)^*h(X_s^x)ds\Big \} \end{aligned}$$

with the convention that $Z_{0,n+1}^x=Z_{n+1}^x$. The technical results in this appendix are critical in proving the results in Appendix A.3. Although some of the results are more-or-less known in the literature (e.g. Picard (1984)), we give the proofs for the completeness of the article.

Lemma A.5

Assume (D1). Then for any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}_0\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\\&\quad - \varphi (X_{n+1}^x)Z_{n+1}^x\Big |^r\Big ]^{1/r}\le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

Proof

We have

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\nonumber \\&-\,\varphi (X_{n+1}^x)Z_{n+1}^x\Big |^r\Big ] \le C(T_1+T_2) \end{aligned}$$

(36)

where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\\&- \varphi (X_{n+1}^x)Z_{n+1}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x)\Big |^r\Big ] \\ T_2:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (X_{n+1}^x)Z_{n+1}^l(X_0^x, X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x)\\&-\,\varphi (X_{n+1}^x)Z_{n+1}^x\Big |^r\Big ]. \end{aligned}$$

The term $T_2$ can be treated with a very similar proof to $T_1$ along the lines of (Crisan 2011, Theorem 21.3), so we will give a proof for $T_1$ only.

One has

$$\begin{aligned} T_1 \le C(T_3 + T_4) \end{aligned}$$

(37)

where

$$\begin{aligned} T_3:= & {} \overline{\mathbb {E}}\Big [\Big |[\varphi (\widetilde{X}_{n+1}^x)-\varphi (X_{n+1}^x)] Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\\&\quad \widetilde{X}_{n+1 -\Delta _l}^x)\Big |^r\Big ]\\ T_4:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (X_{n+1}^x)\Big (Z_{n+1}^l(\widetilde{X}_0^x, \widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\\&\quad -\,Z_{n+1}^l (X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x)\Big )\Big |^r\Big ]. \end{aligned}$$

We now need to appropriately upper-bound $T_3$ and $T_4$. For $T_3$, taking expectations of $Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)$ w.r.t. the process $\{Y_t\}$ and using the fact that h is bounded along with the fact that $\varphi \in \text {Lip}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$ gives the upper-bound

$$\begin{aligned} T_3 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r \overline{\mathbb {E}}[\Vert \widetilde{X}_{n+1}^x-X_{n+1}^x\Vert _2^r]. \end{aligned}$$

Then using standard results on Euler discretization of diffusion processes (e.g. Kloeden and Platen (1992))

$$\begin{aligned} T_3 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r\Delta _l^{r/2}. \end{aligned}$$

(38)

For $T_4$ as $\varphi \in \mathcal {B}_b(\mathbb {R}^{d_x})$, one has

$$\begin{aligned} T_4\le & {} (\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r\overline{\mathbb {E}} \Big [\Big |Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{n+1-\Delta _l}^x)\nonumber \\&-\,Z_{n+1}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x) \Big |^r\Big ]. \end{aligned}$$

(39)

Now, by the Mean Value Theorem (MVT)

$$\begin{aligned}&Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{n+1-\Delta _l}^x)-Z_{n+1}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x)\nonumber \\&\quad = \Big ( H_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{n+1-\Delta _l}^x)\nonumber \\&\qquad -\,H_{n+1}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x) \Big )\nonumber \\&\quad \int _{0}^1 \widetilde{H}_{n+1}^{l,s}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n+1-\Delta _l}^x,X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x) ds\nonumber \\ \end{aligned}$$

(40)

where for any $(l,n,(x_0,x_{\Delta _l},\dots ,x_{n+1-\Delta _l}),(x_0',x_{\Delta _l}',\dots ,x_{n+1-\Delta _l}'),s)\in \{0,1,\dots \}^2\times (\mathbb {R}^{d_x})^{2(n+1)\Delta _l^{-1}}\times [0,1]$

$$\begin{aligned}&H_{n+1}^l(x_0,x_{\Delta _l},\dots ,x_{n+1-\Delta _l})\\&\quad = \log [Z_{n+1}^l(x_0,x_{\Delta _l},\dots ,x_{n+1-\Delta _l})] \\&\widetilde{H}_{n+1}^{l,s}(x_0,x_{\Delta _l},\dots ,x_{n+1-\Delta _l},x_0',x_{\Delta _l}',\dots ,x_{n+1-\Delta _l}'))\\&\quad = \exp \{sH_{n+1}^l(x_0,x_{\Delta _l},\dots ,x_{n+1-\Delta _l})\\&\qquad +\,(1-s)H_{n+1}^l(x_0',x_{\Delta _l}',\dots ,x_{n+1-\Delta _l}')\}. \end{aligned}$$

