Hagen

# Image Memorability Prediction with Vision Transformers

Thomas Hagen  Department of Psychology, University of Oslo, Oslo, Norway
Thomas Espeseth  Department of Psychology, University of Oslo, Oslo, Norway
Department of Psychology, Oslo New University College, Oslo, Norway

###### Abstract

Behavioral studies have shown that the memorability of images is similar
across groups of people, suggesting that memorability is a function of the
intrinsic properties of images, and is unrelated to people’s individual
experiences and traits. Deep learning networks can be trained on such
properties and be used to predict memorability in new data sets. Convolutional
neural networks (CNN) have pioneered image memorability prediction, but more
recently developed vision transformer (ViT) models may have the potential to
yield even better predictions. In this paper, we present the ViTMem, a new
memorability model based on ViT, and evaluate memorability predictions
obtained by it with state-of-the-art CNN-derived models. Results showed that
ViTMem performed equal to or better than state-of-the-art models on all data
sets. Additional semantic level analyses revealed that ViTMem is particularly
sensitive to the semantic content that drives memorability in images. We
conclude that ViTMem provides a new step forward, and propose that ViT-derived
models can replace CNNs for computational prediction of image memorability.
Researchers, educators, advertisers, visual designers and other interested
parties can leverage the model to improve the memorability of their image
material.

###### keywords:

memorability | vision transformers | psychology | semantic information

## Introduction

Everyone knows that our memories depend on the experiences we have had, facts
we have encountered, and the abilities we have to remember them. Combinations
of these factors differ between individuals and give rise to unique memories
in each of us. However, a complementary perspective on memory focuses on the
material that is (to be) remembered rather than the individual that does the
remembering. In one central study, Isola et al. (1) presented more than 2000
scene images in a continuous repeat-detection task. The participants were
asked to respond whenever they saw an identical repeat. The results revealed
that the memorability score (percent correct detections) varied considerably
between images. Most importantly, by running a consistency analysis in which
Spearman’s rank correlation was calculated on the memorability scores from
random splits of the participant group, Isola and colleagues (1) were able to
show that the memorability score ranking was consistent across participants –
some images were memorable and some were forgettable. These results indicate
that the degree to which an image was correctly detected depended on
properties intrinsic to the image itself, not the traits of the observers.
This is important because it shows that one can use the memorability scores in
a stimulus set to predict memory performance in a new group of participants.

These results have been replicated and extended in a number of studies,
revealing that similar findings are obtained with different memory tasks (2),
different retention times (2, 1), different contexts (3), and independent of
whether encoding is intentional or incidental (4). However, although image
memorability has proven to be a robust and reliable phenomenon, it has not
been straightforward to pinpoint the image properties that drive it. What
seems clear though, is that memorability is multifaceted (5, 6). One way to
characterize the underpinnings of memorability is to investigate the
contribution from processes at different levels of the visual processing
stream. For example, at the earliest stages of processing of a visual scene,
visual attributes such as local contrast, orientation, and color are coded. At
an intermediate level, contours are integrated, surfaces, shapes, and depth
cues are segmented, and foreground and background are distinguished. At a
higher level, object recognition is conducted through matching with templates
stored in long term memory.

Positive correlations between brightness and high contrast of objects with
memorability has been found (7), but in general, low-level visual factors such
as color, contrast, and spatial frequency do not predict memorability well (8,
5, 9). This is consistent with results showing that perceptual features are
typically not retained in long term visual memory (10). In contrast to the
low-level features, the evidence for a relation between intermediate to high
level semantic features and memorability is much stronger. For example, images
that contain people, faces, body parts, animals, and food are often associated
with high memorability, whereas the opposite is a typical finding for objects
like buildings and furniture and images of landscapes and parks (3, 7, 11,
12). Other intermediate to high level features such as object interaction with
the context or other objects, saliency factors, and image composition also
contribute to memorability (5). Furthermore, although memorability is not
reducible to high-level features such as aesthetics (1, 12), interestingness
(13, 1), or popularity (12), emotions, particularly of negative valence, seem
to predict higher memorability (9, 12). Finally, memorability seems to be
relatively independent of cognitive control, attention, or priming (14).

Overall, the available evidence indicates that memorability seems to capture
intermediate- to high-level properties of semantics, such as objects or
actions, and image composition, such as layout and clutter, rather than low-
level features (5, 15). This fits well with the central role of semantic
categories in organizing cognition and memory (16). Generally, the priority of
semantic-level information enables us to quickly understand novel scenes and
predict future events (17). For example, when inspecting a novel scene or an
image, we do not primarily focus on low-level perceptual features or pixels,
but prioritize more abstract visual schemas involving spatial regions,
objects, and the relation between them (18). Also, when people are asked to
indicate which regions of an image helps them recognize an image, there is
high consistency between people’s responses (18). Similarly, fixation map data
from eye-tracking have shown that there is a positive correlation between
fixation map consistency and scene memorability, and this relation is
associated with the presence of meaningful objects (3, 7, 19). Bylinskii et
al. (5) suggest that these properties most efficiently signal information of
high utility to our species, for example, emotions, social aspects, animate
objects (e.g., faces, gestures, interactions), unexpected events, and tangible
objects.

### Memorability prediction

The finding that the memorability of an image is governed by properties
intrinsic to the image itself, not only implies that one can predict memory
performance in a new set of participants, as described above, but also that
one can predict the memorability of a novel set of images (i.e., memorability
is an “image computable” feature). Given the availability of computational
algorithms and high-quality training sets of sufficient size, one can predict
memorability in novel sets of images for future (or already conducted)
behavioral or neuroimaging studies. Such memorability prediction could also be
valuable in a number of applied settings (e.g., within education, marketing
and human-computer interaction).

Memorability researchers have employed computer vision models such as
convolutional neural networks (CNNs) from early on (12), and advancements in
the field have allowed researchers to predict image memorability with
increasing precision (20, 21, 22). The inductive bias (the assumptions of the
learning algorithms used to generalize to unseen data) of CNNs is inspired by
knowledge about the primate visual system, and activations in the networks
layers have, with some success, been used to explain neural activations (23).
However, some vulnerabilities of CNNs have been noted. For example, CNNs
appear to depend more on image texture than biological vision systems do (24),
and have problems with recognizing images based on the shape of objects (e.g.,
when texture is suppressed or removed). However, this vulnerability is reduced
when the model’s shape bias is increased through training on shape
representations (25).

The LaMem train/test splits is a well-established benchmark for memorability
prediction (12). The original MemNet (12), which is based on AlexNet (26),
achieved a Spearman rank correlation of 0.64 on this benchmark. There have
been several improvements on this benchmark, the leading approaches utilize
image captioning to enhance memorability predictions. That is, a CNN produces
a textual description of the image, which is then used to provide more high-
level semantic information which is embedded into a semantic vector space
before being combined with CNN image features in a multi-layered perceptron
network. Squalli-Houssaini et al. (21) used this approach to reach a Spearman
correlation of 0.72, with a mean squared error (MSE) of approximately 0.0092
(22). Leonardi et al. (22) used the captioning approach with dual ResNet50s
and a soft attention mechanism to reach a rank correlation of 0.687 with an
MSE of 0.0079. The ResMem model (20), which is a CNN-based residual neural
network architecture (ResNet), uses LaMem, but also takes advantage of a more
recently published dataset named MemCat (11). This is a data set containing
10,000 images based on categories of animals, food, landscape, sports and
vehicles. This data set also has a higher split half correlation than LaMem.
Needell and Bainbridge (20) argue that the LaMem dataset on its own is lacking
in generalizability due to poor sampling of naturalistic images. That is, the
images are more intended as artistic renderings designed to attract an online
audience. Hence by combining MemCat with LaMem this should potentially yield a
more generalizable model. Moreover, the increased size of the combined dataset
might help in driving the model performance further than previous models based
on LaMem. The authors of ResMem also noted the importance of semantic
information and structured their approach to utilize semantic representations
from a ResNet model in order to improve predictions. An added benefit of
ResMem is that it is shared on the python package index, which makes it easily
accessible to researchers in diverse fields.

### Vision transformers

Vision transformers (ViT) have recently been shown to provide similar or
better performance than CNNs in a variety of computer vision tasks (27). This
architecture was first introduced in the natural language processing field
(28) for capturing long-range dependencies in text. This architecture leads to
superior speed/performance balance relativ to ResNet architectures (29).
Moreover, ViTs have been shown to produce errors that are more similar to
human errors (30), suggesting that they could take similar information into
account (see also (31)). A reason for this may be that ViTs are likely to take
more of the global context into account and be more dependent on the shape of
objects rather than their texture (30). While it is not entirely clear why
such properties may yield better predictions of image memorability, it could
still help inform the discourse on the visual characteristics that are
relevant as well as potentially yielding a better model for predicting image
memorability.

Hence, we set out to investigate if vision transformers can yield better
predictions of memorability than the state-of-the-art in image memorability
prediction. In particular, we aimed to (i) benchmark a model based on ViT
against the well-established LaMem train/test splits (12), (ii) train a ViT
against the combined LaMem and MemCat data sets (20) to benchmark against the
ResMem model (20), (iii) train a final ViT model against a more diverse and
deduplicated data set, (iv) validate the final ViT model against additional
independent data sets and (v) inspect semantic level distributions of
memorability scores for behavioral and predicted data.

## Methods

As our model is based on ViT to predict memorability we named it ViTMem.
Because it has been shown that low-level visual features are less important
for image memorability prediction, it would seem appropriate to use image
augmentations in training our ViTMem model to reduce overfitting. This
approach have also been used by others (22), although not to the extent done
here. The augmentations used consisted of horizontal flipping, sharpen, blur,
motion blur, random contrast, hue saturation value, CLAHE, shift scale rotate,
perspective, optical distortion and grid distortion (32). For training all
models we used PyTorch, the ADAM optimizer and mean squared error (squared L2
norm) for the loss function. Images were input as batches of 32 in RGB and
resized to 256x256 pixels before applying augmentations with a probability of
0.7 and center cropping to 224x224 pixels. For creating ViTMem we used
transfer learning on a vision transformer (27) model pretrained on ImageNet 1k
(vit_base_patch16_224_miil) (33). The final classification layer was reduced
to a single output and a sigmoid activation function.

