# Using network structure and community detection to discover important
website features when distinguishing between phishing and legitimate ones

Arash Negahdari Kia, Finbarr Murphy, Zahra Dehghani Mohammadabadi, & Parisa
Shamsi

###### Abstract

In this paper, we uncover the essential features of websites that allow
intelligent models to distinguish between phishing and legitimate sites.
Phishing websites are those that are made with a similar user interface and a
near similar address to trustworthy websites in order to persuade users to
input their private data for potential future misuse by attackers. Detecting
phishing websites with intelligent systems is an important goal to protect
users, companies, and other online services that use the HTTP protocol. An
intelligent model needs to distinguish features that are important as input to
predict phishing sites. In this research, using correlation-based networks, we
provide a novel network-based method to find features that are more important
in phishing detection. The networks are trained and tested on an established
phishing dataset. Three different networks are made by partitioning the
dataset by its data instance labels. The important features are found by
discovering the hubs of these networks, and the results are presented and
analysed. This is the first time using a network-based approach for feature
selection which is a fast and accurate way to do so.

###### Index Terms:

Phishing Detection, Knowledge Graph, Community Detection

##  I Introduction

The internet has become ubiquitous for all private and commercial activity. In
many instances, websites may require personal information such as usernames
and passwords; and there is a general degree of implicit trust associated with
this information transfer. This allows malicious hackers to steal private data
for illicit gain. To do so, they can try to trick people by making them think
that they are passing their information to a trustworthy website by displaying
a fake website with similar characteristics to the legitimate website and a
near similar URL address. Such websites are called phishing websites. Many
people fall into this trap and face substantial negative consequences [1].
According to the IBM threat index 2020, phishing is the most popular cyber-
attack which happens globally [2].

Significant efforts have been made to detect phishing websites and prevent
such fraud. However, there is yet no clear way to distinguish legitimate
websites from phishing websites. Therefore, efforts are focused on methods
that can detect phishing websites with higher accuracy.

Phishing detection is approached in a variety of ways. Most contemporary
approaches for detecting phishing websites are based on machine learning and
intelligent models, like using a classification method on website features.
One way to optimize the results of these approaches is to find the most
salient features of a website to identify its legitimacy [3, 4, 5, 6]. In
another research, the effectiveness of finding features in optimally detecting
phishing websites was investigated by two methods: Wrapper-based feature
selection and correlation-based feature selection [7]. In correlation-based
feature selection, the criterion had been calculated, and important subset of
features were selected according to the criterion. Wrapper-based feature
selection needs the supervised algorithms and labels for each instance of the
dataset for selecting the important features. In the wrapper method, a subset
of features that make the most accurate prediction/classification is selected.
The researchers finally compared the performance of both methods in their
study. In this paper, we propose a heuristic method to find the most important
features of a website to help intelligent models in phishing detection.
Knowledge graph representation has helped us find the most important features
in distinguishing between phishing and legitimate websites. In our approach
nodes of the network represent the features and those nodes that have more
connections have more influence on other nodes, and therefore, they represent
more important features. In section I-A, we give a brief explanation and
examples of some concepts used in our proposed method.

In section I-B, we investigate some preliminaries methods. We explain our
proposed method in section II and discuss its results in section III. Section
IV concludes the paper and presents suggestions for future researches.

###  I-A Related Works

####  I-A1 Phishing Detection

Some considerable research has been undertaken to increase the detection of
phishing websites. The approaches can be classified into the blacklist
approach and the heuristic approach.

  1. 1.

Blacklist Approach:

In the blacklist approach, a list of malicious URLs is formed as a blacklist.
When a user requests a website, the domain will be compared with the list to
find a match. If this match is found, the connection would not be allowed. Its
disadvantage is that the blacklist should be updated frequently, and some
phishing websites may not be discovered.

Another study proposed a blacklist approach that keeps the blacklist up to
date by using search engine results to detect suspicious domains [8]. This
way, the website’s legitimacy can be checked. Another research proposed a
system (PhishNet) using an algorithm to find a close match in the blacklist
[9]. Another study proposed an approach that uses the redirection URLs from
phishing websites for completing the blacklist [10].

  2. 2.

Heuristic Approach:

In the heuristic approach, some techniques like machine learning are used to
find phishing websites based on general phishing features. The advantage of
this approach is that new phishing websites can be detected.

In one study, researchers propose a heuristic method based on a relative
detection based on the website’s logo and legal logo [11]. Another research
proposed a phishing detection technique based on machine learning using an
analysis of the URL’s features, website host and interpretation of the visual
appearance [3]. Mao et al. propose a heuristic phishing detection method using
machine learning techniques to find the similarity between the website’s user
interface and a legitimate website’s user interface [4]. Chiew et al. propose
a heuristic feature-based method that uses machine learning to detect phishing
websites, called the Hybrid Ensemble Feature Selection (HEFS) [5]. HEFS
included two steps: The first by using the Cumulative Distribution Function
gradient algorithm found the number of optimal features and in the second
step, selected a subset of features by the hybrid framework. HEFS had high
performance when using a random forest algorithm. Rao et al. propose a method
based on URL and using TF-IDF property to detect a phishing website. Also, the
dataset that we used in our paper has features based on URLs such as
URL$\\_$length [6]. Zhang et al. also propose a method only using URL
addresses for phishing detection. Techniques used in the method are
bidirectional LSTM, skip-gram, and CNN [12]. Chavan et al. propose a phishing
detection method using deep learning technique and feature engineering and
reduces features of the dataset from 19 to 10 [13]. One of the big problems in
phishing detection is the lack of phishing data against legitimate data.
Shirazi et al. used data augmentation to solve the problem [14].

####  I-A2 Network Structures

Many real-world phenomena can be modelled by networks such as social networks
and information networks.

Social networks show the interaction of people or groups of people in the form
of nodes and edges connecting them [15]. Barabasi et al. discuss scientific
collaborations as complex networks [16]. Nekovee et al. propose a model to
show the spread of rumours [17] and Potts et al. propose a market-based
definition of creative industries, both based on complex social networks [18].
In another research, Schimit used complex networks for modelling a population
to show how people connect and analysed it as a disease spreading model [19].

Information networks are networks showing interactions between some items of
data. Information networks are a type of network showing the interaction
between concepts in the outside world that can be interpreted as nodes and
links [15]. We can refer to the World Wide Web as the best-known information
network. A citation network is a network that is based on paper citations. Son
et al. propose a method for an academic paper recommender system based on
citation networks [20]. Their proposed network is a multilevel simultaneous
citation network, and this method is useful when citation information is not
enough.

In this paper, we analyse the information network built from the phishing and
legitimate websites data for feature selection. The features can be used for
better phishing detection models used in both literature and application like
intrusion detection systems.

####  I-A3 Community Detection

Community detection is a procedure used to group network nodes in a way to
make nodes in each community have dense connections. As a result, a better
understanding of the network’s structure and function is discovered. There is
a broad application of community detection used in the researches. Kanavos et
al. propose an efficient methodology for community detection to analyse the
behaviour of users on an emotional level based on their tweets [21]. In this
paper, we use community detection to cluster the website features in order to
analyse their similarities. We deploy a similarity knowledge graph using
different characteristics and features of phishing/legitimate websites. This
is a unique approach that has not been used in the phishing detection research
area by far. Network modelling has been found useful and effective in
different areas of research, and this is the first time it is used in phishing
research.

###  I-B Preliminaries

####  I-B1 Constructing a Similarity Graph

We define

$Correlation(feature_{i},feature_{j})=1-\frac{6\sum d_{k}^{2}}{n(n^{2}-1)},$
(1)

where $d_{i}$ is the difference between two ranks of each feature for each
instance of the dataset, and $n$ is the number of instances in the dataset. By
using Equation 1, we form the correlation matrix. The dataset features are
categorical, so we use Spearman’s rank-order correlation [22].

The distance between $feature_{i}$ and $feature_{j}$ can be obtained from
Equation 2.

$d_{i,j}=\sqrt{2\ (1-Correlation(feature_{i},feature_{j}))}.$ (2)

Finally, we use Equation 3 to form the similarity matrix.

$similarity\ measure=e^{-d_{i,j}}.$ (3)

This approach of constructing a similarity matrix is used in other researches
such as the researches of Bonanno et al. [23], Wang et al. [24], and Song et
al. [25]. Most of these researches are in the financial data mining and
analysis domain. Our research employs the same models to determine the
existence of phishing.

####  I-B2 Louvain Community Detection

Community detection determines the similarity amongst features of the phishing
dataset. Louvain is a greedy and extendable community detection method that
divides a large network into communities [26], and because of its greedy
nature, it is a fast method in comparison with other methods, specially when
dealing with complex networks [27]. Louvain is based on optimizing the
modularity meaning the detection of communities in a way that nodes in a
community have dense connections, while nodes in different communities have
scattered connections. The Louvain algorithm is described as following:

  1. 1.

Consider each node as a community.

  2. 2.

Merge two communities if it raises the modularity.

  3. 3.

Repeat step 2 until no other changes could be done, and that means the
modularity is optimized.

####  I-B3 Maximum Spanning Tree (MST)

Using the maximum spanning tree helps us find the strongest relationship
structure amongst the features of our phishing dataset when modelled into a
correlation network. A maximum spanning tree ($T(V_{T},E_{T})$) is a subgraph
of an edge-weighted undirected graph ($G(V_{G},E_{G})$) that,

$V_{T}=V_{G},$ (4)

and

$E_{T}\subset E_{G},$ (5)

with the maximum possible total edge weight where $V_{T}$ and $E_{T}$ are sets
of the tree’s vertices and edges and $V_{G}$ and $E_{G}$ are sets of the
graph’s vertices and edges.

We will use Kruskal’s algorithm [28] to form the maximum spanning tree. The
algorithm is described as following:

  1. 1.

Sort the graph’s edges in descending order.

  2. 2.

Pick the first edge.

  3. 3.

Pick the next edge if the set of selected edges up to this step does not form
a cycle. A cycle is a non-empty trail in which the only repeated vertices are
the first and last vertices [29].

  4. 4.

If the number of selected edges is one unit less than the number of the main
graph’s vertices, stop the algorithm. Else, repeat step 3.

Using thresholding instead of a maximum spanning tree may lead to expert bias.
Even using statistical significance testing would need distribution assumption
which would also lead to expert bias.

####  I-B4 Centrality Measures Used In The Research

Centrality measures are used to find the most important nodes in a knowledge
graph. There are many centrality measures defined in the network science. The
meaning of what is important in this context depends on the mathematical
definition of each centrality measure [15]. These measurements help us capture
feature attributes of the phishing dataset. These measurements help us
understand which features are more important than others in the phishing
dataset.

  * •

Degree:

In a graph, the number of edges connected to each node is called its degree.
In an undirected graph (G=(V, E)), the relationship between the number of
edges (E) and the number of nodes (V) is

$\sum_{v\in V}\deg(v)=2|E|.$ (6)

  * •

Hub:

To find the most important nodes in a graph, we can use different measures and
definitions. For example, we can say if a node’s degree is higher, it is more
influential in the graph. In network science, a node with a degree much higher
than the average is called hub [16].

In this research, we consider nodes with a degree higher than two as a hub.

####  I-B5 Gamma Value

Gamma value is a measurement in network structures that shows the scale-
freeness of the network. In some networks, connections between nodes are based
on a power-law distribution called preferential attachment. In these networks,
called scale-free networks, the gamma value in Equation 8 is a parameter in
the range $2<\gamma<3$ [15]. Social networks are a kind of scale-free
networks. In social networks, there are few nodes with dense connections and
many nodes with few connections. In the case of higher gamma values, there
will be fewer hub nodes with higher degree and more nodes connected to the
hubs with less degrees. This means that in higher gamma values, we have some
important features and many other features that relate to these hub features.
Therefore, it may be possible that they can be ignored when constructing
intelligent phishing detection models. Imagine a network with $n$ nodes. If
the number of nodes with degree k is $n_{k}$, then the probability that a node
is of degree k is equal to

$p(k)=\frac{n_{k}}{n}.$ (7)

The proper distribution function for the above expression in a network is as
follows:

$P(k)\sim k^{-\gamma}.$ (8)

In this paper, first, we will calculate the gamma value for each network
structure constructed from the phishing dataset. Subsequently, we provide a
network analysis of the nodes and their connections to discover important
nodes which correspond to important features of websites.

##  II Method

In this section, we explain the design of the study, as shown in Figure 1 and
network construction mechanism in Figure 2. We apply a process on the salient
features of a website and find the most effective ones that can help
intelligent models to detect phishing websites. These features are described
in the Appendix.

Figure 1: An overview of the research methodology

###  II-A Design of the study

In the following, we describe the research methodology of Figure1 for each
sub-procedure which is enumerated in the figure.

  1. 1.

We divided data into three parts. The first group contains all the websites.
In the second group, there are only legitimate websites, and the last group
includes only phishing websites. All the features are the same in the three
parts of the data. The following steps apply to all three groups. For each, we
build a network from the dataset features. This is to capture the
characteristics of different categories of websites along with a network for
all the websites together.

  2. 2.

For building the networks, as it can be seen in Figure 2, first, the
correlation matrix must be calculated as described in section I-B1. Then, we
calculate the distance matrix, and by doing so, we calculate the similarity
matrix for each part of the dataset. The reason for doing this is to construct
a similarity graph where nodes represent the features of websites in the
dataset and the links with their weights represent the similarity between each
pair of features. The code for network construction has been added in the
GitHub account of the paper111The code is available here:
https://github.com/dmresearches/phishing.

The network construction procedure has a successful history of representing
the similarity between data in finance paradigm. A full and comprehensive
description of these correlation networks has been studied in network science
literature [23].