Then, by using (40) in (39) and applying Cauchy–Schwarz

$$\begin{aligned} T_4\le & {} (\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r \overline{\mathbb {E}}\Big [\Big |\int _{0}^1 \widetilde{H}_{n+1}^{l,s}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\nonumber \\&\quad \widetilde{X}_{n+1-\Delta _l}^x,X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x) ds\Big |^{2r}\Big ]^{1/2}\nonumber \\&\times \overline{\mathbb {E}}\Big [\Big |H_{n+1}^l(\widetilde{X}_0^x, \widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)-H_{n+1}^l\nonumber \\&\quad (X_0^x,X_{\Delta _l}^x,\dots ,X_{n+1-\Delta _l}^x)\Big |^{2r}\Big ]^{1/2}. \end{aligned}$$

(41)

Now, taking expectations w.r.t. the process $\{Y_t\}$ and using the fact that h is bounded, there exists a $C<+\infty$ such that

$$\begin{aligned}&\sup _{l\ge 0}\sup _{s\in [0,1]} \overline{\mathbb {E}}\Big [\Big |\widetilde{H}_{n+1}^{l,s}(\widetilde{X}_0^x, \widetilde{X}_{\Delta _l}^x,\\&\quad \dots ,\widetilde{X}_{n+1-\Delta _l}^x,X_0^x,X_{\Delta _l}^x, \dots ,X_{n+1-\Delta _l}^x)\Big |^{2r}\Big ] \le C \end{aligned}$$

so, via Jensen, we need only deal with the right-most expectation on the R.H.S of (41), call it $T_5$. Now

$$\begin{aligned}&H_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{n+1-\Delta _l}^x)-H_{n+1}^l(X_0^x,X_{\Delta _l}^x,\\&\quad \dots ,X_{n+1-\Delta _l}^x)= M^l_{n+1} - R^l_{n+1} \end{aligned}$$

where

$$\begin{aligned} M^l_{n+1}:= & {} \sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\Big [\{h(\widetilde{X}_{k\Delta _l})^* -h(X_{k\Delta _l})^*\}(Y_{(k+1)\Delta _l}-Y_{k\Delta _l})\Big ] \\ R^l_{n+1}:= & {} \frac{\Delta _l}{2}\sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\Big [h (\widetilde{X}_{k\Delta _l})^*h(\widetilde{X}_{k\Delta _l})-h(X_{k\Delta _l})^*h (X_{k\Delta _l}))\Big ] \end{aligned}$$

and we set $M_0=R_0=0$. Thus, applying the $C_{2r}-$inequality, one has

$$\begin{aligned} T_5^2 \le C\Big (\overline{\mathbb {E}}[|M^l_{n+1}|^{2r}] + \overline{\mathbb {E}}[|R^l_{n+1}|^{2r}]\Big ). \end{aligned}$$

(42)

We first focus on the first term on the R.H.S. of (42). Applying $C_{2r}-$inequality $d_y-$times, we have

$$\begin{aligned}&\overline{\mathbb {E}}[|M^l_{n+1}|^{2r}] \le C\sum _{i=1}^{d_y}\overline{\mathbb {E}}\Big [\Big | \sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\Big [\{h^{(i)}(\widetilde{X}_{k\Delta _l})\nonumber \\&\quad -\,h^{(i)}(X_{k\Delta _l})\}(Y_{(k+1)\Delta _l}^{(i)}-Y_{k\Delta _l}^{(i)})\Big ]\Big |^{2r}\Big ]. \end{aligned}$$

(43)

We consider just the $i^{\text {th}}$ summand on the R.H.S., as the argument to be used is essentially exchangeable w.r.t. i. As $\{M^l_{n},\mathcal {F}_{n\Delta _{l}^{-1}}\}_{n\in \{0,1,\dots \}}$ is a Martingale, applying the Burkholder-Gundy-Davis (BGD) inequality, Minkowski inequality, along with $h^{(i)}\in \text {Lip}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$:

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big | \sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\Big [\{h^{(i)}(\widetilde{X}_{k\Delta _l})-h^{(i)}(X_{k\Delta _l})\}(Y_{(k+1)\Delta _l}^{(i)}-Y_{k\Delta _l}^{(i)})\Big ]\Big |^{2r}\Big ]\\&\quad \le C\Delta _l^{r}\Big ( \sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\overline{\mathbb {E}}[\Vert \widetilde{X}_{k\Delta _l}-X_{k\Delta _l}\Vert _2^{2r}]^{1/r} \Big )^{r}. \end{aligned}$$

Then using standard results on Euler discretization of diffusion processes:

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big | \sum _{k=0}^{\Delta _l^{-1}(n+1)-1}\Big [\{h^{(i)} (\widetilde{X}_{k\Delta _l})-h^{(i)}(X_{k\Delta _l})\}(Y_{(k+1)\Delta _l}^{(i)} -Y_{k\Delta _l}^{(i)})\Big ]\Big |^{2r}\Big ]\\&\quad \le C\Delta _l^{r}. \end{aligned}$$

Thus, on returning to (43), we have shown that

$$\begin{aligned} \overline{\mathbb {E}}[|M^l_{n+1}|^{2r}] \le C\Delta _l^{r}. \end{aligned}$$

(44)

Noting that as $h^{(i)}\in \text {Lip}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$ and $h^{(i)}\in \mathcal {B}_b(\mathbb {R}^{d_x})$, it follows that $(h^{(i)})^2\in \text {Lip}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$. So using very similar calculations to those for $M^l_{n+1}$ (except not requiring to apply the BGD inequality), one can prove that

$$\begin{aligned} \overline{\mathbb {E}}[|R^l_{n+1}|^{2r}] \le C\Delta _l^{r}. \end{aligned}$$

(45)

Thus combining (44)-(45) with (42), one has that $T_5 \le C\Delta _l^{r/2}$ and hence that

$$\begin{aligned} T_4 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r\Delta _l^{r/2}. \end{aligned}$$

(46)

Noting (37) and using the bounds (38) and (46)

$$\begin{aligned} T_1 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})^r\Delta _l^{r/2}. \end{aligned}$$

As noted above, a similar bound can be obtained for $T_2$ and noting (36), the proof is hence concluded. $\square$

Lemma A.6

Assume (D1). Then for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n>p$ there exists a $C<+\infty$ such that for any $(x,x_*)\in \mathbb {R}^{d_x}\times \mathbb {R}^{d_x}$

$$\begin{aligned} \overline{\mathbb {E}}\Big [\Big |Z_{p,n}^x-Z_{p,n}^{x_*}\Big |^r\Big ]^{1/r} \le C\Vert x-x_{*}\Vert _2. \end{aligned}$$

Proof

This result can be proved in a similar manner to considering (40) in the proof of Lemma A.5, that is by using the MVT and a Martingale plus remainder method. The main difference is that one must use the result (which can be deduced by Roger (2000, Corollary v.11.7) and the Grönwall’s inequality)

$$\begin{aligned} \sup _{t\in [p,n]}\overline{\mathbb {E}}[\Vert X_t^x-X_{t}^{x_*}\Vert _2^{2r}]^{1/(2r)} \le C\Vert x-x_{*}\Vert _2. \end{aligned}$$

(47)

The proof is omitted due to the similarity to the proof associated to (40). $\square$

Lemma A.7

Assume (D1). Then for any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(\varphi ,x,x_{*})\in \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\varphi (X_{n+1}^x)Z_{n+1}^x- \varphi (X_{n+1}^{x_*})Z_{n+1}^{x_*}\Big |^r\Big ]^{1/r}\\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Vert x-x_*\Vert _2. \end{aligned}$$

Proof

We have

$$\begin{aligned} \overline{\mathbb {E}}\Big [\Big |\varphi (X_{n+1}^x)Z_{n+1}^x- \varphi (X_{n+1}^{x_*})Z_{n+1}^{x_*}\Big |^r\Big ]^{1/r} \le T_1 + T_2 \end{aligned}$$

where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\Big [\Big |\{\varphi (X_{n+1}^x)-\varphi (X_{n+1}^{x_*})\}Z_{n+1}^x\Big |^r\Big ]^{1/r}\\ T_2:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (X_{n+1}^{x_*})\{Z_{n+1}^x-Z_{n+1}^{x_*}\}\Big |^r\Big ]^{1/r}. \end{aligned}$$

So we proceed to control the two terms in $T_1$ and $T_2$.