As we aim to provide an accessible model to the research community, it is also
necessary to compare against the publicly available ResMem model.
Unfortunately, the authors of ResMem did not publish their held-out test set,
hence it is difficult to make a balanced comparison between the currently
published ResMem model and any competing models. We propose to do 10
train/test splits that can be used by future researchers (available at
https://github.com/brainpriority/vitmem_data). Moreover, ResMem was not
benchmarked on LaMem, hence a fair comparison can only be made on the combined
LaMem and MemCat data set.

For the semantic level analysis, we chose to use image captioning (34) as this
provides an efficient method for deriving semantic properties from images at
scale. Importantly, as the image captioning model was trained on human image
descriptions, it is likely to extract content that humans find meaningful in
images, and in particular objects and contexts that are relevant for conveying
such meanings. Hence, nouns derived from such descriptions are likely to be
representative portions of the content that would convey meaning to humans
observing the images.

### Data Sources

For the large-scale image memorability (LaMem) benchmark we used the LaMem
dataset (12). The image set used by ResMem is a combination of the image sets
LaMem (12) and MemCat (11). LaMem containing 58,741 and MemCat 10,000 images,
for a total of 68,741 images. ResMem is reported to have used a held-out test
set with 5000 images, hence we randomly selected 5000 images as our test set
for our 10 train/test splits for this combined data set. For our final model
we aimed to clean up the data and combine more of the available data sets on
image memorability. As number of duplicated images within and between data
sets is unknown and duplicated images may interfere with performance measures,
we aimed to deduplicate the data for this model. Duplicated images were
identified by simply deriving embeddings from an off-the-shelf CNN model, and
then visually inspecting the most similar embeddings. Our analysis of the data
sets LaMem and MemCat showed that LaMem have 229 duplicated images while
MemCat have 4. Moreover, 295 of the images in LaMem is also in MemCat. We
aimed to build a larger and more diverse data set by combining more sources,
and for this we chose CVPR2011 (9) and FIGRIM (3). CVPR2011 had 6 internal
duplicates, 651 duplicates against LaMem, 78 against MemCat og 9 against
FIGRIM. FIGRIM had 20 duplicates against MemCat and 70 against LaMem. All
identified duplicates were removed before merging the data sets. As the images
from FIGRIM and CVPR2011 were cropped, we obtained the original images before
including them in the data set. This resulted in a data set with 71,658
images. For this data set we performed a 10% split for the test set.

## Results

### Results on LaMem data set

On the LaMem data set the ViTMem model reached an average Spearman rank
correlation of 0.711 and an MSE of 0.0076 (see Table 1). Here we compare our
performance to measures obtained by MemNet (12), Squalli-Houssaini et al. (21)
and Leonardi et al. (22).

Table 1: Comparison of model performance on LaMem data set Model | MSE Loss $\downarrow$ | Spearman $\rho$ $\uparrow$
---|---|---
MemNet | Unknown | 0.640
Squalli-Houssaini et al. | 0.0092 | 0.720
Leonardi et al. | 0.0079 | 0.687
ViTMem | 0.0076 | 0.711

### Results on the combined LaMem and MemCat data set

Training on 10 train/test splits on the combined data set the results showed
that ViTMem performed better than the ResMem model (see Table 2). The average
across splits showed a Spearman rank correlation of 0.77 and an MSE of 0.005.

Table 2: Model performance on LaMem and MemCat combiend dataset Model | MSE Loss $\downarrow$ | Spearman $\rho$ $\uparrow$
---|---|---
ResMem | 0.009 | 0.67
ViTMem | 0.005 | 0.77

### Results on combined and cleaned data set

To assess model performance on the larger and cleaned data set, we made a
train/test split and then performed repeated k-fold cross validation with 10
train/test splits on the training set. This resulted in a mean MSE loss of
0.006 and a mean Spearman rank correlation of 0.76 (see Table 3). In order to
provide a model for the community we used the full data set to train the final
model (ViTMem Final Model), which is published on the python package index as
version 1.0.0. This was trained on the full training set and tested on its
corresponding test set. The results showed a Spearman rank correlation of 0.77
and an MSE of 0.006 (see Table 3). The train/test splits are available on
github.

Table 3: Model performance on combined and cleaned data set Model | MSE Loss $\downarrow$ | Spearman $\rho$ $\uparrow$
---|---|---
ViTMem | 0.006 | 0.76
ViTMem Final Model | 0.006 | 0.77

### Validation on independent data sets

To further validate our model, we used memorability scores from an independent
data set by Dubey and colleagues named PASCAL-S (7, 35) consisting of 852
images and cropped objects from the same images. ViTMem achieved a Spearman
correlation of 0.44 on the images and 0.21 on the objects. In comparison
ResMem achieved a correlation of 0.36 on the images and 0.14 on the objects.
Validating against the THINGS data set (15), which consists of 26,106 images
with memorability scores, achieved a Spearman rank correlation of 0.30 for
ViTMem and 0.22 for ResMem.

### Semantic level analysis

In order to better understand how the model predictions relate to the semantic
content of the images, we performed image captioning (34) on the combined
LaMem and MemCat data set and the Places205 data set (36). We extracted nouns
from the resulting image descriptions and averaged behavioral or predicted
memorability scores for each noun (37). That is, the memorability for each
image was assigned to each noun derived from the image captioning procedure.
For the combined LaMem and MemCat data set we averaged behavioral memorability
scores over nouns (see Figure 1), while for the Places205 data set we averaged
predicted memorability scores from the ViTMem model (see Figure 2). A general
interpretation of the visualizations in Figure 1 and 2 is that they appear to
reveal a dimension from nouns usually observed outdoors to more indoor related
nouns and ending with nouns related to animals, and in particular, humans.
This would appear to reflect the distributions observed in previous work (9,
15), and hence help to validate the model in terms of the image content it is
sensitive to. To further investigate how well memorability associated with
nouns were similar across the models we selected nouns occurring more than the
85th percentile in each set (654 nouns for LaMem and MemCat, 2179 nouns for
Places205), this resulted in 633 matched nouns across sets. Analysis of these
showed a Spearman ranked correlation of 0.89 and a R2 of 0.79, p<0.001 (see
Figure 3). This analysis indicates that nouns from image captioning is a
strong predictor of image memorability and that the ViTMem model is able to
generalize the importance of such aspects from the training set to a new set
of images.

Figure 1: Average behavioral image memorability scores for nouns that were
extracted from images in the LaMem and MemCat data sets. The nouns shown are
those that occurred most frequently or that are more frequent in the English
language (38). Figure 2: Average ViTMem predicted image memorability scores
for nouns that were extracted from images in the Places205 data set. The nouns
shown are those that occurred most frequently or that are more frequent in the
English language (38). Figure 3: Average memorability scores for images with
matching nouns in different data sets. The y-axis shows average predicted
memorability scores from ViTMem on the Places205 data set. The x-axis shows
average behavioral memorability scores on the combined LaMem and MemCat data
set.

## Discussion

Using vision transformers we have improved on the state-of-the-art in image
memorability prediction. Results showed that ViTMem performed equal to or
better than state-of-the-art models on LaMem, and better than ResMem on the
LaMem and MemCat hybrid data set. In addition, we assembled a new deduplicated
hybrid data set and benchmarked the ViTMem model against this before training
a final model. The model was further validated on additional data sets, and
performed better than ResMem on these as well. Finally, we ran a semantic
level analysis by using image captioning on the hybrid data set. We ranked the
behavioral memorability scores on the images, labeled with nouns extracted
from the captioning procedure. The results revealed that images labeled by
nouns related to landscapes, cities, buildings and similar, were ranked
lowest, whereas images labeled by nouns related to animate objects and food,
were ranked highest. This finding is consistent with known category effects on
memorability (3, 7, 11, 12, 15) and suggests that the labels extracted from
captioning procedure is strongly related to factors that drive memorability
for those images. Subsequently, we predicted memorability scores on images
from a novel data set (Places205), ran the image captioning procedure, and
ranked the predicted memorability scores on the images, labeled with nouns
extracted from the captioning procedure. Visual inspection of the results
revealed that the ranks were similar across samples and methods. This
impression was confirmed by a strong correlation between matching pairs of
nouns and 79${\%}$ explained variance, suggesting that ViTMem captures the
semantic content that drives memorability in images.

The use of image augmentations in training the ViTMem model in combination
with state-of-the-art performance suggest that such augmentations are not
disrupting the ability of the model to predict image memorability and hence
may further support the importance of semantic level properties in image
memorability. That is, the augmentations modify a range of low-level image
properties but mostly leave the semantic content intact.

In comparison with ResMem, which relies on a CNN-based residual neural network
architecture, ViTMem is based on vision transformers which integrate
information in a more global manner (30). As images are compositions of
several semantically identifiable objects or parts of objects, a more holistic
approach may be more apt at delineating the relative relevance of objects
given their context. That is, we speculate that a broader integration of image
features allows for a more complete evaluation of its constituent features in
relation to each other. Hence, if semantic content is important for predicting
image memorability, the model may have weighed the importance of semantic
components in relation to each other to a larger degree than models based on
CNNs.

ViTMem code and train/test sets are shared on github
(https://github.com/brainpriority/), and a python package named vitmem is
available on the python package index (see supplementary section 1 for a
tutorial). Researchers and interested parties can use the model to predict
memorability in existing or novel stimuli and employ them in research or
applied settings. The ViTMem model will allow researchers to more precisely
predict image memorability. The release of ViTMem follows up ResMem in
providing an accessible method for predicting image memorability. This is
important for studies aiming to control for how easily an image can be
remembered. This will for example allow experimental psychologists and
neuroscientists to better control their research. Similarly, educators,
advertisers and visual designers can leverage the model to improve the
memorability of their content.

Despite state-of-the-art performance in memorability prediction, improvements
may still be possible to achieve. Previous works have shown benefits of
pretraining their networks on data sets of places and objects prior to fine
tuning for memorability prediction (39). Moreover, ViTMem do not take image
captioning into account, which have been successfully done with CNNs (21, 22).
Hence there is potentially more to be gained from incorporating image
semantics and/or pretraining on data sets of objects and places. In addition,
ViTMem is only based on the "base" configuration of the available ViT models.
Model performance may still increase by adopting the “large” or “huge”
configurations of the model.