  3. 3.

For finding the features that are most related to each other, we apply a
community detection algorithm on all three graphs extracted in the previous
step. In this research, we use the Louvain modularity, which is described in
section I-B2.

  4. 4.

We find the maximum spanning tree of all three networks built in part 2 of
Figure1, which the weights of the edges are the similarity between the
features (described in section I-B3). The reason for doing so is to capture
the most important feature relationships in each dataset. The maximum spanning
tree finds the strongest relations among nodes in each graph.

  5. 5.

As described in section I-B4, we find hubs for each maximum spanning tree. By
doing this, we find the most important features for each category of websites.
These features are the most related ones to other features in their dataset.
In other words, these hub features can be seen as the candidates in a feature
selection procedure for the future supervised or semi-supervised prediction of
phishing or legitimate websites.

  6. 6.

Finally, we find the gamma values for each network, as described in section
I-B5. The value shows if the hub features are good representatives of other
features. As described in section I-B5, it is known that in scale-free
networks with high gamma values, there are fewer nodes with high degree and a
lot of other nodes with low degree. This means that the minority nodes with
high degree that we are going to call as hubs are those which relate to many
other nodes which represent the features in our dataset.

Figure 2: An overview of network building steps in the research methodology

In the next section, the graphs, trees, and numerical results achieved from
our proposed method are presented to discover the essential features in
phishing, legitimate, and the whole dataset.

##  III Results and Discussion

In this section, we provide an analysis and discussion based on the results
achieved from the methodology outlined in section II. As discussed, we
construct a maximum spanning tree for features of each network; all websites
presented in Figure 3, legitimate websites presented in Figure 4, and phishing
websites presented in Figure 5.

Figure 3: Maximum spanning tree for the graph extracted from the dataset of
all websites

In each maximum spanning tree, we find hub features as described in section
I-B4. These hubs are listed in tables I, II, III.

TABLE I: Website features discovered as hubs (nodes with high connections) in the Maximum Spanning Tree built from all the websites Hub label | Degree | Community
---|---|---
Shortening Service | 4 | 0
SSLfinal State | 4 | 2
URL Length | 4 | 2
Double Slash Redirecting | 3 | 0
Links Pointing To Page | 3 | 2
Port | 3 | 1
Submitting To Email | 3 | 1
URL Of Anchor | 3 | 2

By finding the maximum spanning tree, we are specifying the strongest
relations among the features. For example, in Figure 3 which shows the maximum
spanning tree of features in the dataset of all websites, if a website has the
feature ”Shortening Service”, it is more probable that it has the feature
”Double Slash Redirecting”, and so it is more probable that it has the feature
”HTTPS Token”. In general, it can be said that the features that are hubs are
more important than other features of the website in distinguishing between
phishing and legitimate, and changes in the values of these features affect
the values of other features.

Figure 4: Maximum spanning tree for the graph extracted from the dataset of
legitimate websites

Maximum spanning tree of features in legitimate websites (Figure 4) have six
hubs listed in table II. These features are the most effective features to
discuss the legitimacy of a website. In the Appendix we describe in which
state of each feature, it is effective in determining the legitimacy of a
website.

TABLE II: Website features discovered as hubs (nodes with high connections) in the Maximum Spanning Tree built from the legitimate websites Hub label | Degree | Community
---|---|---
Double Slash Redirecting | 6 | 2
URL Length | 4 | 1
Ifram | 3 | 0
Links Pointing To Page | 3 | 1
Port | 3 | 0
Submitting To Email | 3 | 0

In Figure 5, the maximum spanning tree of features in phishing websites
dataset is displayed whit six features listed in table III as hubs. That
means, for checking if a website is a phishing one, we can focus on these
features in prediction models and gain a better performance.

Figure 5: Maximum spanning tree for the graph extracted from the dataset of
phishing websites

Some features like ”URL Length” are common in both hub features of the
legitimate websites and the phishing websites. That means the length of the
website’s URL, can help us judge both the legitimacy or being a phishing one.
This shows that the value of the URL length feature is a discriminator of the
classes phishing, or legitimate.

TABLE III: Website features discovered as hubs (nodes with high connections) in the Maximum Spanning Tree built from the phishing websites Hub label | Degree | Community
---|---|---
Port | 4 | 0
Shortening Service | 4 | 2
URL Length | 4 | 1
Age Of Domain | 3 | 1
Double Slash Redirecting | 3 | 2
Page Rank | 3 | 1

In table IV, the gamma values are presented. These gamma values are calculated
(as described in section I-B5) for features in all websites, legitimate
websites, and phishing websites. The higher the gamma value is, the less the
number of hubs and the more the degrees of the hubs would be. So, those hubs
are better candidates for us to predict the website type (legitimate, or
phishing). The gamma value of features in legitimate websites is higher than
the gamma value of features in phishing websites. As a result, we can check
the legitimacy of a website by the hub features in table II with better
performance than checking if a website is phishing using the features in table
III because this gamma value is higher and the tree is more scale-free
(described in section I-B5).

TABLE IV: Gamma values of the maximum spanning trees for three different datasets of all data, legitimate data, and phishing data. Higher gamma indicates features that make a scale-free network, and the more the hub features are effective in website type prediction. Group | Gamma values
---|---
All Of Data | 0.09
Legitimate Data | 0.13
Phishing Data | 0.08

Table I, table II, and table III, also show some of the results of the
community detection algorithm on the networks built in the second part of the
method. The nature of community detection algorithms is to cluster similar
entities into the same cluster/community. Given that, all those features that
appear in the same community can be considered having the same importance in
phishing/legitimate detection systems so that one can be a delegate for
others. As you can see in table II, most features belong to community 0, so
features in this community play a more effective role in detecting legitimate
websites. In table III, most features belong to community 1, so features in
this community play a more effective role in detecting phishing websites.

The above method was also applied on different random subsets of the dataset
and the results stayed the same, which shows the reliability of the results.

For evaluating the proposed method, we follow these steps: Table I shows three
nodes with the degree of 4. These are the most important features when
discussing the legitimacy or illegitimacy of a website. So these are some of
the features we worked with. For other nodes with the degree of 3, we chose
the ones directly connected to a node with the degree of 4, as can be seen in
Figure 3. As the result, we were dealing with five features:
”SSLfinal$\\_$State”, ”Shortening$\\_$Service”, ”URL$\\_$Length”,
”URL$\\_$Of$\\_$Anchor”, and ”Double$\\_$Slash$\\_$Redirecting”. We chose the
eXtreme Gradient Boosting (XGBoost) [30] algorithm for classification by the
selected features since it is one of the strongest ensemble methods. The
accuracy of classification by these five selected features was 0.917. For
comparison, we used the Principal Component Analysis (PCA) [31] method by five
components for feature selection and ran the XGBoost algorithm for the dataset
with the five components. The accuracy by using the PCA method was 0.899.

When talking about cybersecurity, we are facing a complex system that its
elements do not have a linear relationship with each other, and also some of
the relationships are not clear. So the best way to model such a system is
using network-based approaches. The network-based approaches can discover
hidden patterns in the system. Using methods like Principal Component Analysis
(PCA) for feature selection and finding the most important features in
phishing detection would implement the assumption that the features have a
linear relationship with each other, which here is not the case.

##  IV Conclusion

In order to find the most important features for intelligent models to help
them detect phishing websites, we propose a method that finds these features
and discovers the connections between them. In this way, we can prevent data
loss from the phishing website and provide information security for those who
use the HTTP protocol and machines like smart routers that can filter the
malicious HTTP traffic.

In this paper, we built correlation-based networks of features in phishing,
legitimate, and all websites in our dataset. We subsequently identify
important features in each network by finding the hubs in them that had the
most effect on the other features.

By extracting the relation networks out of the datasets and finding the hub
nodes and gamma values for scale-freeness, we showed which features have a
stronger effect on the website class (phishing or legitimate) and which
website class is more dependent on particular features.

In the network made by phishing instances, the important features were, Port,
Shortening Service, URL Length, Age Of Domain, Double Slash Redirecting, and
PageRank. In the network made by legitimate instances, the important features
were, Double Slash Redirecting, URL Length, Ifram, Links Pointing to Page,
Port, and Submitting To email. In the knowledge graph made by the whole
dataset, the important features were, Shortening Service, SSLfinal State, URL
Length, Double Slash Redirecting, Links Pointing To Page, Port, Submitting TO
email, and URL Of Anchor.

The results of our study can be used in smart routers and intrusion detection
systems that try to monitor the HTTP traffic and filter the phishing websites.

Future researches that analyses different supervised models with our reduced
feature sets for phishing detection using different similarity functions other
than correlation can be employed to produce different networks that capture
different information from the dataset.

##  Appendix A Data Gathering

The studying dataset is mainly gathered from archives of ”PhishingTank”,
”MillerSmiles” and Google’s searching operators [32] and includes 11055
samples, 30 features, and labels that show, according to the listed features,
if a website is phishing or not. The rows including the listed features are
categorized as 1, 0, -1. The value 1 defines that a website is legitimate, 0
defines the situation that a website is suspicious to be phishing, and -1
defines that a website is phishing.

In Figure 6, all dataset features can be seen. Also, the percentage that
shows, in each feature, how many of the samples are phishing, suspicious, or
legitimate.

Figure 6: Features in the dataset and the percentage of them appearing in
phishing, legitimate, and suspicious classes.

For understanding the dataset better, we explain each feature briefly. For
this purpose, we used references such as research of [33] and a Computer
Networks reference book [34].

#### Having$\\_$IP$\\_$Address

If a URL includes an IP in the domain name, the website is phishing. It should
be noticed that some times the IP will be turned into a hexadecimal code, so
it will be hard for users to pay attention to it easily.

#### URL$\\_$Length

Phishers use long URLs in order to hide the suspicious part in the address
bar. Studies show that if the length of a URL is less than 54 characters, the
website is legitimate with a high probability. If the length is more than 54
and less than 75 characters, the website is suspicious, and if the length is
more than 75 characters, the website is more probable to be phishing.

#### Shortening$\\_$service

Shortening URL is a method that shortens the URL address significantly. First,
by clicking on the shortened URL, users would be referred to the website which
offers this service, and then enters the main website. In this dataset, a
shortened URL is considered as a probable sign of a phishing website.

#### Having$\\_$At$\\_$symbol

If a URL includes the symbol ”@”, the website is phishing, and if not, it is
legitimate.

#### Double$\\_$Slash$\\_$Redirecting

If a URL includes the symbol ”//”, it is a sign of redirecting users to a new
website. Studies show that if a URL starts with HTTP, the symbol ”//” should
be in the sixth position, and if it starts with HTTPS, the symbol ”//” should
be in the seventh position. Thus, if the symbol ”//” is in the next positions,
the website is phishing. Otherwise, it is legitimate.

#### Prefix-Suffix

It rarely happens that a legitimate URL includes the symbol ”-”. Phishers
usually put a prefix or a suffix isolated by a ”-” in the URL, so users assume
that they are facing a legitimate website.

#### Having$\\_$Sub$\\_$Domain

Consider the URL ”http://www.hud.ac.uk/students”. By omitting ”www.” from it
and counting dots in the remaining part, we can bring up with a rule. If the
number of remaining dots is 1, the website is legitimate, if 2, the website is
suspicious (because it has subdomain) and if the number is greater than 2 (it
has multi-subdomains), the website is more probable to be phishing.

#### SSLfinal$\\_$State

If a URL supports HTTPS, it significantly has improved the probability of
being legitimate. However, the existence of HTTPS is not enough itself, and
for more assurance, features like SSL’s (Secure Sockets Layer) source and its
certification age should be considered. Studies show that if a website uses
HTTPS and the certificate’s sources are valid, and its certificate age is more
than one year, the website is legitimate, and if the website uses HTTPS but
the certificate’s sources are not valid, it is suspicious to be phishing.
Otherwise, it is phishing with a high probability.

#### Domain$\\_$Registration$\\_$Length

Most phishing websites will not be on a specified domain on the World Wide Web
for a long time, while if a website is a legitimate one and wants to be on a
specified domain for a long time, the cost will be prepaid. Thus, if a website
wants to be on a specified domain for a short time, it is more probable that
it is phishing. Otherwise, it is legitimate.

#### Favicon

Favicon is a particular graphical image that would be placed as an icon beside
the address bar. If this icon is loaded from a domain other than the website
domain, the website is more probable to be phishing. Otherwise, it is
legitimate.

#### Port

When a user sends a request to a particular server, along with the request,
the number of expected ports for answering back from the server will be sent.
For protecting the user’s information, websites must have a particular control
on the ports. If all ports are open, hackers can threaten the user’s
information. If all ports are closed except 80 and 443, it reduces the
probability of any break-in, and the website is more probable to be
legitimate.

#### HTTPS$\\_$Token

Phishers will use HTTPS in the URL but not in the right place. They put it
after HTTP to make the website feel legitimate. For example, we can take a
look at a URL like this: http://https-www-paypal-it-webapps-mpp-home.soft-
hair.com/

So, if there is HTTPS in the domain, the website is phishing. Otherwise, it is
legitimate.

#### Request$\\_$URL

In legitimated websites that contain photos, videos, and such things, their
source and the website’s URL should have the same domain. In general, if the
percentage of different cases is less than 22$\%$, the website is legitimate,
and if it is between 22$\%$ and 61$\%$, the website is suspicious. Otherwise,
the website is more probable to be phishing.

#### URL$\\_$Of$\\_$Anchor

If we need to make a link from our website to another website, we use the
”$<$a$>$” tag. Thus, there are two situations:

  1. 1.

The ”$<$a$>$” tag’s domain is different from the website’s domain

  2. 2.