For $T_1$, apply Cauchy–Schwarz to obtain the upper-bound

$$\begin{aligned} T_1 \le \overline{\mathbb {E}}[|Z_{n+1}^{x}|^{2r}]^{1/(2r)}\mathbb {\overline{E}}\Big [\Big |\{\varphi (X_{n+1}^x)-\varphi (X_{n+1}^{x_*})\}\Big |^{2r}\Big ]^{1/(2r)}. \end{aligned}$$

As $\overline{\mathbb {E}}[|Z_{n+1}^{x}|^{2r}]^{1/(2r)}\le C$ and using $\varphi \in \text {Lip}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})$ along with (47) yields

$$\begin{aligned} T_1 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\Vert x-x_{*}\Vert _2. \end{aligned}$$

(48)

For $T_2$ using $\varphi \in \mathcal {B}_b(\mathbb {R}^{d_x})$

$$\begin{aligned} T_2 \le (\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\overline{\mathbb {E}}\Big [\Big |Z_{n+1}^x-Z_{n+1}^{x_*}\Big |^{r}\Big ]^{1/r}. \end{aligned}$$

Applying Lemma A.6 gives

$$\begin{aligned} T_2 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\Vert x-x_{*}\Vert _2. \end{aligned}$$

(49)

Noting (48) and (49) allows one to conclude. $\square$

Lemma A.8

Assume (D1). Then for any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x,x_{*})\in \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\\&\quad -\,\varphi (\widetilde{X}_{n+1}^{x_*})Z_{n+1}^{l-1}(\widetilde{X}_0^{x_*}, \widetilde{X}_{\Delta _{l-1}}^{x_*},\dots ,\widetilde{X}_{n+1-\Delta _{l-1}}^{x_*}) \Big |^r\Big ]^{1/r}\\&\le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Big (\Delta _l^{1/2}+\Vert x-x_*\Vert _2\Big ). \end{aligned}$$

Proof

The expectation in the statement of the Lemma is upper-bounded by $\sum _{j=1}^3 T_j$ where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)\\&\quad -\,\varphi (X_{n+1}^x)Z_{n+1}^x\Big |^r\Big ]^{1/r} \\ T_2:= & {} \overline{\mathbb {E}}\Big [\Big | \varphi (X_{n+1}^x)Z_{n+1}^x- \varphi (X_{n+1}^{x_*})Z_{n+1}^{x_*} \Big |^r\Big ]^{1/r} \\ T_3:= & {} \overline{\mathbb {E}}\Big [\Big |\varphi (\widetilde{X}_{n+1}^{x_*})Z_{n+1}^{l-1}(\widetilde{X}_0^{x_*},\widetilde{X}_{\Delta _{l-1}}^{x_*},\dots ,\widetilde{X}_{n+1-\Delta _{l-1}}^{x_*})\\&\quad -\,\varphi (X_{n+1}^{x_*})Z_{n+1}^{x_*}\Big |^r\Big ]^{1/r}. \end{aligned}$$

The proof is completed by applying Lemma A.5 to $T_1$ and $T_3$, and Lemma A.7 to $T_2$. $\square$

Lemma A.9

Assume (D1). Then for any $(n,p,r)\in \mathbb {N}_0\times \mathbb {N}^2$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}_0\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x) Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\\&\quad -\frac{\overline{\mathbb {E}}[\varphi (X_{n+1}^x)Z_{n}^x|\mathcal {Y}_{n}]}{\overline{\mathbb {E}}[Z_{p}^x|\mathcal {Y}_{p}]}\Bigg |^r\Bigg ]^{1/r} \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