We conclude that ViTMem can be used to predict memorability for images at a
level that is equal to or better than state-of-the-art models, and we propose
that vision transformers provide a new step forward in the computational
prediction of image memorability.

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*format=largeformat

##  1 How to use the vitmem python package

Python needs to be installed on a computer before pip can be used to install
the vitmem package.

To install vitmem from a command prompt run:

    pip install vitmem

To predict image memorability for an image named "image.jpg", run the
following in a python interpreter:

    from vitmem import ViTMem
    model = ViTMem()
    memorability = model("image.jpg")
    print(f"Predicted memorability: {memorability}")


$\begin{split}\widetilde{J}_{2}&\leq\mathbb{E}\Bigg{[}\left|\sum_{k=0}^{N/2-1}\operatorname{e}^{2\gamma
t_{2k}}h\left\|\widehat{Y}_{2k}^{f}-\widehat{Y}_{2k}^{c}\right\|\left\|D^{2}a(\widehat{Y}_{2k}^{f})\big{(}Da(\widehat{Y}_{2k}^{f})\Delta
Z_{2k}\big{)}\Delta W_{2k}\right\|\right|^{p/2}\Bigg{]}\\\ &\leq
d^{5p/4}K_{b}^{p}\mathbb{E}\Bigg{[}\left|\sup_{0\leq n\leq
N/2}\operatorname{e}^{\gamma
t_{2n}}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\right|^{p/2}\left|\sum_{k=0}^{N/2-1}\operatorname{e}^{\gamma
t_{2k}}h\left\|\Delta Z_{2k}\right\|\left\|\Delta
W_{2k}\right\|\right|^{p/2}\Bigg{]}\\\
&\leq\frac{1}{8\zeta}\mathbb{E}\Bigg{[}\sup_{0\leq n\leq
N/2}\operatorname{e}^{\gamma
pt_{2n}}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\Bigg{]}+\zeta
C_{\widetilde{J}_{2}}^{p}p^{p}h^{2p}\operatorname{e}^{\gamma pT}.\end{split}$

Combining all the estimates together, there exists a constant
$\widetilde{C}_{3}>0$, independent of $T$ and $p$, such that

$\mathbb{E}\Bigg{[}\sup_{0\leq n\leq N/2}\operatorname{e}^{\gamma
pt_{2n}}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\Bigg{]}\leq\widetilde{C}_{3}^{p}p^{3p/2}h^{3p/2}\operatorname{e}^{\gamma
pT}.$

As the above inequality holds for any $T>0$, we have

$\sup_{0\leq n\leq
N/2}\mathbb{E}\Bigg{[}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\Bigg{]}\leq\widetilde{C}_{3}^{p}p^{3p/2}h^{3p/2}.$
(52)

Choosing a constant
$\overline{C}_{1}\coloneqq\max\big{(}\widetilde{C}_{1},\;\widetilde{C}_{2},\;\widetilde{C}_{3}\big{)}$,
for all sufficiently small $0<h<\kappa_{1}$, we have

$\sup_{0\leq n\leq
N/2}\mathbb{E}\Bigg{[}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\Bigg{]}\leq\overline{C}_{1}^{p}\min\Big{(}p^{p/2}h^{p/2},\;p^{p}h^{p},\;p^{3p/2}h^{3p/2}\Big{)}.$
(53)

Similarly, using equations (7) and (9), we can write the difference between
the fine and the coarse trajectories at the odd time-step as

$\begin{split}\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n+1}^{c}&=(1-2Sh)(\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c})+h(a(\widehat{Y}_{2n}^{f})-a(\widehat{Y}_{2n}^{c}))+\widehat{I}_{2n},\end{split}$

where

$\widehat{I}_{2n}\coloneqq(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}+\frac{h^{2}}{2}(\mathcal{A}a(\widehat{Y}_{2n}^{f})-\mathcal{A}a(\widehat{Y}_{2n}^{c})).$

Taking the Euclidean norm of the difference and squaring it, using Young’s
inequality, and Assumptions 1, 2, 3, and for all sufficiently small $h>0$, we
have

$\begin{split}\left\|\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n+1}^{c}\right\|^{2}&\leq\Big{\\{}2-6Sh+(2\lambda+dK_{b}^{2})h+Ch^{2}\Big{\\}}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+3\left\|\widehat{I}_{2n}\right\|^{2}\\\
~{}&\leq
C\Big{(}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\left\|\Delta
Z_{2n}\right\|^{2}\Big{)}.\end{split}$

Raising both sides of the above inequality to the power $p/2$, and taking the
expectation, we obtain

$\begin{split}\mathbb{E}\Big{[}\left\|\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n+1}^{c}\right\|^{p}\Big{]}&\leq
2^{p/2-1}C^{p/2}\Big{(}\mathbb{E}\Big{[}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\Big{]}+\mathbb{E}\Big{[}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{p}\left\|\Delta
Z_{2n}\right\|^{p}\Big{]}\Big{)}.\end{split}$

Taking the supremum on both sides of the above inequality, there exist
constants $\kappa_{4},\kappa_{5}>0$ independent of $T$ and $p$, for all
$0<h<\min(\kappa_{3},\kappa_{4}/\sqrt[3]{p})$,

$\sup_{0\leq n\leq
N}\mathbb{E}\Big{[}\left\|\widehat{Y}_{t_{n}}^{f}-\widehat{Y}_{t_{n}}^{c}\right\|^{p}\Big{]}\leq\kappa_{5}\min\Big{(}p^{p/2}h^{p/2},\;p^{p}h^{p},\;p^{3p/2}h^{3p/2}\Big{)}.$

#### Proof of Theorem 4

For the sake of brevity, we show the Radon–Nikodym derivative bound only for
the fine trajectory. The proof argument can be extended similarly for the
coarse trajectory. Recall that the $p-$th power of the Radon–Nikodym
derivative for the fine trajectory is given by

$\begin{split}\mathbb{E}\bigg{[}\left|R_{T}^{f}\right|^{p}\bigg{]}&=\mathbb{E}\Bigg{[}\exp\Bigg{(}-pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{1,n}\right\rangle+\sqrt{3}pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{2,n}\right\rangle\\\
&\qquad-2pS^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\\\
~{}&=\mathbb{E}\Bigg{[}\exp\Bigg{(}-pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{1,n}\right\rangle+\sqrt{3}pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{2,n}\right\rangle\\\
&\quad-4p^{2}S^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}+2(2p-1)pS^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}.\end{split}$

Using Hölder’s inequality, we can write

$\mathbb{E}\bigg{[}\left|R_{T}^{f}\right|^{p}\bigg{]}\leq
J_{1}^{1/2}J_{2}^{1/2},$

where

$\begin{split}J_{1}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}4(2p-1)pS^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]},\\\
J_{2}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}-2pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{1,n}\right\rangle+2\sqrt{3}pS\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{2,n}\right\rangle\\\
&\quad-8p^{2}S^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\leq
1.\end{split}$

since the term $J_{2}$ is a super-martingale. Using Hölder’s inequality, the
term $J_{1}$ can be bounded as

$\begin{split}J_{1}&\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}h\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}=\mathbb{E}\Bigg{[}\prod_{n=0}^{N-1}\exp\Bigg{(}8p^{2}S^{2}h\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\\\
~{}&\leq\prod_{n=0}^{N-1}\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{1/N}=\prod_{n=0}^{N/2-1}(\widehat{J}_{2n+2})^{1/N}\prod_{n=0}^{N/2-1}(\widehat{J}_{2n+1})^{1/N},\end{split}$

where

$\begin{split}\widehat{J}_{2n+2}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]},\\\
\widehat{J}_{2n+1}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n+1}^{c}\right\|^{2}\Bigg{)}\Bigg{]}.\end{split}$
(54)

To bound the term $\widehat{J}_{2n+2}$, we define the set

$\Theta_{2n}\coloneqq\Bigg{\\{}\omega\in\Omega\;\bigg{|}\;\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\geq\nu_{1}\left|\log
h\right|\Bigg{\\}},$ (55)

where $\nu_{1}>0$ is a constant independent of $T$. Using Markov’s inequality,
and the result from Theorem 3, we obtain

$\begin{split}\mathbb{P}\big{(}\Theta_{2n}\big{)}\leq\frac{\mathbb{E}\Bigg{[}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{q}\Bigg{]}}{\nu_{1}^{q}\left|\log
h\right|^{q}}\leq\nu_{2}^{q}\nu_{1}^{-q}q^{q/2}h^{q/2}\left|\log
h\right|^{-q},\end{split}$

where $\nu_{1},\nu_{2}>0$ are constants independent of $T$. Using the law of
total expectation, we obtain

$\begin{split}&\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}\\\
=&\underbrace{\mathbb{E}\Bigg{[}\mathbbm{1}_{\Theta_{2n}}\exp\bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}}_{\eqqcolon\widehat{I}_{1}}+\underbrace{\mathbb{E}\Bigg{[}\mathbbm{1}_{\Theta_{2n}^{\mathsf{c}}}\exp\bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}}_{\eqqcolon\widehat{I}_{2}}.\end{split}$

We begin the analysis by bounding the term $\widehat{I}_{1}$. For all
sufficiently small $h>0$, using Hölder’s inequality with the Hölder exponent
$\bar{p}\coloneqq h^{-1}\left|\log h\right|\implies\bar{q}\coloneqq\left|\log
h\right|/\big{(}\left|\log h\right|-h\big{)}$, we obtain

$\begin{split}\widehat{I}_{1}&=\mathbb{E}\Bigg{[}\mathbbm{1}_{\Theta_{2n}}\exp\bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}\\\
&\leq\mathbb{E}\Big{[}\mathbbm{1}_{\Theta_{2n}}^{\bar{p}}\Big{]}^{\frac{1}{\bar{p}}}\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\bar{q}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}^{\frac{1}{\bar{q}}}\\\
&\leq\bigg{(}\nu_{2}^{q}\nu_{1}^{-q}q^{q/2}h^{q/2}\left|\log
h\right|^{-q}\bigg{)}^{\frac{h}{\left|\log
h\right|}}\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}^{1-\frac{h}{\left|\log
h\right|}}.\end{split}$

Using equation (38), and Young’s inequality, we have

$\begin{split}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}&\leq\Big{(}1-4\big{(}2S-\lambda\big{)}h+Ch\left|\log
h\right|^{-1}\Big{)}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\\\
&+C\Bigg{\\{}h^{3}\left|\log
h\right|\bigg{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\\\
&+h\left|\log h\right|\left\|\Delta W_{2n}\right\|^{2}+\left\|\Delta
Z_{2n}\right\|^{2}+\left\|\Delta Z_{2n+1}\right\|^{2}\Bigg{\\}}\\\
&+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\\\
&+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle\\\
&+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle\\\
&-2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},Da(\widehat{Y}_{2n}^{c})\Delta
W_{2n}\right\rangle,\end{split}$ (56)

where $C>0$ is a constant that depends only on $S,d$ and the constants from
Assumptions 1, 2, and 3. Note that the constant $C$ may change from line to
line in the proof argument.