The ”$<$a$>$” tag is not linking any website (for example: $<$a
href=”$\\#$”$>$)

If the percentage of any of the explained situations is less than 31$\%$ of
the whole HTML code, the website is legitimate. If this percentage is between
31$\%$ and 67$\%$, the website is suspicious. Otherwise, the website is more
probable to be phishing.

#### Links$\\_$In$\\_$Tags

In HTML programming language, programmers use ”$<$Meta$>$”, ”$<$Script$>$” and
”$<$Link$>$” tags in HTML documents. In legitimate websites, it is expected
that these tags have their links in the same website domain. If the percentage
of differences in the domains is less than 17$\%$, the website is legitimate.
If it is between 17$\%$ and 81$\%$, the website is suspicious. Otherwise, it
is more probable that the website is phishing.

#### SFH (Server Form Handler)

It is a field that contains an address that the user receives from the server.
If SFH is empty, the website is phishing. If its domain is different from the
website domain, the website is suspicious. Otherwise, the website is
legitimate.

#### Submitting$\\_$To$\\_$email

If a website wants users to enter their personal information like email
address by using ”mail()” or ”mailto:” method, it is likely that it is just an
effort to access their information. So the website is more probable to be
phishing.

#### Abnormal$\\_$URL

This feature can be extracted from websites of the ”WHOIS” database. If the
URL does not include the host’s name, the website is phishing, and if not, it
is legitimate.

#### Redirect

Legitimate websites redirect at most once. If the number of redirection is
between 2 and 4, the website is suspicious, more than that, it will be
phishing.

#### On$\\_$Mouseover

If the URL in status bar changes with ”onMouseOver”, the website is phishing.
Otherwise, it is legitimate.

#### Right$\\_$Click

If Right$\\_$Click is disabled on the website, it is more probable to be
phishing. Otherwise, it is legitimate.

#### Pop$\\_$Up$\\_$Windows

In legitimate websites, it is not common to ask users to enter their personal
information in a popup window, and these windows are being used for welcoming
or warning users. In general and with a high probability, if users are not
asked to enter a text in pop up windows, the website is legitimate. Otherwise,
it is phishing.

#### Iframe

There is a tag in HTML that allows displaying a website on another website.
Phishers may use this feature and make the frame invisible. Thus, if a website
uses the $<$iframe$>$ tag, it is more probable to be phishing. Otherwise, it
is legitimate.

#### Age$\\_$Of$\\_$Domain

Phishing websites are usually available for a short time. Studies show that
websites older than six months are legitimate. Otherwise, they are phishing.
This feature can be seen on the ”WHOIS” website.

#### DNS$\\_$Record (Domain Name Server Record)

This feature can be recognized from the ”WHOIS” database. If this feature is
empty or is not among the features in ”WHOIS”, the website is phishing.
Otherwise, it is legitimated.

#### Web$\\_$Traffic

”Alexa” is a database that ranks websites based on their views. In the worst
ranking, legitimate websites are among the top 100,000 websites. If the
website is ranked less than 100,000, the website is legitimate. If it is
ranked more than 100,000, it is suspicious, and if it is not among the
”Alexa”’s ranking, it is more probable to be phishing.

#### Page$\\_$Rank

PageRank shows the importance of websites and gets values between 0 and 1. In
the examined dataset, 95$\%$ of the phishing websites did not have PageRank,
and the other 5$\%$ had a PageRank value lower than 0.2. Thus, if the PageRank
value is more than 0.2, the website is legitimate. Otherwise, it is phishing.

#### Google$\\_$Index

Phishing websites are not usually in Google’s index for their short
availability. Thus, if a website is not on Google’s index, it is more probable
to be phishing. Otherwise, it is legitimate.

#### Links$\\_$Pointing$\\_$To$\\_$Page

In general, if there are many links from other websites to the website being
tested, it is more probable that the tested website is legitimate, otherwise
phishing. In this dataset, if there are no websites to point the website, we
consider it phishing. If this number is less than 2, the website is
suspicious. Otherwise, it is legitimate.

#### Statistical$\\_$Report

”Phishtank Stats” and ”StopBadware” are two of the institutes working on
providing statistical reports regarding phishing websites. If the website’s
host is in the list of phishing IPs or domains from those two institutes, the
website is phishing. Otherwise, it is legitimate.

Figure 7: The proportion of phishing and legitimate websites in the dataset

After a brief overview of the features, we review the data labels. We show the
phishing class label by -1 and legitimate class label by 1 in the dataset. The
number of phishing classes is 4898, and the number of legitimate classes is
6157. In Figure 7 the chart that shows the proportion of phishing classes and
legitimate classes in the whole dataset, is presented.

## List of abbreviations

HEFS: Hybrid Ensemble Feature Selection; HTTP: Hypertext Transfer Protocol;
URL: Uniform Resource Locator; HTTPS: Hypertext Transfer Protocol Secure; SSL:
Secure Sockets Layer; IP: Internet Protocol; SFH: Server Form Handler; DNS:
Domain Name System; HTML: HyperText Markup Language; MST: Maximum Spanning
Tree

## References

  * [1] M. Wu, “Fighting phishing at the user interface,” Ph.D. dissertation, Massachusetts Institute of Technology, 2006.
  * [2] M. Alvarez, D. Bales, J. Chung, S. Craig, K. Dahl, C. DeBeck, A. Eitan, B. Faby, R. Gates, D. Harz, L. Kessem, C. Lee, D. McMillen, S. Moore, G. Prassinos, C. Singleton, M. Usher, A. Vila, H. Virani, C. Zaboeva, and J. Zorabedian, “Ibm x-force threat intelligence index 2020,” _United States of America_ , 2020.
  * [3] V. Patil, P. Thakkar, C. Shah, T. Bhat, and S. Godse, “Detection and prevention of phishing websites using machine learning approach,” in _2018 Fourth International Conference on Computing Communication Control and Automation (ICCUBEA)_. IEEE, 2019, pp. 1–5.
  * [4] J. Mao, J. Bian, W. Tian, S. Zhu, T. Wei, A. Li, and Z. Liang, “Phishing page detection via learning classifiers from page layout feature,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2019, no. 1, p. 43, 2019\.
  * [5] K. L. Chiew, C. L. Tan, K. Wong, K. S. Yong, and W. K. Tiong, “A new hybrid ensemble feature selection framework for machine learning-based phishing detection system,” _Information Sciences_ , vol. 484, pp. 153–166, 2019\.
  * [6] R. S. Rao, T. Vaishnavi, and A. R. Pais, “Catchphish: detection of phishing websites by inspecting urls,” _Journal of Ambient Intelligence and Humanized Computing_ , vol. 11, no. 2, pp. 813–825, 2020.
  * [7] R. B. Basnet, A. H. Sung, and Q. Liu, “Feature selection for improved phishing detection,” in _International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems_. Springer, 2012, pp. 252–261.
  * [8] M. Sharifi and S. H. Siadati, “A phishing sites blacklist generator,” in _2008 IEEE/ACS International Conference on Computer Systems and Applications_. IEEE, 2008, pp. 840–843.
  * [9] P. Prakash, M. Kumar, R. R. Kompella, and M. Gupta, “Phishnet: predictive blacklisting to detect phishing attacks,” in _2010 Proceedings IEEE INFOCOM_. IEEE, 2010, pp. 1–5.
  * [10] L.-H. Lee, K.-C. Lee, H.-H. Chen, and Y.-H. Tseng, “Poster: Proactive blacklist update for anti-phishing,” in _Proceedings of the 2014 ACM SIGSAC Conference on Computer and Communications Security_. ACM, 2014, pp. 1448–1450.
  * [11] W. Yao, Y. Ding, and X. Li, “Deep learning for phishing detection,” in _2018 IEEE Intl Conf on Parallel & Distributed Processing with Applications, Ubiquitous Computing & Communications, Big Data & Cloud Computing, Social Computing & Networking, Sustainable Computing & Communications (ISPA/IUCC/BDCloud/SocialCom/SustainCom)_. IEEE, 2018, pp. 645–650.
  * [12] L. Zhang and P. Zhang, “Phishtrim: Fast and adaptive phishing detection based on deep representation learning,” in _2020 IEEE International Conference on Web Services (ICWS)_. IEEE, 2020, pp. 176–180.
  * [13] S. Chavan, A. Inamdar, A. Dorle, S. Kulkarni, and X.-W. Wu, “Phishing detection: Malicious and benign websites classification using machine learning techniques,” in _Proceeding of International Conference on Computational Science and Applications_. Springer, 2020, pp. 437–446.
  * [14] H. Shirazi, S. R. Muramudalige, I. Ray, and A. P. Jayasumana, “Improved phishing detection algorithms using adversarial autoencoder synthesized data,” in _2020 IEEE 45th Conference on Local Computer Networks (LCN)_. IEEE, 2020, pp. 24–32.
  * [15] M. Newman, _Networks: an introduction_. Oxford university press, 2010.
  * [16] A.-L. Barabasi _et al._ , _Network science: Graph Theory, p. 27_. Cambridge university press, 2016.
  * [17] M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili, “Theory of rumour spreading in complex social networks,” _Physica A: Statistical Mechanics and its Applications_ , vol. 374, no. 1, pp. 457–470, 2007.
  * [18] J. Potts, S. Cunningham, J. Hartley, and P. Ormerod, “Social network markets: a new definition of the creative industries,” _Journal of cultural economics_ , vol. 32, no. 3, pp. 167–185, 2008.
  * [19] P. H. Schimit and F. H. Pereira, “Disease spreading in complex networks: a numerical study with principal component analysis,” _Expert Systems with Applications_ , vol. 97, pp. 41–50, 2018.
  * [20] J. Son and S. B. Kim, “Academic paper recommender system using multilevel simultaneous citation networks,” _Decision Support Systems_ , vol. 105, pp. 24–33, 2018.
  * [21] A. Kanavos, I. Perikos, I. Hatzilygeroudis, and A. Tsakalidis, “Emotional community detection in social networks,” _Computers & Electrical Engineering_, vol. 65, pp. 449–460, 2018.
  * [22] S. Kumar and I. Chong, “Correlation analysis to identify the effective data in machine learning: Prediction of depressive disorder and emotion states,” _International journal of environmental research and public health_ , vol. 15, no. 12, p. 2907, 2018.
  * [23] G. Bonanno, G. Caldarelli, F. Lillo, and R. N. Mantegna, “Topology of correlation-based minimal spanning trees in real and model markets,” _Physical Review E_ , vol. 68, no. 4, p. 046130, 2003.
  * [24] G.-J. Wang, C. Xie, and H. E. Stanley, “Correlation structure and evolution of world stock markets: Evidence from pearson and partial correlation-based networks,” _Computational Economics_ , vol. 51, no. 3, pp. 607–635, 2018\.
  * [25] D.-M. Song, M. Tumminello, W.-X. Zhou, and R. N. Mantegna, “Evolution of worldwide stock markets, correlation structure, and correlation-based graphs,” _Physical Review E_ , vol. 84, no. 2, p. 026108, 2011.
  * [26] V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre, “Fast unfolding of communities in large networks,” _Journal of statistical mechanics: theory and experiment_ , vol. 2008, no. 10, p. P10008, 2008.
  * [27] P. Chejara and W. W. Godfrey, “Comparative analysis of community detection algorithms,” in _2017 Conference on Information and Communication Technology (CICT)_. IEEE, 2017, pp. 1–5.
  * [28] J. B. Kruskal, “On the shortest spanning subtree of a graph and the traveling salesman problem,” _Proceedings of the American Mathematical society_ , vol. 7, no. 1, pp. 48–50, 1956.
  * [29] E. A. Bender and S. G. Williamson, _Lists, Decisions and Graphs_. S. Gill Williamson, 2010.
  * [30] T. Chen, “Story and lessons behind the evolution of xg-boost,” 2016.
  * [31] K. Pearson, “Liii. on lines and planes of closest fit to systems of points in space,” _The London, Edinburgh, and Dublin philosophical magazine and journal of science_ , vol. 2, no. 11, pp. 559–572, 1901.
  * [32] R. Mohammad, L. McCluskey, and F. Thabtah, “Uci machine learning repository: phishing websites data set (2015),” 2016. [Online]. Available: https://archive.ics.uci.edu/ml/datasets/Phishing+Websites
  * [33] R. M. Mohammad, F. Thabtah, and L. McCluskey, “Phishing websites features,” _Unpubl. Available Httpe-prints Hud Ac Uk243306RamiPhishingWebsitesFeatures Pdf, 2015_ , 2015.
  * [34] A. S. Tanenbaum _et al._ , “Computer networks, 4-th edition,” _ed: Prentice Hall_ , 2003.