Proof

We have

$$\begin{aligned}&\overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}} [\varphi (\widetilde{X}_{n+1}^x)Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\\&\quad -\frac{\overline{\mathbb {E}} [\varphi (X_{n+1}^x)Z_{n}^x|\mathcal {Y}_{n}]}{\overline{\mathbb {E}}[Z_{p}^x|\mathcal {Y}_{p}]}\Bigg |^r\Bigg ]^{1/r} \le T_1 + T_2 \end{aligned}$$

where

$$\begin{aligned} T_1:= & {} \overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x) Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\\&- \frac{\overline{\mathbb {E}}[\varphi (X_{n+1}^x)Z_{n}^l(X_0^x,X_{\Delta _l}^x, \dots ,X_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x) |\mathcal {Y}_{p}]}\Bigg |^r\Bigg ]^{1/r} \\ T_2= & {} \overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}} [\varphi (X_{n+1}^x)Z_{n}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x)| \mathcal {Y}_{p}]} \\&- \frac{\overline{\mathbb {E}}[\varphi (X_{n+1}^x)Z_{n}^x|\mathcal {Y}_{n}]}{\overline{\mathbb {E}}[Z_{p}^x|\mathcal {Y}_{p}]}\Bigg |^r\Bigg ]^{1/r}. \end{aligned}$$

$T_2$ can be dealt with in a similar way to $T_1$, except one uses approaches similar to Crisan (2011, Theorem 21.3) (which is a similar MVT, Martingale plus remainder method that has been used in the proof of Lemma A.5), so we treat the former only.

Now, we have

$$\begin{aligned} T_1 \le T_3 + T_4 \end{aligned}$$

where

$$\begin{aligned} T_3:= & {} \overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}} [\varphi (\widetilde{X}_{n+1}^x)Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}] \overline{\mathbb {E}}[Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x)| \mathcal {Y}_{p}]}\\&\Big (\overline{\mathbb {E}}[Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots , X_{p-\Delta _l}^x)|\mathcal {Y}_{p}]\\&- \overline{\mathbb {E}}[Z_{p}^l (\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}]\Big ) \Bigg |^r\Bigg ]^{1/r}\\ T_4:= & {} \overline{\mathbb {E}}\Bigg [\Bigg |\frac{1}{\overline{\mathbb {E}} [Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\\&\quad \Big (\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n}^l(\widetilde{X}_0^x, \widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n] \\&-\,\overline{\mathbb {E}}[\varphi (X_{n+1}^x)Z_{n}^l(X_0^x,X_{\Delta _l}^x, \dots ,X_{n-\Delta _l}^x)|\mathcal {Y}_n]\Big ) \Bigg |^r\Bigg ]^{1/r}. \end{aligned}$$

For $T_3$ applying Cauchy–Schwarz and conditional Jensen,

$$\begin{aligned} T_3\le & {} \overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}] \overline{\mathbb {E}}[Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\Bigg |^{2r}\Bigg ]^{1/(2r)}\\&\times \,\overline{\mathbb {E}}\Big [\Big |Z_{p}^l(X_0^x,X_{\Delta _l}^x,\dots ,X_{p-\Delta _l}^x)-Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{p-\Delta _l}^x)\Big |^{2r}\Big ]^{1/(2r)}. \end{aligned}$$

For the left-most expectation on the R.H.S. one can use $\varphi \in \mathcal {B}_b(\mathbb {R}^{d_x})$, the Hölder and Jensen inequalities along with the proof approaches in the proof of Lemma A.10 to establish that the expectation is upper-bounded by a constant C that does not depend upon l, x. For the right-most expectation on the R.H.S. one can use the ideas in (40) to deduce that

$$\begin{aligned} T_3 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\Delta _l^{1/2}. \end{aligned}$$

The proof for $T_4$ is similar, except using the ideas for (37) instead of (40). That is, $T_4 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\Delta _l^{1/2}$. Hence

$$\begin{aligned} T_1 \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{\text {Lip}})\Delta _l^{1/2}. \end{aligned}$$

This completes the argument. $\square$

Remark A.1

One can easily deduce: Assume (D1). Then for any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}_0\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n}^x) Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n-\Delta _l}^x)| \mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x, \dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_{n}]}\\&\quad - \frac{\overline{\mathbb {E}}[\varphi (X_{n}^x)Z_{n}^x|\mathcal {Y}_{n}]}{\overline{\mathbb {E}}[Z_{n}^x|\mathcal {Y}_{n}]}\Bigg |^r\Bigg ]^{1/r} \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