Note that for all $0<h<w(2)/2,\;\;\frac{\left|\log h\right|}{\left|\log
h\right|-h}\leq 2$, where $w(x)$ denotes the product logarithm function. Using
equation (56), the tower property of expectations, for all sufficiently small
$h>0$, there exists a $\gamma_{1}\in(0,2S-\lambda)$, such that the following
inequality holds

$\begin{split}&\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}\\\
~{}\leq&\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-2\gamma_{1}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\\\
&\quad+C\Bigg{\\{}h^{3}\left|\log
h\right|\bigg{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}+h\left|\log
h\right|\left\|\Delta W_{2n}\right\|^{2}+\left\|\Delta
Z_{2n}\right\|^{2}\Bigg{\\}}\\\ &\quad+2\frac{\left|\log h\right|}{\left|\log
h\right|-h}(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\\\ &\quad-2h\frac{\left|\log h\right|}{\left|\log
h\right|-h}(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},Da(\widehat{Y}_{2n}^{c})\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\times\widetilde{J}_{1}\Bigg{]},\end{split}$

where $\widetilde{J}_{1}$ is defined as

$\begin{split}\widetilde{J}_{1}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}\\\
&\qquad\qquad\bigg{(}2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle\\\
&\qquad\qquad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle\\\ &\qquad\qquad+C\left\|\Delta
Z_{2n+1}\right\|^{2}\bigg{)}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}.\end{split}$

Using Hölder’s inequality conditioned on the filtration
$\mathcal{F}_{t_{2n+1}}$ generated upto time $t_{2n+1}$, we get

$\displaystyle\widetilde{J}_{1}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}48p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\times$
$\displaystyle\quad\mathbb{E}\Bigg{[}\exp\Bigg{(}48p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\times$
$\displaystyle\quad\mathbb{E}\Bigg{[}\exp\Bigg{(}24p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}C\left\|\Delta
Z_{2n+1}\right\|^{2}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}.$

We define the random variable
$\chi_{n}\sim\mathcal{N}(0,\operatorname{I_{d}})$ for all $n=0,\ldots,N$.
Recall that the random variable $\xi\coloneqq\sum_{k=1}^{d}(\chi_{n,k})^{2}$
is chi-squared distributed. The moment generating function for the chi-squared
distribution is given by

$\mathbb{E}\big{[}\exp(\theta\xi)\big{]}=(1-2\theta)^{-d/2},\qquad\forall\theta<\frac{1}{2}.$
(57)

Using equation (57), for all sufficiently small $h>0$, the following
inequality holds

$\begin{split}&\mathbb{E}\Bigg{[}\exp\bigg{(}24p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}C\left\|\Delta
Z_{2n+1}\right\|^{2}\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\\\
=&\mathbb{E}\Bigg{[}\exp\bigg{(}16p^{2}S^{2}TCh^{3}\left\|\chi_{2n+1}\right\|^{2}\bigg{)}\Bigg{]}^{1/3}\leq\frac{1}{\big{(}1-32p^{2}S^{2}TCh^{3}\big{)}^{d/6}}\leq\exp(CTh^{3}),\end{split}$

for all $T\leq C_{0}h^{-3}$, where $C_{0}$ is a constant that is independent
of $h$. Using Assumptions 1, and 2, and the moment generating function for the
standard Normal distribution, we have

$\begin{split}&\mathbb{E}\Bigg{[}\exp\bigg{(}48p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}(1-4Sh+2S^{2}h^{2})\\\
&\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\\\
=&\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{48}{\sqrt{3}}p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}h^{3/2}(1-4Sh+2S^{2}h^{2})\\\
&\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\chi_{2n+1}\right\rangle\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\leq\exp\bigg{(}CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}.\end{split}$

Similarly, we have

$\displaystyle\mathbb{E}\Bigg{[}\exp\bigg{(}48p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n+1}}\Bigg{]}^{1/3}\leq\exp\bigg{(}CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}.$

Combining the above estimates, we obtain

$\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]}$
$\displaystyle~{}\leq$
$\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-2\gamma_{1}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch^{3}\left|\log
h\right|\bigg{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\bigg{)}\Bigg{)}\times$
$\displaystyle\quad\exp\Bigg{(}CTh^{3}+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\times\widetilde{J}_{2}\Bigg{]},$

where $\widetilde{J}_{2}$ is defined as

$\begin{split}\widetilde{J}_{2}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}\bigg{(}Ch\left|\log h\right|\left\|\Delta
W_{2n}\right\|^{2}+C\left\|\Delta Z_{2n}\right\|^{2}\\\
&\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\\\
&\quad-2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},Da(\widehat{Y}_{2n}^{c})\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}.\end{split}$

Using Hölder’s inequality conditioned on the filtration $\mathcal{F}_{t_{2n}}$
generated until time $t_{2n}$, we obtain

$\displaystyle\widetilde{J}_{2}$
$\displaystyle\leq\mathbb{E}\bigg{[}\exp\bigg{(}64p^{2}S^{2}TCh\left|\log
h\right|\left\|\Delta
W_{2n}\right\|^{2}\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\bigg{]}^{1/4}\times\mathbb{E}\bigg{[}\exp\bigg{(}64p^{2}S^{2}TC\left\|\Delta
Z_{2n}\right\|^{2}\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\bigg{]}^{1/4}$
$\displaystyle\times\mathbb{E}\bigg{[}\exp\bigg{(}64p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log h\right|-h}(1-Sh)(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\bigg{]}^{1/4}\times$
$\displaystyle\mathbb{E}\bigg{[}\exp\bigg{(}-64p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},Da(\widehat{Y}_{2n}^{c})\Delta
W_{2n}\right\rangle\bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\bigg{]}^{1/4}.$

Using a similar argument as before and combining all the estimates, we obtain

$\begin{split}&\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\\\
~{}\leq&\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-2\gamma_{1}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch^{3}\left|\log
h\right|\bigg{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\bigg{)}\Bigg{)}\times\\\
&\qquad\exp\Bigg{(}CTh^{2}\left|\log
h\right|+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}.\end{split}$

Note that for all $1\leq
T\leq\gamma_{1}h^{-2}/\big{(}2\max\big{(}1,\;C\big{)}\big{)}$, and for all
sufficiently small $h>0$, there exists a $\widehat{\gamma}_{1}>0$ such that
the following inequality holds

$\begin{split}&\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\\\
~{}\leq&\exp\big{(}CTh^{2}\left|\log h\right|\big{)}\\\
&\qquad\qquad\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-2\widehat{\gamma}_{1}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch^{3}\left|\log
h\right|\bigg{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\bigg{)}\Bigg{)}\Bigg{]}.\end{split}$

Using Hölder’s inequality once again, we obtain

$\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]}$
$\displaystyle\leq$ $\displaystyle\exp\big{(}CTh^{2}\left|\log
h\right|\big{)}\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\Bigg{]}^{1-2\widehat{\gamma}_{1}h}$
$\displaystyle\qquad\qquad\mathbb{E}\Bigg{[}\exp\bigg{(}CTh^{2}\left|\log
h\right|\bigg{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\bigg{)}\Bigg{]}^{2\widehat{\gamma}_{1}h}.$

Using Theorem 1, Hölder’s inequality, Taylor series expansion of the
exponential function, Stirling’s approximation, and Fatou’s lemma, the
following inequality holds

$\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}CTh^{2}\left|\log
h\right|\bigg{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}\Bigg{)}\Bigg{]}^{2\widehat{\gamma}_{1}h}$
$\displaystyle~{}\leq$
$\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}CTh^{2}\left|\log
h\right|\left\|\widehat{Y}_{2n}^{f}\right\|^{2}\Bigg{)}\Bigg{]}^{\widehat{\gamma}_{1}h}\mathbb{E}\Bigg{[}\exp\Bigg{(}CTh^{2}\left|\log
h\right|\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{\widehat{\gamma}_{1}h}$
$\displaystyle~{}=$
$\displaystyle\left(1+\sum_{k=1}^{\infty}\frac{(CTh^{2}\left|\log
h\right|)^{k}\mathbb{E}\bigg{[}\left\|\widehat{Y}_{2n}^{f}\right\|^{2k}\bigg{]}}{k!}\right)^{\widehat{\gamma}_{1}h}\left(1+\sum_{k=1}^{\infty}\frac{(CTh^{2}\left|\log
h\right|)^{k}\mathbb{E}\bigg{[}\left\|\widehat{Y}_{2n}^{c}\right\|^{2k}\bigg{]}}{k!}\right)^{\widehat{\gamma}_{1}h}$
$\displaystyle~{}\leq$
$\displaystyle\left(1+\sum_{k=1}^{\infty}\frac{(CT\operatorname{e}h^{2}\left|\log
h\right|)^{k}}{\sqrt{2\pi
k}}\right)^{\widehat{\gamma}_{1}h}\left(1+\sum_{k=1}^{\infty}\frac{(CT\operatorname{e}h^{2}\left|\log
h\right|)^{k}}{\sqrt{2\pi k}}\right)^{\widehat{\gamma}_{1}h}\leq
2^{2\widehat{\gamma}_{1}h},$

for all $T\leq C_{0}h^{-2}\left|\log h\right|^{-1}$. Combining all the above
estimates, $\widehat{I}_{1}$ yields the upper bound