& \;  +\E_t\bigg[   \int_t^T \e^{cr}\Big( L_{\rm u}  |\sigma^\t_r \partial u_r^s|^2+4 L_z  |\sigma^\t_r z_r|^2 +  2 L_v  |\sigma^\t_r v_r^r|^2+2  L_v  |\sigma^\t_r v_r^s|^2+  L_{\rm v}|\sigma^\t_r \partial v_r^s|^2\Big) \d r \bigg].
\end{align*}

where we recall the notation $L_\star=\max\{L_y,L_u,L_{\rm u},L_z,L_v,L_{\rm v}\}$. Thus, for any $c>0$ we obtain

\begin{align*}
& \mathrm{max}\big\{ \e^{\frac{c}2 t} |\Yc_t| ,\, \e^{\frac{c}2 t} |\Uc_t| ,\,\e^{\frac{c}2 t} |U_t^s|,\,\e^{\frac{c}2 t} |\partial U_t^s| \big\} \\
\leq &\ \|\xi\|_{\Lc^{\infty,c}} +  \|\tilde h\|_{\L^{1,\infty,2,c}}+ 2\big(  \|\eta \|_{\Lc^{\infty,2,c}} + \|\tilde g\|_{\L^{1,\infty,2,c}} \big)    +(1+T+TL_{{\rm u}} )\big( \|\partial_s \eta \|_{\Lc^{\infty,2,c}}+ \|\nabla \tilde g\|_{\L^{1,\infty,2,c}} \big)   \\
& \;   + (4+T+L_{\rm u} T)  L_{\star} T   \Big( \|y\|^2_{\Sc^{2\infty,c}}+\|u\|^2_{\Sc^{\infty,2,c}}+ \|\partial u\|^2_{\Sc^{\infty,2,c}}\Big)  + (4+T+L_{\rm u} T)  L_{\star}   \Big( \|z\|^2_{\H^{2,c}_{{\rm BMO}}}+\|v\|^2_{\overline \H^{2,2,c}_{{\rm BMO}}}+ \|\partial v\|^2_{\H^{2,2,c}_{{\rm BMO}}}\Big),
\end{align*}

* We show $(\Zc,\Vc,\Nc,\Mc)\in \big(\H^{2,c}_{{\rm BMO}}\big)^2\times \big(\M^{2,c}\big)^2$ and $\| V\|_{ \H^{2,2,c}_{{\rm BMO}}}^2+\|M\|_{{\M}^{2,2,c}}^2+\| \partial V\|_{ \H^{2,2,c}_{{\rm BMO}}}^2+\|\partial M\|_{{\M}^{2,2,c}}^2<\infty $.

From $(iii)$, <Ref><ref> and <ref><ref>, together with Young's inequality, yield that, for any $\eps_i>0$, $i\in\{1,2\}$, and defining $  C_{\eps_{1}}:=\eps_1^{-1} 7T  L_{\rm u}^2$, and $  C_{\eps_{2}}:= \eps_2^{-1} 7T$, we have

\begin{align*}
2 \Yc_r \cdot h_r-c |\Yc_r|^2 & \leq 2 \|\Yc\|_{\Sc^{\infty,c}}  \big( L_y  |y_r|^2+L_z  |\sigma^\t_r z_r|^2+L_u  |u_r^r|^2+L_v  |\sigma^\t_r v_r^r|^2+ |\tilde h_r|\big)\\
&\quad + \eps_1(7T)^{-1}  |\partial U_r^r|^2+  (  \widetilde C_{\eps_{1}}-c) |\Yc_r|^2 \\[0,5em]
2 \Uc_r \cdot  g_r-c |\Uc_r|^2 & \leq  2 \|\Uc\|_{\Sc^{\infty,c}}  \big( L_y  | y_r|^2+L_z  |\sigma^\t_r z_r|^2+ | u_r^r|^2+L_v  |\sigma^\t_r v_r^r|^2+ |\tilde g_r|\big)\\
&\quad + \eps_2(7T)^{-1}  |\partial U_r^r|^2 + (  \widetilde C_{\eps_{2}}-c) |\Uc_r|^2 , \\[0,5em]
2 U_r^s \cdot  g_r(s)-c |U_r^s|^2 & \leq   2 \|U\|_{\Sc^{\infty,2,c}}  \big( L_u  |  u_r^s|^2  + L_{ v} |\sigma^\t_r  v_r^s|^2+L_y  |  y_r|^2+L_z  |\sigma^\t_r z_r|^2+ |\tilde g_r(s)| \big)-c |U_r^s|^2 , \\[0,5em]
2 \partial U_r^s \cdot \nabla g_r(s)-c |\partial U_r^s|^2&  \leq  2\|\partial U\|_{\Sc^{\infty,c,2}}  \big(L_{\rm u}  |  \partial u_r^s|^2  + L_{ v} |\sigma^\t_r \partial v_r^s|^2+ L_u  |  u_r^s|^2  + L_{\rm v} |\sigma^\t_r  v_r^s|^2+ |\nabla \tilde g_r(s)|  \big)\\
&\quad  +2\|\partial U\|_{\Sc^{\infty,c,2}}  \big( L_y  | y_r|^2+L_z  |\sigma^\t_r z_r|^2\big) -c  |\partial U_r^s|^2
\end{align*}

These inequalities in combination with the analogous version of <Ref> (which holds for $c>2L_{\rm u}$), Young's inequality,  and Itô's formula, as in (<ref>), show that for any $\eps_i>0$, $i\in\{3,...,24\}$

\begin{align*}
&\sum_{i=1}^4 \e^{ct}  |\Yf_t^i|^2+\E_t\bigg[  \int_t^T \e^{cr}  |\sigma^\t_r  \Zf_r^i|^2 \d r+\int_t^T\e^{c r-} \d \Tr [\Nf^i]_r\bigg]+ \E_t\bigg[ \int_t^T \e^{cr}\big(  |\Yc_r|^2 (c-C_{\eps_{1}})+|\Uc_r|^2 (c- C_{\eps_{2}} )\big)\d r\bigg] \\
&\; +\sup_{s\in [0,T]}  \E_t\bigg[ \int_t^T c \e^{cr}|U_r^s|^2 \d r\bigg] +\sup_{s\in [0,T]}  \E_t\bigg[ \int_t^T c \e^{cr} |\partial U_r^s|^2  \d r\bigg]\\
= & \  + \E_t\Big[  \e^{cT}\big( |\xi|^2+ |\eta(T)|^2+|\eta(s)|^2+|\partial_s \eta(s)|^2\big)\Big] + ( \eps_1+\eps_2)  \Big( \|\partial_s \eta \|_{\Lc^{\infty,2,c}}^2  +\|\nabla \tilde g\|^2_{\L^{1,\infty,2,c}} \Big) \\
&\; +( \eps_1+\eps_2) \Big( L_\star T^2 \|y\|_{\Sc^{\infty,c}}^4+L_\star T^2 \|u\|_{\Sc^{\infty,c}}^4+L_\star T^2 \|\partial u\|_{\Sc^{\infty,c}}^4+ 2 L_\star^2  \Big( \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}} + \|v\|^4_{\H^{2,2,c}_{{\rm BMO}}}+ \|z\|^4_{\H^{2,c}_{{\rm BMO}}}\Big) \Big)\\
&\; +\big(\eps_{3}^{-1}+ \eps_{7}^{-1}+\eps_{8}^{-1}+\eps_{9}^{-1} +\eps_{10}^{-1}\big) \|\Yc\|^2_{\Sc^{\infty,c}}  +\big(\eps_{4}^{-1}+\eps_{11}^{-1}+ \eps_{12}^{-1}+ \eps_{13}^{-1}+\eps_{14}^{-1}\big) \|\Uc\|^2_{\Sc^{\infty,c}} \\
&\; +\big(\eps_{5}^{-1}+ \eps_{15}^{-1}+\eps_{16}^{-1}+\eps_{17}^{-1} +\eps_{18}^{-1}\big)  \|U\|^2_{\Sc^{\infty,c,2}} +\big(\eps_{6}^{-1}+\eps_{19}^{-1}+\eps_{20}^{-1} +\eps_{21}^{-1}+\eps_{22}^{-1}+\eps_{23}^{-1}+\eps_{24}^{-1}\big)  \|\partial U\|^2_{\Sc^{\infty,c,2}} \\
&\; +\E_t\bigg[\eps_3  \bigg|\int_t^T  \e^{cr}  |\tilde h_r|\d r\bigg|^2 + \eps_{4} \bigg|\int_t^T  \e^{cr} |\tilde g_r|\d r\bigg|^2 \bigg]   + \eps_{5} \bigg|\int_t^T \e^{cr} |\tilde g_r(s)|\d r\bigg|^2+ \eps_{6} \bigg|\int_t^T \e^{cr} |\nabla \tilde g_r|\d r\bigg|^2 \bigg]\\
&\; +( \eps_{7} +\eps_{11}+\eps_{15}+\eps_{19} )L_\star T^2\|y\|^4_{\Sc^{\infty,c}}
+( \eps_{9} +\eps_{13} +\eps_{17}+\eps_{21})L_\star T^2 \|u\|^4_{\Sc^{\infty,2,c}}+ \eps_{23}L_\star T^2 \|\partial u\|^4_{\Sc^{\infty,2,c}}\\
&\; +( \eps_{8}+\eps_{12}+\eps_{16}+\eps_{20} ) L_z^2\E_t\bigg[  \bigg|\int_t^T \e^{cr} |\sigma^\t_r z_r|^2 \d r\bigg|^2\bigg]+( \eps_{10}+\eps_{14} ) L_v^2\E_t\bigg[  \bigg|\int_t^T \e^{cr} |\sigma^\t_r v_r^r|^2 \d r\bigg|^2\bigg]\\
&\;  + (\eps_{18}+\eps_{22}) L_v^2\E_t\bigg[  \bigg|\int_t^T \e^{cr} |\sigma^\t_r v_r^s|^2 \d r\bigg|^2\bigg] +\eps_{24} L_{\rm v}^2\E_t\bigg[  \bigg|\int_t^T \e^{cr} |\sigma^\t_r \partial v_r^s|^2 \d r\bigg|^2\bigg]
\end{align*}

We now let $\tau \in \Tc_{0,T}$. In light of (<ref>), for

\begin{align}\label{Eq:cZwelldefinedq}
\begin{split}
c\geq \max &\   \{  \eps_1^{-1} 7T  L_{\rm u}^2,  \eps_2^{-1} 7T , 2L_{\rm u}\} ,
\end{split}
\end{align}

<Ref> yields

\begin{align*}
&\sum_{i=1}^4 \e^{ct}  |\Yf_t^i|^2+\E_t\bigg[  \int_t^T \e^{cr}  |\sigma^\t_r  \Zf_r^i|^2 \d r+\int_t^T\e^{c r-} \d \Tr [\Nf^i]_r\bigg]\\
=&\ \|\xi\|_{\Lc^{\infty,c}}^2+2 \|\eta \|_{\Lc^{\infty,2,c}}^2 \! + (1+ \eps_1+\eps_2)\|\partial_s \eta \|_{\Lc^{\infty,2,c}}^2\! +  \eps_3  \|  \tilde h\|^2_{\L^{1,\infty,c}} \!+ ( \eps_{4} +\eps_{5}) \|  \tilde g\|^2_{\L^{1,\infty,2,c}} \! + ( \eps_1+\eps_2+ \eps_{6}) \| \nabla  \tilde g\|^2_{\L^{1,\infty,2,c}}   \\
&\; + L_\star^2  T^2( \eps_1+\eps_2+\eps_{7} +\eps_{11}+\eps_{15}+\eps_{19}  )  \|y\|^4_{\Sc^{\infty,c}} + L_\star^2  T^2( \eps_1+\eps_2+ \eps_{9}+\eps_{13}+\eps_{17}+\eps_{21} )  \|u\|^4_{\Sc^{\infty,2,c}}\\
&\; +L_\star^2  T^2(\eps_1+\eps_2+\eps_{23})\|\partial u\|_{\Sc^{\infty,2,c}}^4 +2 L_\star^2  ( \eps_1+\eps_2+ \eps_{8}+\eps_{12}+\eps_{16}+\eps_{20} )  \|z\|^4_{\H^{2,c}_{{\rm BMO}}}  \\
&\; +2 L_\star^2  ( \eps_1+\eps_2+ \eps_{10}+\eps_{14}+\eps_{18}+\eps_{22} ) \|v\|^4_{\overline \H^{2,2,c}_{{\rm BMO}}} + 2 L_\star^2  ( \eps_1+\eps_2+\eps_{24}) \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}}\\
&\; +\big(\eps_{3}^{-1}+ \eps_{7}^{-1}+\eps_{8}^{-1}+\eps_{9}^{-1} +\eps_{10}^{-1}\big) \|\Yc\|^2_{\Sc^{\infty,c}}  +\big(\eps_{4}^{-1}+\eps_{11}^{-1}+ \eps_{12}^{-1}+\eps_{13}^{-1} + \eps_{14}^{-1}\big) \|\Uc\|^2_{\Sc^{\infty,c}} \\
&\; +\big(\eps_{5}^{-1}+ \eps_{15}^{-1}+\eps_{16}^{-1}+\eps_{17}^{-1}+\eps_{18}^{-1} \big)  \|U\|^2_{\Sc^{\infty,c,2}} +\big(\eps_{6}^{-1}+\eps_{19}^{-1}+\eps_{20}^{-1} +\eps_{21}^{-1}+\eps_{22}^{-1}+\eps_{23}^{-1}+\eps_{24}^{-1}\big)  \|\partial U\|^2_{\Sc^{\infty,c,2}}
\end{align*}