Corollary A.2

Assume (D1). Then for any $(n,p,r)\in \mathbb {N}_0\times \mathbb {N}^2$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$

$$\begin{aligned}&\overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots , \widetilde{X}_{p-\Delta _l}^x)|\mathcal {Y}_{p}]}\\&\quad - \frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n}^{l-1}(\widetilde{X}_0^x, \widetilde{X}_{\Delta _{l-1}}^x,\dots ,\widetilde{X}_{n-\Delta _{l-1}}^x)|\mathcal {Y}_n]}{\overline{\mathbb {E}}[Z_{p}^{l-1}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _{l-1}}^x, \dots ,\widetilde{X}_{p-\Delta _{l-1}}^x)|\mathcal {Y}_{p}]} \Bigg |^r\Bigg ]^{1/r}\\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

Proof

Can be easily proved by using Lemma A.9. $\square$

Remark A.2

Using a similar strategy to Lemma A.9 and Corollary A.2, one can establish the following under (D1). For any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$:

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x) Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n-\Delta _l}^x) |\mathcal {Y}_n]\nonumber \\&\quad - \overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n}^{l-1}(\widetilde{X}_0^x, \widetilde{X}_{\Delta _{l-1}}^x,\dots ,\widetilde{X}_{n-\Delta _{l-1}}^x)| \mathcal {Y}_n]\Big |^r\Big ]^{1/r}\nonumber \\\le & {} C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

(50)

For any $(n,r)\in \mathbb {N}^2$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$:

$$\begin{aligned}&\overline{\mathbb {E}}\Bigg [\Bigg |\frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)|\mathcal {Y}_{n+1}]}{\overline{\mathbb {E}}[Z_{n}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n-\Delta _l}^x)|\mathcal {Y}_{n}]}\\&\quad - \frac{\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^{l-1}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _{l-1}}^x,\dots ,\widetilde{X}_{n+1-\Delta _{l-1}}^x)|\mathcal {Y}_{n+1}]}{\overline{\mathbb {E}}[Z_{n}^{l-1}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _{l-1}}^x,\dots ,\widetilde{X}_{n-\Delta _{l-1}}^x)|\mathcal {Y}_{n}]} \Bigg |^r\Bigg ]^{1/r}\\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

For any $(n,r)\in \mathbb {N}_0\times \mathbb {N}$, there exists a $C<+\infty$ such that for any $(l,\varphi ,x)\in \mathbb {N}\times \mathcal {B}_b(\mathbb {R}^{d_x})\cap {\mathrm{Lip}}_{\Vert \cdot \Vert _2}(\mathbb {R}^{d_x})\times \mathbb {R}^{d_x}$:

$$\begin{aligned}&\overline{\mathbb {E}}\Big [\Big |\overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^l(\widetilde{X}_0^x,\widetilde{X}_{\Delta _l}^x,\dots ,\widetilde{X}_{n+1-\Delta _l}^x)|\mathcal {Y}_{n+1}]\\&\quad - \overline{\mathbb {E}}[\varphi (\widetilde{X}_{n+1}^x)Z_{n+1}^{l-1}(\widetilde{X}_0^x,\widetilde{X}_{\Delta _{l-1}}^x,\dots ,\widetilde{X}_{n+1-\Delta _{l-1}}^x)|\mathcal {Y}_{n+1}]\Big |^r\Big ]^{1/r}\\&\quad \le C(\Vert \varphi \Vert +\Vert \varphi \Vert _{{\mathrm{Lip}}})\Delta _l^{1/2}. \end{aligned}$$

1.5 Results for the particle filter

Lemma A.10

Assume (D1). Then for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n\ge p$, there exists a $C<+\infty$ such that for any $l\in \mathbb {N}_0$, $(i,\varphi )\in \{1,\dots ,N\}\times \mathcal {B}_b(E_l)$

$$\begin{aligned}&\max \{\overline{\mathbb {E}}[\pi _p^l(\mathbf {Q}_{p,n}^l(\varphi ))^{-r}], \overline{\mathbb {E}}[\pi _p^{l,N}(\mathbf {Q}_{p,n}^l(\varphi ))^{-r}],\\&\quad \overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r]^{1/r}\} \le C\Vert \varphi \Vert . \end{aligned}$$