$\begin{split}\widehat{I}_{1}&\leq\Big{(}\nu_{2}^{q}\nu_{1}^{-q}q^{q/2}h^{q/2}\left|\log
h\right|^{-q}\Big{)}^{\frac{h}{\left|\log
h\right|}}\mathbb{E}\Bigg{[}\exp\bigg{(}8p^{2}S^{2}T\frac{\left|\log
h\right|}{\left|\log
h\right|-h}\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\bigg{)}\Bigg{]}^{1-\frac{h}{\left|\log
h\right|}}\\\ ~{}&\leq\Big{(}2^{2\widehat{\gamma}_{1}h}\exp(CTh^{2}\left|\log
h\right|)\Big{)}^{1-\frac{h}{\left|\log
h\right|}}\Big{(}\nu_{2}^{q}\nu_{1}^{-q}q^{q/2}h^{q/2}\left|\log
h\right|^{-q}\Big{)}^{\frac{h}{\left|\log h\right|}}\times\\\
&\qquad\qquad\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{(1-2\widehat{\gamma}_{1}h)\Big{(}1-\frac{h}{\left|\log
h\right|}\Big{)}}.\end{split}$

Note that $\exp(x^{2})\geq 1,\;\forall x\in\mathbb{R}$. For
$q=h^{-1}\left|\log h\right|^{2}$, for all sufficiently large
$\nu_{1}>\nu_{2}\operatorname{e}$, for all $T\leq C_{0}h^{-2}\left|\log
h\right|^{-1}$, and for all sufficiently small $h>0$, we have

$\begin{split}\widehat{I}_{1}&\leq\Big{(}2^{2\widehat{\gamma}_{1}h}\exp(C)\Big{)}^{1-\frac{h}{\left|\log
h\right|}}\Bigg{(}\frac{\nu_{2}}{\nu_{1}}\Bigg{)}^{\left|\log
h\right|}\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{1-2\widehat{\gamma}_{1}h}\\\
~{}&\leq\widehat{C}_{1}h\Big{(}\widehat{J}_{2n}\Big{)}^{1-2\widehat{\gamma}_{1}h},\end{split}$

where $\widehat{C}_{1}\in(0,1)$ is a constant independent of $T$ and $h$.

We now bound the term $\widehat{I}_{2}$ similarly. Using Assumptions 1 and 2,
and the mean value theorem, we have

$\begin{split}a(\widehat{Y}_{2n+1}^{f})-a(\widehat{Y}_{2n}^{f})=Da(\widehat{Y}_{2n}^{f})(\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n}^{f})+R_{2},\end{split}$

where $\left\|R_{2}\right\|\leq
d^{3/2}K_{b}\left\|\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n}^{f}\right\|^{2}$.

Using equation (39), Hölder’s inequality, and Young’s inequality, there exists
$\gamma_{2}\in(0,2S-\lambda)$, for all sufficiently small $h>0$, such that

$\displaystyle\allowdisplaybreaks\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}$
$\displaystyle\leq(1-4(2S-\lambda)h+Ch\left|\log
h\right|^{-1})\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+C\bigg{\\{}h^{3}\left|\log
h\right|\bigg{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}$
$\displaystyle\quad+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}+h^{2}\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n+1}\right\|^{2}+\left\|\Delta
Z_{2n}\right\|^{2}\bigg{\\}}$
$\displaystyle\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle$
$\displaystyle\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},R_{2}\right\rangle$
$\displaystyle\quad-h^{2}(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},\Delta
a(\widehat{Y}_{2n}^{f})\right\rangle$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle$
$\displaystyle\leq(1-4\gamma_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+C\bigg{\\{}h^{3}\left|\log
h\right|\bigg{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}$
$\displaystyle\quad+h^{2}\left\|\Delta W_{2n}\right\|^{2}+\left\|\Delta
Z_{2n+1}\right\|^{2}+\left\|\Delta Z_{2n}\right\|^{2}\bigg{\\}}$
$\displaystyle\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle$
$\displaystyle\quad+Ch\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\bigg{(}h^{2}\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}+\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n}\right\|^{2}\bigg{)}$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle.$

Using the above inequality, equation (55), and Young’s inequality, the term
$\widehat{I}_{2}$ yields

$\displaystyle\widehat{I}_{2}$
$\displaystyle=\mathbb{E}\Bigg{[}\mathbbm{1}_{\Theta_{2n}^{\mathsf{c}}}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+2}^{f}-\widehat{Y}_{2n+2}^{c}\right\|^{2}\Bigg{)}\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\big{(}\mathbbm{1}_{\Theta_{2n}^{\mathsf{c}}}\big{)}^{2}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-4\gamma_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}$
$\displaystyle\quad+C\bigg{\\{}h^{3}\left|\log
h\right|\big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\big{)}+h^{2}\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n+1}\right\|^{2}+\left\|\Delta
Z_{2n}\right\|^{2}\bigg{\\}}$
$\displaystyle\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle$
$\displaystyle\quad+Ch\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\bigg{\\{}h^{2}\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}+\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n}\right\|^{2}\bigg{\\}}$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n+1}^{f})-Da(\widehat{Y}_{2n}^{f}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n+1}\right\rangle$
$\displaystyle\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\Bigg{]}.$

Following a similar argument as before, using Hölder’s inequality conditioned
on the filtration $\mathcal{F}_{t_{2n+1}}$ generated until time $t_{2n+1}$,
using equation (57), and for all sufficiently small $h>0$, we obtain

$\displaystyle\widehat{I}_{2}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-4\gamma_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}$
$\displaystyle\quad+C\bigg{\\{}h^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}+h^{2}\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n}\right\|^{2}\bigg{\\}}$
$\displaystyle\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle$
$\displaystyle\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\times$
$\displaystyle\qquad\exp\Bigg{(}C\nu_{1}Th^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}+C\nu_{1}Th\left|\log
h\right|\left\|\Delta Z_{2n}\right\|^{2}$
$\displaystyle\qquad\qquad+CTh\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\left\|\Delta
W_{2n}\right\|^{2}\Bigg{)}\times\exp\Bigg{(}CTh^{3}+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}.$

Similarly, using the tower property, taking the expectation conditioned on the
filtration $\mathcal{F}_{t_{2n}}$ generated upto time $t_{2n}$, we obtain

$\displaystyle\widehat{I}_{2}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-4\gamma_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\bigg{)}\Bigg{)}$
$\displaystyle\qquad\qquad\times\exp\Bigg{(}C\nu_{1}Th^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\Bigg{)}$
$\displaystyle\qquad\qquad\times\exp\Bigg{(}CTh^{3}+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\times\widetilde{J}_{3}\Bigg{]},$

where $\widetilde{J}_{3}$ is defined as

$\begin{split}\widetilde{J}_{3}&\coloneqq\mathbb{E}\Bigg{[}\exp\Bigg{(}C\nu_{1}Th\left|\log
h\right|\left\|\Delta
Z_{2n}\right\|^{2}+CTh\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\left\|\Delta
W_{2n}\right\|^{2}\Bigg{)}\\\
&\quad\times\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}C\bigg{\\{}h^{2}\left\|\Delta
W_{2n}\right\|^{2}+\left\|\Delta Z_{2n}\right\|^{2}\bigg{\\}}\\\
&\quad+2(1-Sh)(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\\\
&\quad+2h(1-4Sh+2S^{2}h^{2})\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}.\end{split}$

Using Hölder’s inequality, we obtain

$\displaystyle\widetilde{J}_{3}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}C\nu_{1}Th\left|\log
h\right|\left\|\Delta
Z_{2n}\right\|^{2}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}$
$\displaystyle\quad\times\underbrace{\mathbb{E}\Bigg{[}\exp\Bigg{(}CTh\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\left\|\Delta
W_{2n}\right\|^{2}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}}_{\widetilde{J}_{4}}$
$\displaystyle\quad\times\mathbb{E}\Bigg{[}\exp\Bigg{(}CTh^{2}\left\|\Delta
W_{2n}\right\|^{2}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}\times\mathbb{E}\Bigg{[}\exp\Bigg{(}CT\left\|\Delta
Z_{2n}\right\|^{2}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}$
$\displaystyle\quad\mathbb{E}\Bigg{[}\exp\Bigg{(}96p^{2}S^{2}T(1-Sh)(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}$
$\displaystyle\quad\mathbb{E}\Bigg{[}\exp\Bigg{(}96p^{2}S^{2}Th(1-4Sh+2S^{2}h^{2})$
$\displaystyle\qquad\qquad\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
W_{2n}\right\rangle\bigg{)}\Bigg{)}\;\bigg{|}\;\mathcal{F}_{t_{2n}}\Bigg{]}^{1/6}.$

Note that for all $T\leq C_{0}h^{-2}\left|\log h\right|^{-1}$, and for all
sufficiently small $h>0$, using Young’s inequality, we obtain

$\begin{split}\widetilde{J}_{4}&\leq\exp\Bigg{(}CTh^{2}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|\Bigg{)}\leq\exp\Bigg{(}CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch\Bigg{)}.\end{split}$

Combining all the above estimates, using equation (57), we obtain

$\begin{split}\widetilde{J}_{3}\leq\exp\Bigg{(}C\nu_{1}Th^{4}\left|\log
h\right|+CTh^{3}+Ch+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}.\end{split}$

Note that for all $1\leq T\leq\gamma_{2}h^{-2}/(2\max(1,\;C))$, and all
sufficiently small $h>0$, there exists a $\widehat{\gamma}_{2}>0$ such that
the following inequality holds

$\begin{split}\widehat{I}_{2}&\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-4\gamma_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+Ch^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\bigg{)}\Bigg{)}\\\
&\quad\exp\Bigg{(}C\nu_{1}Th^{4}\left|\log
h\right|+CTh^{3}+Ch\Bigg{)}\exp\Bigg{(}C\nu_{1}Th^{3}\left|\log
h\right|\Big{(}1+\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\Bigg{)}\Bigg{]}\\\
~{}&\leq\exp\Big{(}C\nu_{1}\big{(}Th^{3}\left|\log
h\right|+h\big{)}\Big{)}\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T(1-4\widehat{\gamma}_{2}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\\\
&\qquad\qquad\qquad\qquad\qquad+C\nu_{1}Th^{3}\left|\log
h\right|\Big{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\Bigg{)}\Bigg{]}.\end{split}$