which in turn leads to

\begin{align}\label{Eq:thm:wdq:ineq:final}
\begin{split}
&\frac{1}{10}\Big(\|\Yc\|^2_{\Sc^{\infty,c}} +\|\Uc\|^2_{\Sc^{\infty,c}} +\|U\|^2_{\Sc^{\infty,2,c}}+\|\partial U\|^2_{\Sc^{\infty,2,c}} +    \|\Zc\|_{\H^{2,c}_{ {\rm BMO}}}^2  \\
&\quad + \|V\|_{\overline \H^{2,2,c}_{{\rm BMO}}}^2+ \|\partial V\|_{\H^{2,2,c}_{{\rm BMO}}}^2 + \|\Nc\|_{{\M}^{2,c}}^2+  \|M\|_{{\M}^{2,2,c}}^2+  \|\partial M\|_{{\M}^{2,2,c}}^2
\Big)   \\
\leq&\ \|\xi\|_{\Lc^{\infty,c}}^2+2 \|\eta \|_{\Lc^{\infty,2,c}}^2 \! + (1+ \eps_1+\eps_2)\|\partial_s \eta \|_{\Lc^{\infty,2,c}}^2\! +  \eps_3  \|  \tilde h\|^2_{\L^{1,\infty,c}} \\
&\; + ( \eps_{4} +\eps_{5}) \|  \tilde g\|^2_{\L^{1,\infty,2,c}} \! + ( \eps_1+\eps_2+ \eps_{6}) \| \nabla  \tilde g\|^2_{\L^{1,\infty,2,c}}   \\
&\; + L_\star^2  T^2( \eps_1+\eps_2+\eps_{7} +\eps_{11}+\eps_{15}+\eps_{19}  )  \|y\|^4_{\Sc^{\infty,c}} + L_\star^2  T^2( \eps_1+\eps_2+ \eps_{9}+\eps_{13}+\eps_{17}+\eps_{21} )  \|u\|^4_{\Sc^{\infty,2,c}}\\
&\; +L_\star^2  T^2(\eps_1+\eps_2+\eps_{23})\|\partial u\|_{\Sc^{\infty,2,c}}^4 +2 L_\star^2  ( \eps_1+\eps_2+ \eps_{8}+\eps_{12}+\eps_{16}+\eps_{20} )  \|z\|^4_{\H^{2,c}_{{\rm BMO}}}  \\
&\; +2 L_\star^2  ( \eps_1+\eps_2+ \eps_{10}+\eps_{14}+\eps_{18}+\eps_{22} ) \|v\|^4_{\overline \H^{2,2,c}_{{\rm BMO}}} + 2 L_\star^2  ( \eps_1+\eps_2+\eps_{24}) \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}}\\
&\; +\big(\eps_{3}^{-1}+ \eps_{7}^{-1}+\eps_{8}^{-1}+\eps_{9}^{-1} +\eps_{10}^{-1}\big) \|\Yc\|^2_{\Sc^{\infty,c}}  +\big(\eps_{4}^{-1}+\eps_{11}^{-1}+ \eps_{12}^{-1}+\eps_{13}^{-1} + \eps_{14}^{-1}\big) \|\Uc\|^2_{\Sc^{\infty,c}} \\
&\; +\big(\eps_{5}^{-1}+ \eps_{15}^{-1}+\eps_{16}^{-1}+\eps_{17}^{-1}+\eps_{18}^{-1} \big)  \|U\|^2_{\Sc^{\infty,c,2}} +\big(\eps_{6}^{-1}+\eps_{19}^{-1}+\eps_{20}^{-1} +\eps_{21}^{-1}+\eps_{22}^{-1}+\eps_{23}^{-1}+\eps_{24}^{-1}\big)  \|\partial U\|^2_{\Sc^{\infty,c,2}}
\end{split}
\end{align}

From (<ref>) we conclude $(Z,N)\in \H^{2,c}_{{\rm BMO}}\times  {\M}^{2,c}$, $\| V\|_{\overline \H^{2,2,c}_{{\rm BMO}}}^2+\| \partial V\|_{\H^{2,2,c}_{{\rm BMO}}}^2+ \|M\|_{{\M}^{2,2,c}}^2+ \|\partial M\|_{{\M}^{2,2,c}}^2<\infty$.

Defining $C_{\eps}$ analogously and if for some $\gamma\in(0,\infty)$

\begin{align}\label{Eq:thm:wpq:smalldatacond}
\begin{split}
I_0^\eps\leq \gamma R^2/10,
\end{split}
\end{align}

we obtain back in (<ref>)

\begin{align*}
& \|(Y,Z,N,U,V,M,\partial U,\partial V,\partial M)\|^2_{\Hc^{c}}\\
\leq &\  C_{\eps}^{-1} \Big( 10 I_0^\eps  +10L_\star^2 \max\{2,T^2\}  \big(
( \eps_1+\eps_2+\eps_{7} +\eps_{11}+\eps_{15}+\eps_{19}  )  \|y\|^4_{\Sc^{\infty,c}} +( \eps_1+\eps_2+ \eps_{9}+\eps_{13}+\eps_{17}+\eps_{21} )  \|u\|^4_{\Sc^{\infty,2,c}}\\
&\qquad +(\eps_1+\eps_2+\eps_{23})\|\partial u\|_{\Sc^{\infty,2,c}}^4 +  ( \eps_1+\eps_2+ \eps_{8}+\eps_{12}+\eps_{16}+\eps_{20} )  \|z\|^4_{\H^{2,c}_{{\rm BMO}}}  \\
&\qquad + ( \eps_1+\eps_2+ \eps_{10}+\eps_{14}+\eps_{18}+\eps_{22} ) \|v\|^4_{\overline \H^{2,2,c}_{{\rm BMO}}} +   ( \eps_1+\eps_2+\eps_{24}) \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}}\Big)  \\
\leq  &\   C_{\eps}^{-1} R^2  \bigg(\gamma  +10L_\star^2 \max\{2,T^2\} R^2   \bigg( \eps_1+\eps_2+\sum_{i=7}^{24} \eps_{i} \bigg)   \bigg)
\end{align*}

Therefore, to obtain $\Tf(\Bc_R)\subseteq \Bc_R$, that is to say that the image under $\Tf$ of the ball of radius $R$ is contained in the ball of radius $R$, it is necessary to find $R^2$ such that the term in parentheses above is less or equal than $C_{\eps}$, i.e.

\[
R^2 \leq \frac1{1 0 L_\star^2\max\{ 2, T^2\} }\frac{ C_{\eps} -  \gamma }{  \eps_1+\eps_2+ \sum_{i=7}^{24} \eps_{i} }
\]

which after optimising the choice of $\eps$'s renders

\begin{align}\label{Eq:Rwelldefinedq}
R^2 < \frac{1}{2^6\cdot 3\cdot 5^2\cdot 7\cdot L^2_\star\cdot \max\{ 2, T^2\} }
\end{align}

* The continuity of the applications $([0,T],\Bc([0,T])) \longrightarrow (\Sc^{\infty,c},\|\cdot \|_{ \Sc^{\infty,c}})\big($resp. $(\H_{{\rm BMO}}^{2,c},\|\cdot \|_{\H_{{\rm BMO}}^{2,c}}),\, ({\M}^{2,c},\|\cdot \|_{{\M}^{2,c}} )\big)  : s \longmapsto \varphi^s $ for $\varphi=U^s,\partial U^s\, ($resp. $V^s,\partial V^s, M^s,\partial M^s).$ follows analogously as in the proof <Ref>.

We conclude, $\Tf(\Bc_R)\subseteq \Bc_R$ for all $R$ satisfying (<ref>).

Step 2: We now argue that $\Tf$ is a contraction in $\Bc_R\subseteq \Hc$ for the norm $\| \cdot \|_{\Hc^c}$. Let

\begin{align*}
\delta h_t&:=h_t(y^1_t,z^1_t,u_t^{1,t}, v_t^{1,t},\partial U_t^{1,t})-h_t(y^2_t,z^2_t,u_t^{2,t}, v_t^{2,t},\partial U_t^{2,t}),\\
\delta   g_t&:=g_t(t,u_t^{1,t},v^{1,t}_t,y_t^1,z_t^1)-\partial U_t^{1,t} -g_t(t,u_t^{2,t},v^{2,t}_t,y_t^2,z_t^2)+\partial U_t^{2,t},\\
\delta \tilde h_t&:=h_t(y^1_t,z^1_t,u_t^{1,t}, v_t^{1,t},\partial U_t^{2,t})-h_t(y^2_t,z^2_t,u_t^{2,t}, v_t^{2,t},\partial U_t^{2,t}),\\
\delta \tilde g_t&:=g_t(t,u^{1,t}_t,v^{1,t}_t,y_t^1,z_t^1) -g_t(t,u^{2,t}_t,v^{2,t}_t,y_t^2,z_t^2),\\
\delta  \tilde g_t(s)&:=g_t(s,u^{1,s}_t,v^{1,s}_t,y_t^1,z_t^1)-g_t(s,u^{2,s}_t,v^{2,s}_t,y_t^2,z_t^2),\\
\delta \nabla  \tilde g_t(s)&:=\nabla g_t(s,\partial u^{1,s}_t,\partial v^{1,s}_t,u^{1,s}_t,v^{1,s}_t,y_t^1,z_t^1)-g_t(s,\partial u^{2,s}_t,\partial v^{2,s}_t,u^{2,s}_t,v^{2,s}_t,y_t^2,z_t^2).
\end{align*}

Applying Itô's formula we obtain that for any $t\in[0,T]$

\begin{align*}
&\sum_{i=1}^4 \e^{ct}  |\delta \Yf_t^i|^2+\int_t^T \e^{cr}  |\sigma^\t_r  \delta \Zf_r^i|^2 \d r+\int_t^T\e^{c r-} \d \Tr [\delta \Nf^i]_r  +\delta \widetilde  \Mf_t -\delta \widetilde  \Mf_T\\
=&\  \int_t^T  \e^{c  r} \bigg( 2  \delta \Yc_r \cdot \delta h_r +  2 \delta \Uc_r  \cdot \delta  g_r +  2 \delta U_r^s  \cdot \delta \tilde g_r(s)+ 2 \delta \partial U_r^s \cdot \delta \nabla \tilde g_r(s)\bigg) \d r \\
\leq &\ \int_t^T  \e^{c  r} \bigg(2 | \delta \Yc_r| \big( L_{\rm u} |\delta \partial U_r^r|+|\delta  \tilde h_r|\big)  +2 | \delta \Uc_r| \big(  |\delta \partial U_r^r|+|\delta  \tilde g_r| \big) +2 | \delta U_r^s|  |\delta  \tilde g_r(s)|  +2 | \delta \partial U_r^s|  |\delta  \nabla \tilde g_r(s)| -c \sum_{i=1}^4    |\delta \Yf_r^i|^2 \bigg) \d r
\end{align*}

where $\delta \widetilde \Mf$ denotes the corresponding martingale term. Let $\tau \in \Tc_{0,T}$, as in <Ref> we obtain for $c>2L_{\rm u}$

\begin{align}\label{Eq:ineqdeltaUtt:quadratic}
\begin{split}
\E_\tau\bigg[ \int_\tau^T \frac{ \e^{cr}}{3   T}  |\delta \partial U_r^r|^2\d r\bigg] &\leq    \sup_{s\in [0,T]}  \es_{\tau \in \Tc_{0,T}}  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \nabla \tilde g_r(s)|\d r \bigg] \bigg|^2
\end{split}
\end{align}

We now take conditional expectation with respect to $\Fc_\tau$ in the expression above and use <Ref> in combination with (<ref>). We then obtain from Young's inequality that for any $\tilde \eps_i\in (0,\infty)$, $i\in \{1,2\}$,

\begin{align}\label{Eq:c:contraction1q}
\begin{split}
c\geq  \max&\  \{ \tilde \eps_1^{-1} 3T L_{\rm u}^2,\;  3T\tilde \eps_2^{-1},\; 2 L_{\rm u}  \},
\end{split}
\end{align}

it follows that

\begin{align}\label{Eq:contractionItoq}\begin{split}
& \sum_{i=1}^4 \e^{ct}  |\delta \Yf_t^i|^2+\E_\tau\bigg[\int_t^T \e^{cr}  |\sigma^\t_r  \delta \Zf_r^i|^2 \d r+\int_t^T\e^{c r-} \d \Tr [\delta \Nf^i]_r  \bigg] \\
\leq &\  \tilde \eps_3^{-1} \|\delta Y\|^2_{\Sc^{\infty,c}}+ \tilde \eps_4^{-1} \|\delta \Uc\|^2_{\Sc^{\infty,2,c}} + \tilde \eps_{5}^{-1} \|\delta U\|^2_{\Sc^{\infty,2,c}} + \tilde \eps_{6}^{-1} \|\delta \partial U\|^2_{\Sc^{\infty,2,c}} \\
&\; + ( \tilde \eps_1+\tilde \eps_2+\tilde \eps_6)  \sup_{s\in [0,T]}  \es_{\tau \in \Tc_{0,T}}  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \nabla \tilde g_r(s)|\d r \bigg] \bigg|^2 +  \tilde \eps_3 \es_{\tau \in \Tc_{0,T}}  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \tilde h_t |\d r \bigg] \bigg|^2 \\
&\; + \tilde \eps_4     \es_{\tau \in \Tc_{0,T}}  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \tilde g_t|\d r \bigg] \bigg|^2 + \tilde \eps_{5}  \sup_{s\in [0,T]}  \es_{\tau \in \Tc_{0,T}}  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \tilde g_t(s)|\d r \bigg] \bigg|^2
\end{split}
\end{align}