Proof

We start by considering $\overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r]^{1/r}$. Applying Jensen’s inequality, we have the upper-bound:

$$\begin{aligned} \overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r] \le \Vert \varphi \Vert ^r\overline{\mathbb {E}}[\mathbf {G}_p^l(U_p^{l,i})^r \prod _{q=p+1}^{n-1} \mathbf {G}_q^l(U_q)^r] \end{aligned}$$

where $U_{p+1},\dots ,U_{n}$ is a Markov chain of initial distribution $M^l(U_p^{l,i},\cdot )$ and transition $M^l$. As h is bounded, we have the upper-bound

$$\begin{aligned}&\overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r]\\&\quad \le C\Vert \varphi \Vert ^r\overline{\mathbb {E}}\Big [ \exp \Big \{r\sum _{s_1=0}^{\Delta _l^{-1}-1}\sum _{s_2=1}^{d_y}h^{(s_2)}(X_{p+s_1\Delta _l}^{l,i})(Y_{p+(s_1+1)\Delta _l}^{(s_2)}-Y_{p+s_1\Delta _l}^{(s_2)})\Big \}\\&\qquad \times \exp \Big \{r\sum _{q=p+1}^{n-1}\sum _{s_1=0}^{\Delta _l^{-1}-1}\sum _{s_2=1}^{d_y}h^{(s_2)}(X_{q+s_1\Delta _l}^l)(Y_{q+(s_1+1)\Delta _l}^{(s_2)}-Y_{q+s_1\Delta _l}^{(s_2)})\Big \} \Big ]. \end{aligned}$$

Taking expectations w.r.t. the process $\{Y_t\}$ we have

$$\begin{aligned}&\overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r]\\&\quad \le C\Vert \varphi \Vert ^r\overline{\mathbb {E}}\Big [ \exp \Big \{\frac{r^2}{2}\sum _{s_1=0}^{\Delta _l^{-1}-1}\sum _{s_2=1}^{d_y}h^{(s_2)}(X_{p+s_1\Delta _l}^{l,i})^2\Delta _l\Big \}\\&\quad \times \, \exp \Big \{\frac{r^2}{2}\sum _{q=p+1}^{n-1}\sum _{s_1=0}^{\Delta _l^{-1}-1}\sum _{s_2=1}^{d_y}h^{(s_2)}(X_{q+s_1\Delta _l}^l)^2\Delta _l\Big \} \Big ]. \end{aligned}$$

Then using the fact that h is bounded, it clearly follows

$$\begin{aligned} \overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,i})^r]^{1/r} \le C\Vert \varphi \Vert . \end{aligned}$$

For the terms $\overline{\mathbb {E}}[\pi _p^l(\mathbf {Q}_{p,n}^l(\varphi ))^{-r}]$ and $\overline{\mathbb {E}}[\pi _p^{l,N}(\mathbf {Q}_{p,n}^l(\varphi ))^{-r}]$ one can apply (the conditional) Jensen’s inequality and essentially the same argument as above and hence the proof is omitted.

Proposition A.2

Assume (D1). Then for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n\ge p$, there exists a $C<+\infty$ such that for any $l\in \{0,1,\dots \}$, $(N,\varphi )\in \mathbb {N}\times \mathcal {B}_b(E_l)$

$$\begin{aligned} \overline{\mathbb {E}}[|(\pi _p^{l,N}-\pi _p^l)(\mathbf {Q}_{p,n}^l(\varphi ))|^r]^{1/r} \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

Proof

The proof is by induction on p for any fixed $n\ge p$. If $p=0$ one can apply the conditional Marcinkiewicz-Zygmund inequality to yield:

$$\begin{aligned} \overline{\mathbb {E}}[|(\pi _p^{l,N}-\pi _p^l)(\mathbf {Q}_{p,n}^l(\varphi ))|^r]^{1/r} \le \frac{C}{\sqrt{N}} \overline{\mathbb {E}}[\mathbf {Q}_{p,n}^l(\varphi )(U_p^{l,1})^r]^{1/r}. \end{aligned}$$

Then one has the result by Lemma A.10.