For all $T\leq C_{0}h^{-2}\left|\log h\right|^{-1}$, and for all sufficiently
small $h>0$, using Hölder’s inequality, we have

$\begin{split}\widehat{I}_{2}&\leq\exp\big{(}C\nu_{1}h\big{)}\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{1-4\widehat{\gamma}_{2}h}\times\\\
&\qquad\qquad\mathbb{E}\Bigg{[}\exp\Bigg{(}C\nu_{1}Th^{2}\left|\log
h\right|\Big{(}\left\|\widehat{Y}_{2n}^{f}\right\|^{2}+\left\|\widehat{Y}_{2n}^{c}\right\|^{2}\Big{)}\Bigg{)}\Bigg{]}^{4\widehat{\gamma}_{2}h}\\\
~{}&\leq
2^{4\widehat{\gamma}_{2}h}\exp(\widehat{C}_{2}\nu_{1}h)\Big{(}\widehat{J}_{2n}\Big{)}^{1-4\widehat{\gamma}_{2}h},\end{split}$

where $\widehat{C}_{2}>0$ is a constant that is independent of $T$ and $h$.
Combining the terms $\widehat{I}_{1}$ and $\widehat{I}_{2}$, we obtain

$\begin{split}\widehat{J}_{2n+2}\leq\widehat{C}_{1}h\Big{(}\widehat{J}_{2n}\Big{)}^{1-2\widehat{\gamma}_{1}h}+2^{4\widehat{\gamma}_{2}h}\exp\big{(}\widehat{C}_{2}\nu_{1}h\big{)}\Big{(}\widehat{J}_{2n}\Big{)}^{1-4\widehat{\gamma}_{2}h}.\end{split}$

Note that for
$\widetilde{\gamma}\coloneqq\min\big{(}2\widehat{\gamma}_{1},4\widehat{\gamma}_{2}\big{)}$,
the following inequality holds

$\begin{split}\widehat{J}_{2n+2}\leq\Big{(}\widehat{C}_{1}h+2^{4\widehat{\gamma}_{2}h}\exp\big{(}\widehat{C}_{2}\nu_{1}h\big{)}\Big{)}\Big{(}\widehat{J}_{2n}\Big{)}^{1-\widetilde{\gamma}h},\end{split}$

where $\widehat{J}_{2n}$ is given by equation (54). By recursively bounding
the term $\widehat{J}_{2n}$ using a similar argument as above, by defining the
sets $\Theta_{2k}$ for all $k=0,\ldots,n-1$, we obtain

$\begin{split}\widehat{J}_{2n+2}&\leq\Big{(}\widehat{C}_{1}h+2^{4\widehat{\gamma}_{2}h}\exp\big{(}\widehat{C}_{2}\nu_{1}h\big{)}\Big{)}^{\sum_{k=0}^{n}(1-\widetilde{\gamma}h)^{k}}\leq\Big{(}\widehat{C}_{1}h+2^{4\widehat{\gamma}_{2}h}\exp\big{(}\widehat{C}_{2}\nu_{1}h\big{)}\Big{)}^{\frac{1}{\widetilde{\gamma}h}}\\\
~{}&\leq
2^{4\widehat{\gamma}_{2}/\widetilde{\gamma}}\exp\big{(}\widehat{C}_{2}\nu_{1}/\widetilde{\gamma}\big{)}\Big{(}2^{-4\widehat{\gamma}_{2}h}\widehat{C}_{1}h\exp\big{(}-\widehat{C}_{2}\nu_{1}h\big{)}+1\Big{)}^{\frac{1}{\widetilde{\gamma}h}}.\end{split}$

Note that $\operatorname{e}^{-cx}\leq 2^{-cx}\leq 1-cx/2$ for all $x\in[0,1]$.
Using this, we obtain

$\begin{split}\widehat{J}_{2n+2}&\leq
2^{4\widehat{\gamma}_{2}/\widetilde{\gamma}}\exp\big{(}\widehat{C}_{2}\nu_{1}/\widetilde{\gamma}\big{)}\Big{(}2^{-4\widehat{\gamma}_{2}h}\widehat{C}_{1}h\exp\big{(}-\widehat{C}_{2}\nu_{1}h\big{)}+1\Big{)}^{\frac{1}{\widetilde{\gamma}h}}\\\
~{}&\leq
2^{4\widehat{\gamma}_{2}/\widetilde{\gamma}}\exp\big{(}\widehat{C}_{2}\nu_{1}/\widetilde{\gamma}\big{)}\Big{(}\widehat{C}_{1}h\Big{(}1-2\widehat{\gamma}_{2}h\Big{)}\Big{(}1-\frac{\widehat{C}_{2}\nu_{1}}{2}h\Big{)}+1\Big{)}^{\frac{1}{\widetilde{\gamma}h}}\\\
~{}&\leq
2^{4\widehat{\gamma}_{2}/\widetilde{\gamma}}\exp\big{(}\widehat{C}_{2}\nu_{1}/\widetilde{\gamma}\big{)}\Big{(}1+\widehat{C}_{3}h\Big{)}^{\frac{1}{\widetilde{\gamma}h}}\\\
~{}&\leq
2^{4\widehat{\gamma}_{2}/\widetilde{\gamma}}\exp\big{(}\widehat{C}_{2}\nu_{1}/\widetilde{\gamma}\big{)}\exp\big{(}\widehat{C}_{3}h\Big{)}^{\frac{1}{\widetilde{\gamma}h}}\leq\eta_{1},\end{split}$

where $\eta_{1}>0$ is a constant that is independent of $T$ and $h$, and
$\widehat{C}_{3}>0$ is a constant that is independent of $h$.

Using Assumptions 1, 3, 2, and equations (7) and (9), and Young’s inequality,
there exists a $\gamma\in(0,2S-\lambda)$ such that the following inequality
holds

$\begin{split}\widehat{J}_{2n+1}&=\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n+1}^{f}-\widehat{Y}_{2n+1}^{c}\right\|^{2}\Bigg{)}\Bigg{]}\\\
~{}&\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\bigg{(}(1-2\gamma
h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\bigg{)}+C\left\|\Delta
Z_{2n}\right\|^{2}\\\
&\qquad+2(1-2Sh)\left\langle\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c},(Da(\widehat{Y}_{2n}^{f})-Da(\widehat{Y}_{2n}^{c}))\Delta
Z_{2n}\right\rangle\Bigg{)}\Bigg{]}.\end{split}$

Using Hölder’s inequality conditioned on the filtration $\mathcal{F}_{t_{2n}}$
generated until time $t_{2n}$, and using a similar argument as before, the
following inequality holds

$\begin{split}\widehat{J}_{2n+1}&\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T(1-2\gamma
h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\exp\Bigg{(}CTh^{3}+CT^{2}h^{3}\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{)}\Bigg{]}.\end{split}$

For all $1\leq T\leq\gamma h^{-2}/(2\max(1,C))$, and all sufficiently small
$h>0$, there exists $\widehat{\gamma}>0$ such that, using Hölder’s inequality,
the following inequality holds

$\begin{split}\widehat{J}_{2n+1}&\leq\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T(1-2\widehat{\gamma}h)\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}+CTh^{3}\Bigg{)}\Bigg{]}\\\
&\leq\exp(CTh^{2})\mathbb{E}\Bigg{[}\exp\Bigg{(}8p^{2}S^{2}T\left\|\widehat{Y}_{2n}^{f}-\widehat{Y}_{2n}^{c}\right\|^{2}\Bigg{)}\Bigg{]}^{1-2\widehat{\gamma}h}.\end{split}$

For all $T\leq C_{0}h^{-2}\left|\log h\right|^{-1}$, and all sufficiently
small $h>0$, we then obtain

$\begin{split}\widehat{J}_{2n+1}&\leq\exp(CTh^{2})\eta_{1}^{1-2\widehat{\gamma}h}\leq\eta_{2},\end{split}$

where $\eta_{2}>0$ is a constant that is independent of $T$ and $h$.

#### Proof of Theorem 5

Following a similar argument as in [9], using Hölder’s inequality, and
Jensen’s inequality, we have

$\begin{split}&\mathbb{E}\Bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\right|^{p}\Bigg{]}\\\
&\quad=\mathbb{E}\Bigg{[}\Bigg{|}\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{f})+\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})+\varPhi(\widehat{Y}_{T}^{c})-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\Bigg{|}^{p}\Bigg{]}\\\
~{}&\quad\leq
3^{p-1}\Bigg{\\{}\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})\right|^{p}\bigg{]}}_{\widetilde{J}_{1}}+\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})\right|^{2p}\bigg{]}^{1/2}\mathbb{E}\bigg{[}\left|1-R_{T}^{f}\right|^{2p}\bigg{]}^{1/2}}_{\widetilde{J}_{2}}\\\
&\qquad\qquad\qquad+\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{c})\right|^{2p}\bigg{]}^{1/2}\mathbb{E}\bigg{[}\left|1-R_{T}^{c}\right|^{2p}\bigg{]}^{1/2}}_{\widetilde{J}_{3}}\Bigg{\\}}.\end{split}$

From Assumption 4, and using the results from Theorem 3, the term
$\widetilde{J}_{1}$ yields the bound

$\begin{split}\widetilde{J}_{1}=\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})\right|^{p}\bigg{]}\leq
K^{p}\mathbb{E}\bigg{[}\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|^{p}\bigg{]}\leq
C^{p}p^{3p/2}h^{3p/2}.\end{split}$

Similarly, we have

$\begin{split}\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})\right|^{2p}\bigg{]}^{1/2}\leq
C^{p}p^{p/2},\qquad\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{c})\right|^{2p}\bigg{]}^{1/2}\leq
C^{p}p^{p/2}.\end{split}$