We now estimate the terms on the right side of (<ref>). Note that in light of <Ref><ref> we have

\begin{align*}
&\, \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \nabla \tilde g_t(s)|\d r \bigg] \bigg|^2  \\
\leq &\,  \bigg|  \E_\tau \bigg[  \int_\tau^T \e^{c r} \Big( L_{\rm u} | \delta \partial u^s_r|\big( | \partial u_r^{1,s}|+| \partial u_r^{2,s}|\big)+ L_{\rm v} |\sigma^\t_r \delta \partial v^s_r|\big( |\sigma^\t_r \partial v_r^{1,s}|+|\sigma^\t_r \partial v_r^{2,s}|\big)\\
&\hspace{6em}  + L_u|  \delta u^s_r|\big( | u_r^{1,s}|+| u_r^{2,s}|\big) + L_v|\sigma^\t_r \delta v^s_r|\big( |\sigma^\t_r v_r^{1,s}|+|\sigma^\t_r v_r^{2,s}|\big) \\
&\hspace{6em}+ L_y|  \delta y_r|\big( | y_r^1|+| y_r^2|\big) + L_z|\sigma^\t_r \delta z_r|\big( |\sigma^\t_r  z_r^1|+|\sigma^\t_r z_r^2|\big)\Big) 	\d r  \bigg]  \bigg|^2 \\
\leq &\ 6 L_\star^2   \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta \partial u^s_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |  \partial u_r^{1,s}|+|  \partial u_r^{2,s}|\big)^2\d r\bigg]\\
&+6 L_\star^2   \E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta \partial v^s_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |\sigma^\t_r \partial v_r^{1,s}|+|\sigma^\t_r \partial v_r^{2,s}|\big)^2\d r\bigg]\\
&+6 L_\star^2   \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta  u^s_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |    u_r^{1,s}|+|   u_r^{2,s}|\big)^2\d r\bigg]\\
& + 6 L_\star^2 \E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta  v^s_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |\sigma^\t_r   v_r^{1,s}|+|\sigma^\t_r   v_r^{2,s}|\big)^2\d r\bigg]\\
&+6 L_\star^2   \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta  y_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |    y_r^{1}|+|   y_r^{2}|\big)^2\d r\bigg]\\
&+  6 L_\star^2  \E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta z_r|^2\d r \bigg] \E_\tau \bigg[  \int_\tau^T \e^{c r} \big( |\sigma^\t_r z_r^1|+|\sigma^\t_r z_r^2|\big)^2\d r\bigg]\\
\leq&\  6 L_\star^2 R^2\max\{ 2, T\}  \bigg( \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta \partial u^s_r|^2\d r \bigg]+  \E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta \partial v^s_r|^2\d r \bigg] + \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta  u^s_r|^2\d r \bigg]\\
&\hspace{8em} +\E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta  v^s_r|^2\d r \bigg]+ \E_\tau \bigg[  \int_\tau^T \e^{c r}   | \delta  y_r|^2\d r \bigg]+\E_\tau \bigg[  \int_\tau^T \e^{c r}   |\sigma^\t_r \delta z_r|^2\d r \bigg]\bigg)\\
\leq&\  6 L_\star^2 R^2\max\{ 2, T^2\}  \Big( \|\delta \partial u\|_{\Sc^{\infty,2,c}}^2+ \| \delta \partial v\|_{\H^{2,2,c}_{{\rm BMO}}}^2 +\|\delta  u\|_{\Sc^{\infty,2,c}}^2+ \| \delta v\|_{\overline \H^{2,2,c}_{{\rm BMO}}}^2 +\|\delta y\|_{\Sc^{\infty,c}}^2+\| \delta z\|_{\H^{2,c}_{{\rm BMO}}}^2\Big)
\end{align*}

where in the second inequality we used (<ref>) and Cauchy–Schwartz's inequality. Similarly

\begin{align*}
&\max\bigg\{ \bigg|\E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta\tilde  h_r|\d r \bigg]  \bigg|^2 , \bigg|\E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta \tilde g_r(s)|\d r \bigg]  \bigg|^2, \bigg|\E_\tau \bigg[  \int_\tau^T \e^{c r} |\delta g_r|\d r \bigg]  \bigg|^2\bigg\}\\
&\ \leq 4 L_{\star}^2 R^2\max\{ 2, T^2\}  \Big(\|\delta y\|_{\Sc^{\infty,c}}^2+ \| \delta z\|_{\H^{2,c}_{{\rm BMO}}}^2+ \|\delta  u\|_{\Sc^{\infty,2,c}}^2+ \| \delta v \|_{\overline \H^{2,c}_{{\rm BMO}}}^2\Big)
\end{align*}

Overall, we obtain back in (<ref>) that

\begin{align*}
& \sum_{i=1}^4 \e^{ct}  |\delta \Yf_t^i|^2+\E_\tau\bigg[\int_t^T \e^{cr}  |\sigma^\t_r  \delta \Zf_r^i|^2 \d r+\int_t^T\e^{c r-} \d \Tr [\delta \Nf^i]_r  \bigg] \\
\leq &\   \tilde \eps_3^{-1} \|\delta Y\|^2_{\Sc^{\infty,c}}+ \tilde \eps_4^{-1} \|\delta \Uc\|^2_{\Sc^{\infty,2,c}} + \tilde \eps_{5}^{-1} \|\delta U\|^2_{\Sc^{\infty,2,c}} + \tilde \eps_{6}^{-1} \|\delta \partial U\|^2_{\Sc^{\infty,2,c}} \\
&\; +     6 ( \tilde \eps_1+\tilde\eps_2+\tilde\eps_{6}) L_\star^2 R^2\max\{ 2, T^2\}  \Big( \|\delta \partial u\|_{\Sc^{\infty,2,c}}^2+ \| \delta \partial v\|_{\H^{2,2,c}_{{\rm BMO}}}^2 +\|\delta  u\|_{\Sc^{\infty,2,c}}^2+ \| \delta v\|_{\overline \H^{2,2,c}_{{\rm BMO}}}^2 +\|\delta y\|_{\Sc^{\infty,c}}^2+\| \delta z\|_{\H^{2,c}_{{\rm BMO}}}^2\Big) \\
& \; +4( \tilde \eps_3+\tilde\eps_4+\tilde\eps_{5})  L_{\star}^2 R^2\max\{ 2, T^2\}   \   \Big(\|\delta y\|_{\Sc^{\infty,c}}^2+ \| \delta z\|_{\H^{2,c}_{{\rm BMO}}}^2+ \|\delta  u\|_{\Sc^{\infty,2,c}}^2+ \| \delta v \|_{\overline \H^{2,c}_{{\rm BMO}}}^2\Big)
\end{align*}

If we define, for $\tilde \eps_i>10, i\in \{3,4,5,6\}$, $ C_{\tilde \eps}:= {\rm min}\big\{ 1-10/ \tilde \eps_{3 },\; 1-10/ \tilde \eps_{4 },\;1-10/ \tilde \eps_{5},\;1-10/ \tilde \eps_{6}\big\}$, we deduce,

\begin{align}\label{Eq:thm:contq:final}\begin{split}
\| \delta \mathfrak{H}\|_{\Hc^c}^2 \leq  20 C_{\tilde \eps}^{-1} L_{\star}^2 R^2\max\{ 2, T^2\}  (3 \tilde \eps_1 + 3 \tilde \eps_2 +2 \tilde \eps_3+2 \tilde \eps_4+2 \tilde \eps_{5}+3 \tilde \eps_{6 })\|\delta \mathfrak{h}\|_{\Hc^c}.
\end{split}
\end{align}

Minimising for $\tilde \eps_1$ and $\tilde\eps_2$ fixed, we find that letting

\[
R^2 <  \frac{1}{2^6\cdot 3\cdot 5^2\cdot 7\cdot L^2_\star\cdot \max\{ 2, T^2\} } ,\; c\geq   \max  \{  \eps_1^{-1} 7T  L_{\rm u}^2,  \eps_2^{-1} 7T , \tilde \eps_1^{-1} 3T L_{\rm u}^2,\;  3T\tilde \eps_2^{-1},  2 L_{\rm u}  \}
\]

we have that

\[
\|\delta \mathfrak{H}\|_{\Hc^c}^2  < \  \frac{20}{2^4\cdot  3\cdot 7\cdot 10^2}3(\sqrt{30+(\tilde\eps_1+\tilde\eps_2)}+\sqrt{30})^2  \|\delta \mathfrak{h}\|_{\Hc^c}   = \frac{ (\sqrt{30+(\tilde\eps_1+\tilde\eps_2)}+\sqrt{30})^2}{2^3\cdot 7\cdot 10} \|\delta \mathfrak{h}\|_{\Hc^c}.
\]

Thus, letting choosing $(\sqrt{30+(\tilde\eps_1+\tilde\eps_2)}+\sqrt{30})^2 \leq 2^3\cdot 7\cdot 10$, $\Tf$ is contractive.

Step 3: We consolidate our results.. In light of
(<ref>) and (<ref>), taking $\eps_i=\tilde\eps_i, i\in \{1,2\}$, $c$ must satisfy

\begin{align}\label{eq:cfinalq}
c\geq \max   \{  \eps_1^{-1} 7T  L_{\rm u}^2,  \eps_2^{-1} 7T , \tilde \eps_1^{-1} 3T L_{\rm u}^2,\;  3T\tilde \eps_2^{-1},\; 2 L_{\rm u}  \}= \max   \{  \eps_1^{-1} 7T  L_{\rm u}^2,  \eps_2^{-1} 7T ,\; 2 L_{\rm u}  \}
\end{align}

All together we find that given $\gamma\in(0,\infty)$, $\eps_i\in(0,\infty)$, $i\in\{1,2\}$, $c\in (0,\infty)$,  such that $ \eps_1+\eps_2 \leq (4\sqrt{35}-\sqrt{30})^2-30$, $\Tf$ is a well–defined contraction in $\Bc_{ R}\subseteq \Hc^c$ for the norm $\| \cdot \|_{\Hc^c}$ provided: $(i)$ $\gamma $, $\eps_i$, $i\in \{1,2\}$, and the data of the problem satisfy (<ref>); $(ii)$ $c$ satisfies (<ref>).

[Artzner et al., 1999]
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath.
Coherent measures of risk.
Mathematical Finance, 90 (3):0 203–228, 1999.

[Bielagk et al., 2017]
J. Bielagk, A. Lionnet, and G. Dos Reis.
Equilibrium pricing under relative performance concerns.
SIAM Journal on Financial Mathematics, 80
(1):0 435–482, 2017.

[Briand and Hu, 2006]
P. Briand and Y. Hu.
BSDE with quadratic growth and unbounded terminal value.
Probability Theory and Related Fields, 1360
(4):0 604–618, 2006.

[Briand and Hu, 2008]
P. Briand and Y. Hu.
Quadratic BSDEs with convex generators and unbounded terminal
Probability Theory and Related Fields, 1410
(3-4):0 543–567, 2008.

[Buckdahn et al., 2000]
R. Buckdahn, M. Quincampoix, and A. Rǎşcanu.
Viability property for a backward stochastic differential equation
and applications to partial differential equations.
Probability Theory and Related Fields, 0
(116):0 485—504, 2000.

[Cheridito and Nam, 2015]
P. Cheridito and K. Nam.
Multidimensional quadratic and subquadratic BSDEs with special
Stochastics: An International Journal of Probability and
Stochastic Processes, 870 (5):0 871–884, 2015.

[Delbaen et al., 2011]
F. Delbaen, Y. Hu, and X. Bao.
Backward SDEs with superquadratic growth.
Probability Theory and Related Fields, 150:0 145–192,

[Dellacherie and Meyer, 1982]
C. Dellacherie and P.-A. Meyer.
Probabilities and potential B: theory of martingales,
volume 72 of Mathematics studies.
North–Holland, 1982.

[Ekeland and Lazrak, 2006]
I. Ekeland and A. Lazrak.
Being serious about non-commitment: subgame perfect equilibrium in
continuous time.
ArXiv preprint arXiv:0604264, 2006.

[Ekeland and Lazrak, 2010]
I. Ekeland and A. Lazrak.
The golden rule when preferences are time inconsistent.
Mathematics and Financial Economics, 40 (1):0
29–55, 2010.

[El Karoui and Hamadène, 2003]
N. El Karoui and S. Hamadène.
Bsdes and risk-sensitive control, zero-sum and nonzero-sum game
problems of stochastic functional differential equations.
Stochastic Processes and their Applications, 1070
(1):0 145–169, 2003.

[El Karoui et al., 1997]
N. El Karoui, S. Peng, and M.-C. Quenez.
Backward stochastic differential equations in finance.
Mathematical Finance, 70 (1):0 1–71, 1997.

[Élie and Possamaï, 2019]
R. Élie and D. Possamaï.
Contracting theory with competitive interacting agents.
SIAM Journal on Control and Optimization, 570
(2):0 1157–1188, 2019.

[Espinosa and Touzi, 2015]
G.-É. Espinosa and N. Touzi.
Optimal investment under relative performance concerns.
Mathematical Finance, 250 (2):0 221–257,

[Frei, 2014]
C. Frei.
Splitting multidimensional BSDEs and finding local equilibria.
Stochastic Processes and their Applications, 1240
(8):0 2654–2671, 2014.

[Frei and Dos Reis, 2011]
C. Frei and G. Dos Reis.
A financial market with interacting investors: does an equilibrium
Mathematics and Financial Economics, 40 (3):0
161–182, 2011.

[Hamaguchi, 2021]
Y. Hamaguchi.
Extended backward stochastic Volterra integral equations and their
applications to time-inconsistent stochastic recursive control problems.
Mathematical Control and Related Fields, 110
(2):0 433–478, 2021.

[Harter and Richou, 2019]
J. Harter and A. Richou.
A stability approach for solving multidimensional quadratic BSDEs.
Electronic Journal of Probability, 240 (4):0
1–51, 2019.

[Hernández and Possamaï, 2020]
C. Hernández and D. Possamaï.
Me, myself and I: a general theory of non–Markovian
time-inconsistent stochastic control for sophisticated agents.
ArXiv preprint arXiv:2002.12572, 2020.

[Hernández and Possamaï, 2021]
C. Hernández and D. Possamaï.
A unified approach to well-posedness of type-I backward stochastic
Volterra integral equations.
Electronic Journal of Probability, 260 (89):0
1–35, 2021.

[Hu and Peng, 1991]
Y. Hu and S. Peng.
Adapted solution of a backward semilinear stochastic evolution
Stochastic Analysis and Applications, 90 (4):0
445–459, 1991.

[Hu and Peng, 2006]
Y. Hu and S. Peng.
On the comparison theorem for multidimensional BSDEs.
Comptes Rendus Mathématique, 3430 (2):0
135–140, 2006.