For the induction step, we have the standard decomposition via Minkowski

$$\begin{aligned} \overline{\mathbb {E}}[|(\pi _p^{l,N}-\pi _p^l)(\mathbf {Q}_{p,n}^l(\varphi ))|^r]^{1/r} \le T_1 + T_2 + T_3 \end{aligned}$$

(51)

where

$$\begin{aligned} T_1= & {} \overline{\mathbb {E}}[|(\pi _p^{l, N}-\Phi _p^l(\pi _{p-1}^{l, N}))(\mathbf {Q}_{p, n}^l(\varphi ))|^r]^{1/r}\\ T_2= & {} \overline{\mathbb {E}}\Big [\Big |\frac{\Phi _p^l(\pi _{p-1}^{l, N})(\mathbf {Q}_{p, n}^l(\varphi ))}{\pi _{p-1}^l(\mathbf {G}_{p-1}^l)}\{(\pi _{p-1}^l-\pi _{p-1}^{l, N})(\mathbf {G}_{p-1}^l)\}\Big |^r\Big ]^{1/r}\\ T_3= & {} \overline{\mathbb {E}}\Big [\Big |\frac{(\pi _{p-1}^{l, N}-\pi _{p-1}^{l})(\mathbf {Q}_{p-1, n}^l(\varphi ))}{\pi _{p-1}^l(\mathbf {G}_{p-1}^l)}\Big |^r\Big ]^{1/r}. \end{aligned}$$

By the same argument as for the initialization

$$\begin{aligned} T_1 \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

(52)

For $T_2$ applying Hölder

$$\begin{aligned} T_2\le & {} \overline{\mathbb {E}}[\pi _{p-1}^l(\mathbf {G}_{p-1}^l)^{-3r}]^{1/(3r)} \overline{\mathbb {E}}[\Phi _p^l(\pi _{p-1}^{l,N})\\&\quad (\mathbf {Q}_{p,n}^l(\varphi ))^{3r}]^{1/(3r)}\overline{\mathbb {E}} [|(\pi _{p-1}^l-\pi _{p-1}^{l,N})(\mathbf {G}_{p-1}^l)|^{3r}]^{1/(3r)}. \end{aligned}$$

For the left-most term on the R.H.S. one can apply Lemma A.10. For the middle term on the R.H.S. one can apply the conditional Jensen inequality and Lemma A.10. For the right-most term on the R.H.S. one can apply the induction hypothesis. Hence

$$\begin{aligned} T_2 \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

(53)

For $T_3$, one can use Cauchy–Schwarz, Lemma A.10 and the induction hypothesis to yield

$$\begin{aligned} T_3 \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

(54)

Combining (52)–(54) with (51) concludes the proof. $\square$

Remark A.3

It is simple to extend Proposition A.2 to the case of the filter. That is, for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n\ge p$, there exists a $C<+\infty$ such that for any $l\in \{0,1,\dots \}$, $(N,\varphi )\in \mathbb {N}\times \mathcal {B}_b(E_l)$

$$\begin{aligned} \overline{\mathbb {E}}[|(\eta _p^{l,N}-\eta _p^l)(\mathbf {Q}_{p,n}^l(\varphi ))|^r]^{1/r} \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

Remark A.4

It is straightforward to extend Proposition A.2 to the following result, under (D1): for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n\ge p$, there exists a $C<+\infty$ such that for any $l\in \mathbb {N}_0$, $(N,\varphi )\in \mathbb {N}\times \mathcal {B}_b(E_l)$

$$\begin{aligned} \overline{\mathbb {E}}[|(\pi _p^{l,N}-\pi _p^l)(\mathbf {Q}_{p,n}^l(\mathbf {G}_{n}^l\varphi ))|^r]^{1/r} \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

Corollary A.3

Assume (D1). Then for any $(p,n,r)\in \mathbb {N}_0^2\times \mathbb {N}$, $n\ge p$, there exists a $C<+\infty$ such that for any $l\in \mathbb {N}_0$, $(N,\varphi )\in \mathbb {N}\times \mathcal {B}_b(E_l)$

$$\begin{aligned} \overline{\mathbb {E}}[|\pi _p^{l,N}(\mathbf {D}_{p,n}^l(\mathbf {G}_{n}^l\varphi ))|^r]^{1/r} \le \frac{C\Vert \varphi \Vert }{\sqrt{N}}. \end{aligned}$$

Proof

Noting that $\pi _p^l(\mathbf {D}_{p,n}^l(\mathbf {G}_{n}^l\varphi ))=0$ a.s., the result follows immediately by Cauchy–Schwarz, Lemma A.10 and Proposition A.2 (see Remark A.4). $\square$