To bound the term $\widetilde{J}_{2}$, we define $E^{f}$ as

$E^{f}\coloneqq-\sum_{n=0}^{N-1}\left\langle\widehat{S}^{f}_{n},\Delta
V_{1,n}\right\rangle+\sqrt{3}\sum_{n=0}^{N-1}\left\langle\widehat{S}^{f}_{n},\Delta
V_{2,n}\right\rangle-2h\sum_{n=0}^{N-1}\left\|\widehat{S}^{f}_{n}\right\|^{2},$

where $\widehat{S}^{f}_{n}$ is as defined in equation (13). Using the mean
value theorem, there exists a $\xi=\theta x+(1-\theta)y$ for all
$x,y\in\mathbb{R}$, and any $\theta\in(0,1)$, such that

$R_{T}^{f}=\exp(E^{f})=1+\exp(\xi)E^{f},$

where $\left|\xi\right|\leq\left|x\right|$ for all $\theta\in(0,1)$. Using
Hölder’s inequality, we obtain

$\begin{split}\mathbb{E}\bigg{[}\left|1-R_{T}^{f}\right|^{2p}\bigg{]}&=\mathbb{E}\bigg{[}\big{(}\exp(\xi)\left|E^{f}\right|\big{)}^{2p}\bigg{]}\leq\mathbb{E}\bigg{[}\exp(4p\xi)\bigg{]}^{1/2}\mathbb{E}\bigg{[}\left|E^{f}\right|^{4p}\bigg{]}^{1/2}.\end{split}$

Using the result from Theorem 4, we have

$\mathbb{E}\bigg{[}\exp(4p\xi)\bigg{]}\leq\mathbb{E}\bigg{[}\max(\exp(4pE^{f}),1)\bigg{]}\leq\mathbb{E}[\exp(4pE^{f})]+1\leq\eta_{1},$

for all $T\leq C_{0}h^{-2}\left|\log h\right|^{-1}$, where $C_{0}$ is a
constant that is independent of $h$.

From Jensen’s inequality, and Hölder’s inequality, we obtain

$\begin{split}\mathbb{E}\bigg{[}\left|E^{f}\right|^{4p}\bigg{]}\leq
2^{4p-1}S^{4p}\mathbb{E}\Bigg{[}\left|\sum_{n=0}^{N-1}\left\langle\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c},\Delta
V_{3,n}\right\rangle\right|^{4p}\Bigg{]}+2^{8p-1}S^{8p}h^{4p}\mathbb{E}\Bigg{[}\left|\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\right|^{4p}\Bigg{]},\end{split}$

where $\Delta V_{3,n}\coloneqq\sqrt{3}\Delta V_{2,n}-\Delta V_{1,n}$. Using
Theorem 3, and Burkholder–Davis–Gundy’s inequality, there exists a constant
$C_{\text{BDG}}>0$ independent of $T$ and $p$, such that

$\begin{split}\mathbb{E}\bigg{[}\left|E^{f}\right|^{4p}\bigg{]}&\leq
2^{12p-1}S^{4p}C_{\text{BDG}}^{2p}p^{2p}h^{2p}\mathbb{E}\Bigg{[}\left|\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\right|^{2p}\Bigg{]}\\\
&\qquad+2^{8p-1}S^{8p}h^{4p}\mathbb{E}\Bigg{[}\left|\sum_{n=0}^{N-1}\left\|\widehat{Y}_{n}^{f}-\widehat{Y}_{n}^{c}\right\|^{2}\right|^{4p}\Bigg{]}\\\
~{}&\leq C_{3}^{6p}T^{2p}p^{8p}h^{6p}+C_{3}^{8p}T^{4p}p^{12p}h^{12p}\\\
~{}&\leq
C_{3}^{8p}T^{2p}p^{8p}h^{6p}\big{(}1+T^{2p}p^{4p}h^{6p}\big{)}.\end{split}$

Then, for all $T\leq C_{0}h^{-3}$, and for all sufficiently small $h>0$, there
exists a constant $C_{5}>0$ independent of $T$ and $p$, such that

$\mathbb{E}\bigg{[}\left|E^{f}\right|^{4p}\bigg{]}\leq
C_{5}^{8p}T^{2p}p^{8p}h^{6p},$

which then yields

$\mathbb{E}\bigg{[}\left|1-R_{T}^{f}\right|^{2p}\bigg{]}\leq\sqrt{3}C_{5}^{4p}T^{p}p^{4p}h^{3p}.$

A similar result can obtained for the term
$\mathbb{E}\bigg{[}\left|1-R_{T}^{c}\right|^{2p}\bigg{]}$ to yield a bound for
the term $\widetilde{J}_{3}$.

Combining the estimates for the terms $\widetilde{J}_{1}$,
$\widetilde{J}_{2}$, and $\widetilde{J}_{3}$, for all $T\leq
C_{0}h^{-2}\left|\log h\right|^{-1}$, $p\geq 1$, there exists a constant
$\kappa_{7}>0$, independent of $T$, $p$, for all sufficiently small $h>0$, we
obtain

$\begin{split}\mathbb{E}\Bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\right|^{p}\Bigg{]}\leq\kappa_{7}^{p}T^{p/2}p^{\frac{5}{2}p}h^{\frac{3}{2}p}.\end{split}$
(58)

#### Proof of Theorem 6

Following the same steps as in Theorem 5, using Hölder’s inequality, and
Jensen’s inequality, we have

$\begin{split}&\mathbb{E}\Bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\right|^{p}\Bigg{]}\\\
&\quad=\mathbb{E}\Bigg{[}\Bigg{|}\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{f})+\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})+\varPhi(\widehat{Y}_{T}^{c})-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\Bigg{|}^{p}\Bigg{]}\\\
~{}&\quad\leq
3^{p-1}\Bigg{\\{}\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})\right|^{p}\bigg{]}}_{\widetilde{J}_{1}}+\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})\right|^{2p}\bigg{]}^{1/2}\mathbb{E}\bigg{[}\left|1-R_{T}^{f}\right|^{2p}\bigg{]}^{1/2}}_{\widetilde{J}_{2}}\\\
&\qquad\qquad\qquad+\underbrace{\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{c})\right|^{2p}\bigg{]}^{1/2}\mathbb{E}\bigg{[}\left|1-R_{T}^{c}\right|^{2p}\bigg{]}^{1/2}}_{\widetilde{J}_{3}}\Bigg{\\}}.\end{split}$

The terms $\widetilde{J}_{2}$ and $\widetilde{J}_{3}$ can be bounded in a
similar way as in Theorem 5. To bound $\widetilde{J}_{1}$, we begin the
analysis by bound the term $\left|\mathbbm{1}_{\widehat{Y}_{T}^{f}\in
G}-\mathbbm{1}_{\widehat{Y}_{T}^{c}\in G}\right|$. Following a similar
methodology as in [15], we obtain

$\begin{split}\left|\mathbbm{1}_{\widehat{Y}_{T}^{f}\in
G}-\mathbbm{1}_{\widehat{Y}_{T}^{c}\in
G}\right|&\leq\mathbbm{1}_{\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|>d_{\partial
G}(\widehat{Y}_{T}^{f})}\\\
&=\mathbbm{1}_{\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|>d_{\partial
G}(\widehat{Y}_{T}^{f})}\bigg{(}\mathbbm{1}_{d_{\partial
G}(\widehat{Y}_{T}^{f})\leq\delta}+\mathbbm{1}_{d_{\partial
G}(\widehat{Y}_{T}^{f})>\delta}\bigg{)}\\\ &\leq\mathbbm{1}_{d_{\partial
G}(\widehat{Y}_{T}^{f})\leq\delta}+\mathbbm{1}_{\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|>\delta}.\end{split}$

Using Hölder’s inequality, taking the expected value on both sides of the
inequality, and using Markov’s inequality, we obtain

$\begin{split}\mathbb{E}\bigg{[}\left|\mathbbm{1}_{\widehat{Y}_{T}^{f}\in
G}-\mathbbm{1}_{\widehat{Y}_{T}^{c}\in G}\right|^{p}\bigg{]}&\leq
2^{p-1}\bigg{(}\mathbb{P}\Big{(}d_{\partial
G}(\widehat{Y}_{T}^{f})\leq\delta\Big{)}+\mathbb{P}\Big{(}\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|>\delta\Big{)}\bigg{)}\\\
~{}&\leq
2^{p-1}\bigg{(}C_{b}\delta+\delta^{-q}\mathbb{E}\Big{[}\left\|\widehat{Y}_{T}^{f}-\widehat{Y}_{T}^{c}\right\|^{q}\Big{]}\bigg{)}.\end{split}$

Using the result from Theorem 3, for the choice of
$\delta=q^{\frac{3q+2}{2(q+1)}}h^{\frac{3}{2}\frac{q}{q+1}}$,
$\widetilde{J}_{1}$ yields the bound

$\begin{split}\widetilde{J}_{1}=\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})-\varPhi(\widehat{Y}_{T}^{c})\right|^{p}\bigg{]}&=\mathbb{E}\bigg{[}\left|\mathbbm{1}_{\widehat{Y}_{T}^{f}\in
G}-\mathbbm{1}_{\widehat{Y}_{T}^{c}\in G}\right|^{p}\bigg{]}\\\ ~{}&\leq
C^{p}\bigg{(}\delta+\delta^{-q}q^{3q/2}h^{3q/2}\bigg{)}\\\ ~{}&\leq
C^{p}(1+q)q^{\frac{1}{2}\frac{q}{q+1}}h^{\frac{3}{2}\frac{q}{q+1}}.\end{split}$

Note that for all $h\in(0,1)$, and for all $\xi\geq\frac{3}{2}\frac{1}{q+1}$,
$q\geq 1$, the following inequality holds true

$\widetilde{J}\leq
C^{p}(1+q)q^{\frac{1}{2}\frac{q}{q+1}}h^{\frac{3}{2}\frac{q}{q+1}}\leq
C^{p}(1+q)q^{\frac{1}{2}\frac{q}{q+1}}h^{\frac{3}{2}-\xi}.$

Combining the estimates for the terms $\widetilde{J}_{1}$,
$\widetilde{J}_{2}$, and $\widetilde{J}_{3}$, for all $T>1$, $p,q\geq 1$, and
for all $\xi\geq\frac{3}{2}\frac{1}{q+1}$ there exists a constant
$\kappa_{8}>0$, independent of $T$, $p$, and $q$, for all sufficiently small
$h>0$, we obtain