[Hu and Tang, 2016]
Y. Hu and S. Tang.
Multi-dimensional backward stochastic differential equations of
diagonally quadratic generators.
Stochastic Processes and their Applications, 1260
(4):0 1066–1086, 2016.

[Hu et al., 2021]
Y. Hu, S. Tang, and F. Wang.
Quadratic G-BSDEs with convex generators and unbounded terminal
ArXiv preprint arXiv:2101.11413, 2021.

[Jackson and Žitković, 2020]
J. Jackson and G. Žitković.
A characterization of solutions of quadratic BSDEs and a new
approach to existence.
ArXiv preprint arXiv:2004.05412, 2020.

[Jackson and Žitković, 2021]
J. Jackson and G. Žitković.
Existence and uniqueness for non-Markovian triangular quadratic
ArXiv preprint arXiv:2101.12302, 2021.

[Jacod and Shiryaev, 2003]
J. Jacod and A.N. Shiryaev.
Limit theorems for stochastic processes, volume 288 of
Grundlehren der mathematischen Wissenschaften.
Springer–Verlag Berlin Heidelberg, 2003.

[Jamneshan et al., 2017]
A. Jamneshan, M. Kupper, and P. Luo.
Multidimensional quadratic BSDEs with separated generators.
Electronic Communications in Probability, 220
(58):0 1–10, 2017.

[Jouini et al., 2004]
E. Jouini, M. Meddeb, and N. Touzi.
Vector-valued coherent risk measures.
Finance and Stochastics, 80 (4):0 531–552,

[Karatzas and Shreve, 1998]
I. Karatzas and S.E. Shreve.
Brownian motion and stochastic calculus, volume 113 of
Graduate texts in mathematics.
Springer–Verlag New York, 2nd edition, 1998.

[Kazamaki, 1994]
N. Kazamaki.
Continuous exponential martingales and BMO, volume 1579 of
Lecture notes in mathematics.
Springer–Verlag Berlin Heidelberg, 1994.

[Kazi-Tani et al., 2015]
N. Kazi-Tani, D. Possamaï, and C. Zhou.
Quadratic BSDEs with jumps: a fixed-point approach.
Electronic Journal of Probability, 200 (66):0
1–28, 2015.

[Kobylanski, 2000]
M. Kobylanski.
Backward stochastic differential equations and partial differential
equations with quadratic growth.
The Annals of Probability, 280 (2):0 558–602,

[Kramkov and Pulido, 2016]
D.O. Kramkov and S. Pulido.
Stability and analytic expansions of local solutions of systems of
quadratic BSDEs with applications to a price impact model.
SIAM Journal on Financial Mathematics, 70
(1):0 567–587, 2016a.

[Kramkov and Pulido, 2016]
D.O. Kramkov and S. Pulido.
A system of quadratic BSDEs arising in a price impact model.
The Annals of Applied Probability, 260 (2):0
794–817, 2016b.

[Kulikov, 2008]
A.V. Kulikov.
Multidimensional coherent and convex risk measures.
Theory of Probability & Its Applications, 520
(4):0 614–635, 2008.

[Kupper et al., 2019]
M. Kupper, P. Luo, and L. Tangpi.
Multidimensional Markovian FBSDEs with super-quadratic growth.
Stochastic Processes and their Applications, 1290
(3):0 902–923, 2019.

[Landriault et al., 2018]
D. Landriault, B. Li, D. Li, and V.R. Young.
Equilibrium strategies for the mean–variance investment problem over
a random horizon.
SIAM Journal on Financial Mathematics, 90
(3):0 1046–1073, 2018.

[Lepeltier and San Martín, 1998]
J.-P. Lepeltier and J. San Martín.
Existence for BSDE with superlinear–quadratic coefficient.
Stochastics: An International Journal of Probability and
Stochastic Processes, 630 (3–4):0 227–240, 1998.

[Lin, 2002]
J. Lin.
Adapted solution of a backward stochastic nonlinear Volterra
integral equation.
Stochastic Analysis and Applications, 200
(1):0 165–183, 2002.

[Luo, 2020]
P. Luo.
A type of globally solvable bsdes with triangularly quadratic
Electronic Journal of Probability, 25:0 1–23, 2020.

[Luo, 2021]
P. Luo.
Comparison theorem for diagonally quadratic bsdes.
Discrete & Continuous Dynamical Systems, 410
(6):0 2543–2557, 2021.

[Meyer, 1966]
P.-A. Meyer.
Probability and potentials.
Blaisdell Publishing Company, 1966.

[Pardoux and Peng, 1990]
É. Pardoux and S. Peng.
Adapted solution of a backward stochastic differential equation.
System and Control Letters, 140 (1):0 55–61,

[Protter, 2005]
P.E. Protter.
Stochastic integration and differential equations, volume 21
of Stochastic modelling and applied probability.
Springer–Verlag Berlin Heidelberg, 2nd edition, 2005.

[Ren, 2010]
Y. Ren.
On solutions of backward stochastic Volterra integral equations
with jumps in Hilbert spaces.
Journal of Optimization Theory and Applications, 1440
(2):0 319–333, 2010.

[Shi and Wang, 2012]
Y. Shi and T. Wang.
Solvability of general backward stochastic Volterra integral
Journal of the Korean Mathematical Society, 490
(6):0 1301–1321, 2012.

[Stroock and Varadhan, 1997]
D.W. Stroock and S.R.S. Varadhan.
Multidimensional diffusion processes, volume 233 of
Grundlehren der mathematischen Wissenschaften.
Springer–Verlag Berlin Heidelberg, 1997.

[Tevzadze, 2008]
R. Tevzadze.
Solvability of backward stochastic differential equations with
quadratic growth.
Stochastic Processes and their Applications, 1180
(3):0 503–515, 2008.

[Viens and Zhang, 2019]
F. Viens and J. Zhang.
A martingale approach for fractional Brownian motions and related
path dependent PDEs.
The Annals of Applied Probability, 290 (6):0
3489–3540, 2019.

[Wang, 2020]
H. Wang.
Extended backward stochastic Volterra integral equations,
quasilinear parabolic equations, and Feynman–Kac formula.
Stochastics and Dynamics, 210 (01):0 215004,

[Wang and Yong, 2021]
H. Wang and J. Yong.
Time-inconsistent stochastic optimal control problems and backward
stochastic Volterra integral equations.
ESAIM: Control, Optimisation and Calculus of Variations,
270 (22):0 1–40, 2021.

[Wang et al., 2019]
H. Wang, J. Sun, and J. Yong.
Recursive utility processes, dynamic risk measures and quadratic
backward stochastic Volterra integral equations.
Applied Mathematics and Optimization, to appear, 2019.

[Wang and Yong, 2019]
T. Wang and J. Yong.
Backward stochastic Volterra integral equations—representation of
adapted solutions.
Stochastic Processes and their Applications, 1290
(12):0 4926–4964, 2019.

[Wang and Zhang, 2007]
Z. Wang and X. Zhang.
Non-Lipschitz backward stochastic Volterra type equations with
Stochastics and Dynamics, 70 (4):0 479–496,

[Wei et al., 2017]
Q. Wei, J. Yong, and Z. Yu.
Time-inconsistent recursive stochastic optimal control problems.
SIAM Journal on Control and Optimization, 550
(6):0 4156–4201, 2017.

[Xing and Žitković, 2018]
H. Xing and G. Žitković.
A class of globally solvable Markovian quadratic BSDE systems and
The Annals of Probability, 460 (1):0 491–550,

[Yong, 2006]
J. Yong.
Backward stochastic Volterra integral equations and some related
Stochastic Processes and their Applications, 1160
(5):0 779–795, 2006.

[Yong, 2008]
J. Yong.
Well-posedness and regularity of backward stochastic Volterra
integral equations.
Probability Theory and Related Fields, 1420
(1–2):0 21–77, 2008.

[Yong, 2012]
J. Yong.
Time-inconsistent optimal control problems and the equilibrium HJB
Mathematical Control and Related Fields, 20
(3):0 271–329, 2012.

§ PROOFS OF SECTION <REF>

First note that for $Z\in \H^2_{{\rm BMO}}(\R^{n\times \tilde d})$, $Z\bullet X$ is a continuous local martingale, thus we have that

\[ \| Z\bullet  X\|_{{\rm BMO}^{2,c}}=\sup_{\tau\in\Tc_{0,T}} \Big \| \E\big[ \big \langle\e^{\frac{c}2} Z\bullet X\big\rangle_T- \big \langle \e^{\frac{c}2} Z\bullet X\big \rangle_\tau \big|\Fc_\tau\big]\Big\|_\infty<\infty.\]

Therefore, letting $X_t:=\E[ \langle \e^{\frac{c}2 }Z\bullet X\rangle_T- \langle \e^{\frac{c}2 } Z\bullet X\rangle_t |\Fc_t]$, we have: $(i)$ $|X_t|\leq  \| Z\bullet X\|_{{\rm BMO}^{2,c}}=\| Z\|_{\H^{2,c}_{{\rm BMO}}}^2$;
$(ii)$ $A=\langle \e^{\frac{c}2 } Z\bullet X\rangle$. Indeed, note  $X_t=\E\big[  \big\langle \e^{\frac{c}2 }Z\bullet X\big\rangle_T  \big|\Fc_t\big]- \big\langle \e^{\frac{c}2 } Z\bullet X\big\rangle_t $. The result then follows immediately from the energy inequality, i.e.

\[ \E\bigg[ \bigg(\int_0^T\e^{cr }|\sigma^\t_r Z_r|^2 \d r\bigg)^p\bigg]=\E[ (A)_\infty^p] \leq  p !\| Z\|_{\H^{2,c}_{{\rm BMO}}}^{2p}.\]

To obtain the second part of the statement, recall that by definition of $\Ho(\R^{n\times \tilde d})$, $s\longmapsto \partial Z^s$ is the density of $s\longmapsto Z^s$ with respect to the Lebesgue measure and $\Zc$ is given as in <Ref>. By definition of $\Zc$, Fubini's theorem and Young's inequality we have that for $\eps>0$

\begin{align*}
\int_t^T \e^{cu} | \sigma^\t_uZ_u^u|^2-\e^{cu}  |\sigma^\t_u Z_u^t|^2 \d u& = \int_t^T \int_r^T 2 \e^{cu}  \Tr\Big [ {Z_u^r}^\t{\sigma_u}   {\sigma^\t_u} \partial Z_u^r   \Big ] \d u \d r\\
& \leq \int_t^T\int_r^T \eps \e^{cu}  |\sigma_u^\t  Z^r_u|^2+ \eps^{-1}\e^{cu}   |\sigma_u^\t \partial Z^r_u |^2 \d u \d r .
\end{align*}

This proves the first first statement. For the second claim, we may use (<ref>) and (<ref>) to obtain

\begin{align*}
\E_t\bigg[ \bigg( \int_t^T \e^{cu} |\sigma^\t_u \Zc_u|^2 \d u\bigg)^2\bigg] & \leq 3\bigg(\E_t\bigg[ \bigg( \int_t^T\e^{cu}  | \sigma^\t_uZ_u^t|^2 \d u\bigg)^2\bigg]\\
&\quad + T \int_t^T\E_t\bigg[ \bigg(\int_t^T \e^{cu}   |\sigma_u^\t  Z^r_u|^2\d u \bigg)^2\bigg] \d r  + T \int_t^T \E_t\bigg[ \bigg( \int_t^T   \e^{cu}  |\sigma_u^\t \partial Z^r_u |^2 \d u \bigg)^2\bigg]\d r\bigg)\\
& \leq 6 ( (1+T^2)\|Z\|_{\H^{2,2,c}_{\rm BMO}}^4+T^2 \|\partial Z\|_{\H^{2,2,c}_{\rm BMO}}^4)
\end{align*}

The inequality for the $\H^2$ norm is argued similarly taking expectations.

§ PROOFS OF SECTION <REF>

We next lemma helps derive appropriate auxiliary estimates of the terms $U_t^t$ and $\partial U_t^t$ as in <Ref>.

Let $\partial U$ satisfy the equation

\[
\partial U_t^s= \partial_s \eta (s,X_{\cdot\wedge,T})+\int_t^T   \nabla g_r(s,X,\partial U_r^s,\partial v_r^s,U_r^s,v_r^s, \Yc_r, z_r) \d r-\int_t^T  \partial {V_r^s}^\t  \d X_r-\int_t^T \d  \partial M^s_r,
\]

and $c\geq \max\{ 2L_u, 2L_{\rm u}\}$, the following estimates hold for $t\in [0,T]$

\begin{align*}
\E_t\bigg[ \int_t^T \frac{ \e^{cr}}{7   T}  |\partial U_r^r|^2\d r\bigg]&  \leq    \|\partial_s \eta \|_{\Lc^{\infty,2,c}}^2+ \|\nabla \tilde g\|^2_{\L^{1,\infty,2,c}} +  T L_y^2  \E_t\bigg[   \int_t^T \e^{cr}|Y_r|^2\d r\bigg]+  T L_u^2 \sup_{s\in [0,T]} \E_t\bigg[   \int_t^T \e^{cr}|U_r^s|^2\d r\bigg]\\
& \quad +   2 L_{\star}^2    \Big( \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}} + \|z\|^4_{\H^{2,c}_{{\rm BMO}}}+\|v\|^4_{\H^{2,2,c}_{{\rm BMO}}}\Big)\\
\E_t\bigg[ \int_t^T \frac{ \e^{\frac{c}2 r}}{ T}  |\partial U_r^r|\d r\bigg] & \leq    \|\partial_s \eta \|_{\Lc^{\infty,2,c}}+ \|\nabla \tilde g\|_{\L^{1,\infty,2,c}} +   L_y  \E_t\bigg[   \int_t^T \e^{\frac{c}2 r}|Y_r|\d r\bigg]+   L_u \sup_{s\in [0,T]} \E_t\bigg[   \int_t^T \e^{cr}|U_r^s|\d r\bigg]\\
&\quad  +    L_{\star}    \Big(\|\partial v\|^2_{\H^{2,2,c}_{{\rm BMO}}}+  \|v\|^2_{\H^{2,2,c}_{{\rm BMO}}}+ \|z\|^2_{\H^{2,c}_{{\rm BMO}}}\Big)
\end{align*}

By Meyer–Itô's formula for $\e^{\frac{c}2  t} |\partial U_t^s|$, see <cit.>

\begin{align}\label{eq:eq1}
\begin{split}
&\e^{\frac{c}2  t}|\partial U_t^s|+ L_T^0 -\int_t^T \e^{\frac{c}2  r} \sgn( \partial U_r^s)\cdot  \partial {V_r^s}^\t  \d X_r -\int_t^T \e^{\frac{c}2  r-} \sgn( \partial U_{r-}^s)\cdot  \d \partial M_r^s \\
&\; =\e^{\frac{c}2  T}  |\partial_s \eta (s)|  + \int_t^T  \e^{\frac{c}2  r} \bigg(  \sgn(  \partial U_r^s)  \cdot \nabla g_r(s,\partial U_r^s,\partial v_r^s,U_r^s,v_r^s,\Yc_r,z_r)-\frac{c}2  |\partial U_r^s| \bigg) \d r ,\; t\in[0,T],
\end{split}
\end{align}

where $L^0:=L^0(\partial U^s)$ denotes the non-decreasing and pathwise-continuous local time of the semi-martingale $\partial U^s$ at $0$, see <cit.>. We also notice that for any $s\in [0,T]$ the last two terms on the left-hand side are martingales, recall that $\partial V^s\in \H^2$ by <cit.>.