$\mathbb{E}\bigg{[}\left|\varPhi(\widehat{Y}_{T}^{f})R_{T}^{f}-\varPhi(\widehat{Y}_{T}^{c})R_{T}^{c}\right|^{p}\bigg{]}\leq\kappa_{8}^{p}T^{p/2}p^{2p}(1+q)q^{\frac{1}{2}\frac{q}{q+1}}h^{\frac{3}{2}-\xi}.$
(59)

#### Proof of Theorem 7

The MSE, using Hölder’s inequality, can be split into three components based
on the source of error

$\begin{split}\mathbb{E}\Big{[}\big{(}\widehat{\varPhi}-\mathbb{E}[\varPhi(X_{\infty})]\big{)}^{2}\Big{]}&=\mathbb{E}\Big{[}\big{(}(\widehat{\varPhi}-\mathbb{E}[\widehat{\varPhi}])+(\mathbb{E}[\widehat{\varPhi}]-\mathbb{E}[\varPhi(X_{T})])+(\mathbb{E}[\varPhi(X_{T})]-\mathbb{E}[\varPhi(X_{\infty})])\big{)}^{2}\Big{]}\\\
~{}&\leq\underbrace{\mathbb{V}(\widehat{\varPhi})}_{\widetilde{J}_{1}}+\underbrace{2\big{(}\mathbb{E}[\widehat{\varPhi}]-\mathbb{E}[\varPhi(X_{T})]\big{)}^{2}}_{\widetilde{J}_{2}}+\underbrace{2\big{(}\mathbb{E}[\varPhi(X_{T})]-\mathbb{E}[\varPhi(X_{\infty})]\big{)}^{2}}_{\widetilde{J}_{3}},\end{split}$

where $\mathbb{V}(\widehat{\varPhi})$ denotes the variance of the MLMC
estimator. Recall that we want the MSE to be bounded from above by
$\epsilon^{2}$. This can be achieved by splitting the error contribution
equally among the three error components, i.e.

$\widetilde{J}_{k}\leq\frac{\epsilon^{2}}{3},\qquad\forall k\in\\{1,2,3\\}.$

From equation (3), we know that

$\begin{split}\left|\mathbb{E}[\varPhi(X_{T})]-\mathbb{E}[\varPhi(X_{\infty})]\right|\leq\mu^{\star}\operatorname{e}^{-\lambda^{\star}T}.\end{split}$

Then for the choice of

$T=\left\lceil\frac{1}{\lambda^{\star}}\log(\epsilon^{-1})+\frac{1}{\lambda^{\star}}\log(\sqrt{6}\mu^{\star})\right\rceil,$

we achieve the bound $\widetilde{J}_{3}\leq\epsilon^{2}/3$.

We consider two different cases based on the regularity of the payoff function
Lipschitz continuous payoff function: Using Jensen’s inequality and the
results from Theorem 5, we have

$\left|\mathbb{E}\big{[}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\big{]}\right|\leq
C_{1}\sqrt{T}(2^{-\ell}h_{0})^{\frac{3}{2}},\qquad\forall\ell\geq 1$

where $C_{1}>0$ is a constant. From Theorem 4, for the choice of
$h_{0}=C_{2}/\sqrt{T\log T}$, we obtain

$\begin{split}\left|\mathbb{E}[\widehat{\varPhi}]-\mathbb{E}[\varPhi(X_{T})]\right|&=\left|\sum_{\ell=L+1}^{\infty}\mathbb{E}\Big{[}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\Big{]}\right|\\\
~{}&\leq C_{1}C_{2}^{3/2}T^{-1/4}\big{(}\log
T\big{)}^{-3/4}\sum_{\ell=L+1}^{\infty}2^{-\frac{3}{2}\ell}.\end{split}$ (60)

where $C_{2}>0$ is a constant independent of $T$. Then for the choice of $L$
given by

$L=\left\lceil\frac{2}{3}\log_{2}\Big{(}\epsilon^{-1}T^{-1/4}\big{(}\log
T\big{)}^{-3/4}\Big{)}+\frac{2}{3}\log_{2}(\sqrt{6}\widetilde{C}_{1})\right\rceil,$

we achieve the bound $\widetilde{J}_{2}\leq\epsilon^{2}/3$, where
$\widetilde{C}_{1}\coloneqq C_{1}C_{2}^{3/2}/\big{(}2^{3/2}-1\big{)}$, and $L$
is total number of levels of the multilevel estimator.

Let $\mathcal{V}_{\ell}$ denote the variance of the MLMC estimator under the
change-of-measure on level $\ell$. Using the results from Theorem 5, there
exists a constant $c_{1}>0$ such that

$\begin{split}\mathcal{V}_{\ell}\coloneqq\mathbb{V}\Big{(}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\Big{)}\leq
c_{1}T(2^{-\ell}h_{0})^{3},\qquad\forall\ell\geq 1,\end{split}$ (61)

Let $\mathcal{C}_{\ell}$ denote the computational cost needed to generate
numerical realizations of both the fine and the coarse trajectories on each
level $\ell$

$\mathcal{C}_{\ell}\leq c_{2}\frac{T}{2^{-\ell}h_{0}},$

where $c_{2}>0$ is a constant.

To bound the first error component $\widetilde{J}_{1}$, we choose the number
of samples on each level of the multilevel estimator in such a way that that
the total cost is minimized under the constraint that the total variance of
the estimator is $\epsilon^{2}/3$. From the standard MLMC theory [14, Theorem
2.1], for the choice of $N_{\ell}$ given by

$N_{\ell}=\left\lceil
3\epsilon^{-2}\Bigg{(}\sum_{\ell=0}^{L}\sqrt{\mathcal{V}_{\ell}\mathcal{C}_{\ell}}\Bigg{)}\sqrt{\frac{\mathcal{V}_{\ell}}{\mathcal{C}_{\ell}}}\right\rceil,$

we achieve the bound $\widetilde{J}_{1}\leq\epsilon^{2}/3$. The total
computational cost involved in the implementation of the higher-order change-
of-measure scheme is

$\operatorname{Cost}=3\epsilon^{-2}\Bigg{(}\sum_{\ell=0}^{L}\sqrt{\mathcal{V}_{\ell}\mathcal{C}_{\ell}}\Bigg{)}^{2}+\sum_{\ell=0}^{L}\mathcal{C}_{\ell},$

where the total computational cost relates to the total number of operations
required to numerically sample the fine and the coarse trajectories, and to
compute the Radon–Nikodym derivatives over all levels, for all the Monte Carlo
samples.

Note that $\mathcal{V}_{0}=\mathcal{O}(1)$ and
$\mathcal{C}_{0}=\mathcal{O}(Th_{0}^{-1})$. Then for the choice of
$h_{0}=C_{2}/\sqrt{T\log T}$, we obtain

$\begin{split}\operatorname{Cost}&\leq
6\epsilon^{-2}\Bigg{(}\mathcal{V}_{0}\mathcal{C}_{0}+\bigg{(}\sum_{\ell=1}^{L}\sqrt{\mathcal{V}_{\ell}\mathcal{C}_{\ell}}\bigg{)}^{2}\Bigg{)}+\sum_{\ell=0}^{L}\mathcal{C}_{\ell}\\\
&=\mathcal{O}\Big{(}\epsilon^{-2}\left|\log\epsilon\right|^{3/2}(\log\left|\log\epsilon\right|)^{1/2}\Big{)}.\end{split}$

Discontinuous payoff function: From Theorem 6 and for all $\xi>0$, we have

$\left|\mathbb{E}\big{[}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\big{]}\right|\leq
C_{1}\sqrt{T}(2^{-\ell}h_{0})^{\frac{3}{2}-\xi},\qquad\forall\ell\geq 1$

where $C_{1}>0$ is a constant. From Theorem 4, for the choice of
$h_{0}=C_{2}T^{-\frac{1}{\frac{3}{2}-\xi}}$, we obtain

$\begin{split}\left|\mathbb{E}[\widehat{\varPhi}]-\mathbb{E}[\varPhi(X_{T})]\right|&=\left|\sum_{\ell=L+1}^{\infty}\mathbb{E}\Big{[}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\Big{]}\right|\\\
~{}&\leq
C_{1}C_{2}^{3/2-\xi}T^{-1/2}\sum_{\ell=L+1}^{\infty}2^{-(\frac{3}{2}-\xi)\ell}.\end{split}$
(62)

Then for the choice of $L$ given by

$L=\left\lceil\frac{1}{\frac{3}{2}-\xi}\log_{2}\Big{(}\epsilon^{-1}T^{-1/2}\Big{)}+\frac{1}{\frac{3}{2}-\xi}\log_{2}(\sqrt{6}\widetilde{C}_{1})\right\rceil,$

we obtain the bound $\widetilde{J}_{2}\leq\epsilon^{2}/3$, where
$\widetilde{C}_{1}\coloneqq C_{1}C_{2}^{3/2-\xi}/(2^{3/2-\xi}-1)$.

From Theorem 6, we know that there exists a constant $c_{1}>0$ for all $\xi>0$
such that the following holds

$\begin{split}\mathcal{V}_{\ell}\coloneqq\mathbb{V}\Big{(}\varPhi(\widehat{Y}_{T}^{f,\ell})R_{T}^{f,\ell}-\varPhi(\widehat{Y}_{T}^{c,\ell})R_{T}^{c,\ell}\Big{)}\leq
c_{1}T(2^{-\ell}h_{0})^{\frac{3}{2}-\xi},\qquad\forall\ell\geq 1,\end{split}$
(63)

Using a similar argument as for the uniformly Lipschitz continuous payoff
function, the expected total computational cost for a discontinuous payoff
function is

$\begin{split}\operatorname{Cost}&\leq
6\epsilon^{-2}\Bigg{(}\mathcal{V}_{0}\mathcal{C}_{0}+\bigg{(}\sum_{\ell=1}^{L}\sqrt{\mathcal{V}_{\ell}\mathcal{C}_{\ell}}\bigg{)}^{2}\Bigg{)}+\sum_{\ell=0}^{L}\mathcal{C}_{\ell}\\\
&=\mathcal{O}\Big{(}\epsilon^{-2}\left|\log\epsilon\right|^{5/3+\xi}\Big{)}\end{split}$

for all $\xi>0$.

Acknowledgements. We would like to thank Mike Giles for his suggestions for
improving the manuscript.

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