In light of <Ref>, letting $ \nabla g_r(s):=\nabla g_r(s,\partial U_r^s,\partial v_r^s,U_r^s,v_r^s,Y_r, z_r)$, we have that $\d t\otimes \d \P\ae$

\begin{align}\label{Eq:ineqLipUts0}
\begin{split}
|\nabla g_r(s) |\leq & L_{\rm u} |\partial U_r^s|   +L_{\rm v} |\sigma^\t_r  \partial v_r^s|^2+L_u |U_r^s| +L_v |\sigma^\t_r  v_r^s|^2+L_y |Y_r|+L_z |\sigma^\t_r  z_r|^2+  |\nabla \tilde g_r(s)|,
\end{split}
\end{align}

We now take conditional expectation with respect to $\Fc_t$ in <Ref>. We may use (<ref>) and the fact $\tilde L^0$ is non–decreasing to derive that for $c>2 L_{\rm u}$ and $ t\in[0,T]$

\begin{align}\label{Eq:ineqUst}
\e^{\frac{c}2 t}| \partial U_t^s| & \leq  \E_t \bigg[ \e^{\frac{c}2 T} |\partial \eta(s)|+\int_t^T \e^{\frac{c}2 r} \big(  |\nabla \tilde g_r(s)| +L_{\rm v} |\sigma^\t_r  \partial v_r^s|^2+L_u |U_r^s|  +L_v |\sigma^\t_r   v_r^s|^2+L_y |Y_r|+L_z |\sigma^\t_r   z_r|^2\big)  \d r \bigg].
\end{align}

Squaring in (<ref>), we may use (<ref>) and Jensen's inequality to derive that for $t\in [0,T]$

\begin{align*}
\frac{\e^{ct}}{7} |\partial U_t^t|^2  \leq    &\  \E_t\bigg[ \e^{cT} |\partial_s \eta(t)|^2+ \bigg(\int_t^T \e^{\frac{c}2 r} |\nabla \tilde g_r(t)|\d r\bigg)^2+ T L_{u}^2 \int_t^T \e^{c r}    |U_r^t|^2 \d r + T L_y^2 \int_t^T \e^{c r}    |Y_r|^2 \d r\\
& + L_{\rm v}^2 \bigg(\int_t^T \e^{\frac{c}2 r} |\sigma^\t_r \partial v_r^t|^2\d r \bigg)^2+ L_v^2 \bigg(\int_t^T \e^{\frac{c}2 r} |\sigma^\t_r v_r^t|^2\d r\bigg)^2+ L_z^2 \bigg(\int_t^T \e^{\frac{c}2 r} |\sigma^\t_rz_r|^2\d r\bigg)^2 \bigg].
\end{align*}

By integrating the previous expression and taking conditional expectation with respect to $\Fc_t$, it follows from the tower property that for any $t\in[0,T]$

\begin{align*}
\frac{  1}7\E_t\bigg[\int_t^T \e^{cr}|\partial U_r^r|^2\d r\bigg]\leq &\ \E_t\bigg[ \int_t^T   \e^{cT} |\partial_s \eta(r)|^2\d r\bigg]+\E_t\bigg[ \int_t^T\bigg[ \bigg(  \int_r^T \e^{\frac{c}2 u}|\nabla \tilde g_u(r)|\d u\bigg)^2\d r\bigg] \\
&   + T L_{u}^2 \E_t\bigg[  \int_t^T   \int_r^T \e^{c u}    |U_u^r|^2 \d u\bigg] \d r +  T  L_y^2  \E_t\bigg[  \int_t^T   \int_r^T \e^{cu} |Y_u|^2\d u \bigg]\d r\bigg]  \\
& +  L_{\rm v}^2     \int_t^T\E_t\bigg[    \bigg(\int_r^T \e^{\frac{c}2 u} |\sigma^\t_u \partial v_u^r|^2\d u\bigg)^2\bigg] \d r +  L_v^2     \int_t^T\E_t\bigg[    \bigg(\int_r^T \e^{\frac{c}2 u} |\sigma^\t_u v_u^r|^2\d u\bigg)^2\bigg] \d r\\
& + L_z^2    \int_t^T\E_t\bigg[  \bigg(\int_r^T \e^{\frac{c}2 u} | \sigma^\t_u z_u|^2\d u\bigg)^2\bigg] \d r\\
\leq &\ T \sup_{r\in [0,T]} \bigg\{ \|  \e^{cT} |\eta(r)|^2]\|_\infty+  \bigg \|  \int_r^T \e^{\frac{c}2 u} |\nabla \tilde g_u(r)|\d u \bigg \|_\infty^2 \bigg\} +T^2L_y^2 \E_t\bigg[   \int_t^T \e^{cu}|Y_u|^2\d u\bigg]\\
&  + T^2 L_{u}^2 \sup_{r\in [0,T]}  \bigg\{ \E_t\bigg[  \int_t^T  \e^{c u}    |U_u^r|^2 \d u\bigg]  \bigg\} +    T L_{\rm v}^2    \sup_{r\in [0,T]} \bigg\{ \E_t\bigg[    \bigg(\int_t^T \e^{\frac{c}2 u} |\sigma^\t_u \partial v_u^r|^2\d u\bigg)^2\bigg] \d r\bigg\}    \\
&+    T L_v^2    \sup_{r\in [0,T]} \bigg\{ \E_t\bigg[    \bigg(\int_t^T \e^{\frac{c}2 u} |\sigma^\t_u v_u^r|^2\d u\bigg)^2\bigg] \d r\bigg\}  + TL_z^2    \E_t\bigg[  \bigg(\int_t^T \e^{\frac{c}2 u} | \sigma^\t_u z_u|^2\d u\bigg)^2\bigg] \d r,
\end{align*}

and by (<ref>) we obtain for $c>2L_u$, and any $t\in[0,T]$

\begin{align*}
\E_t\bigg[ \int_t^T \frac{ \e^{cr}}{7   T}  |\partial U_r^r|^2\d r\bigg]& \leq    \|\partial_s \eta \|_{\Lc^{\infty,2,c}}^2+ \|\nabla \tilde g\|^2_{\L^{1,\infty,2,c}} +  T L_y^2  \E_t\bigg[   \int_t^T \e^{cr}|Y_r|^2\d r\bigg]+ + T L_{u}^2 \sup_{r\in [0,T]}   \E_t\bigg[  \int_t^T  \e^{c u}    |U_u^r|^2 \d u\bigg]    \\
&\; +   2 L_\star^2    \Big( \|\partial v\|^4_{\H^{2,2,c}_{{\rm BMO}}} + \|z\|^4_{\H^{2,c}_{{\rm BMO}}}+\|v\|^4_{\H^{2,2,c}_{{\rm BMO}}}\Big) .
\end{align*}

Evaluating at $s=t$ in (<ref>) and integrating with respect to $t$ we derive the second estimate.

$({\rm OPT1})=1/(2^{4}5 )$, where

\begin{align*}
\sup\; &\frac{ \min\big \{  \alpha(\eps_3,\eps_{12},\eps_{13}),\;  \alpha(\eps_4,\eps_{14},\eps_{15}) ,\; \alpha(\eps_{5},\eps_{16},\eps_{17}) ,\; \alpha(\eps_{6},\eps_{18},\eps_{19},\eps_{20}) \big \} -\gamma}{ \eps_1+\eps_2+\sum_{i=12}^{20} \eps_i  }\\
{\rm s.t.}\;  &  \alpha(\eps_8,\eps_{12},\eps_{13})=1-10(\eps_{8}^{-1}+\eps_{12}^{-1}+\eps_{13}^{-1})   \in (0,1], \;
\alpha(\eps_9,\eps_{14},\eps_{15})=1-10(\eps_{9}^{-1}+\eps_{14}^{-1}+\eps_{15}^{-1})   \in (0,1], \\
& \alpha(\eps_{10},\eps_{16},\eps_{17}) =1-10(\eps_{10}^{-1}+\eps_{16}^{-1}+\eps_{17}^{-1})   \in (0,1], \;  \alpha(\eps_{11},\eps_{18},\eps_{19},\eps_{20})=1-10(\eps_{11}^{-1}+\eps_{18}^{-1}+\eps_{19}^{-1}+\eps_{20}^{-1})  \in (0,1], \\
& \gamma\in (0,\infty);\;  \eps_i\in (0,\infty), \forall i .
\end{align*}

We begin by noticing that as a function of $(\gamma,\eps_1,\eps_2,\eps_3,\eps_4,\eps_{5},\eps_{6})$ the objective is bounded by the value when $(\gamma,\eps_1,\eps_2,\eps_3,\eps_4,\eps_{5},\eps_{6})\longrightarrow (0,0,0,\infty,\infty,\infty,\infty)$. Thus, we will maximise

\[\frac{\min\{ 1-10( \eps_{12}^{-1}+\eps_{13}^{-1}) ,\;1-10( \eps_{14}^{-1}+\eps_{15}^{-1})  ,\;1-10( \eps_{16}^{-1}+\eps_{17}^{-1})   ,\;1-10(\eps_{18}^{-1}+\eps_{19}^{-1}+\eps_{20}^{-1})  \} }{\sum_{i=12}^{20} \eps_i  }   . \]

From this we observe that the optimal value is positive. Indeed, there is a feasible solution with positive value, and the $\min$ in the objective function does not involve common $\eps_i$ terms, so the minima is attained at one of the terms. Since the value function is symmetric in each of the variables inside each term of the mean we can assume with out lost of generality

\[
\eps_{12}=\eps_{13}=2\alpha_1,\;\eps_{14}=\eps_{15}=2\alpha_2,\; \eps_{16}=\eps_{17}=2\alpha_3, \; \eps_{18}=\eps_{19}=\eps_{20}=3\alpha_4,\; \{\alpha_1,\alpha_2,\alpha_3,\alpha_4\} \in (0,\infty)^4
\]

So we can write the objective function as $\min\{ 1-10\alpha_1^{-1} ,\;1-10\alpha_2^{-1}  ,\;1-10\alpha_3^{-1}   ,\;1-10\alpha_4^{-1}  \} /( 4\alpha_1+4\alpha_2+4\alpha_3+9\alpha_4)$. Now, without lost of generality the $\min$ is attained by the first quantity. This is,
the optimisation problem becomes

\begin{align*}
\sup\;\frac{ 1-10\alpha_1^{-1}  }{ 4\alpha_1+4\alpha_2+4\alpha_3+9\alpha_4   }\;  {\rm s.t.}\;   \alpha_1\leq \min\{\alpha_2,\alpha_3,\alpha_4\}, 1-10\alpha_i^{-1}\in (0,1], \alpha_i\in (0,\infty), i\in \{1,2,3,4\}.
\end{align*}

Now, as the objective function is decreasing in $\alpha_2,\alpha_3,\alpha_4$, and $\alpha_1\leq \min\{\alpha_2,\alpha_3,\alpha_4\}$, we see $\alpha_1=\alpha_2=\alpha_3=\alpha_4$. Thus

\begin{align*}
\sup\;\frac{ 1-10\alpha_1^{-1}  }{ 21 \alpha_1   }\;  {\rm s.t.}\;   1-10\alpha_1^{-1}\in (0,1], \alpha_1\in (0,\infty).
\end{align*}

Let $f(\alpha_1):= \frac{ \alpha_1 -10}{21 \alpha_1^2}$. By first order analysis

\[ \partial_{\alpha_1}f(\alpha_1) = \frac{-\alpha_1(\alpha_1-20)}{21 \alpha_1^4 }=0, \text{ yields, }\alpha_1\in \{0,20 \}
\]

By inspecting the sign of the derivative, one sees that $\alpha_1=0$ corresponds to a minima and $\alpha_1=20$ is the maximum and it is feasible. Thus we obtain that

\begin{align*}
f\big(\alpha_1^\star\big)=\frac{1}{2^3\cdot 3\cdot 5\cdot 7},
\end{align*}

We conclude the maxima when $(\eps_{12},\eps_{13},\eps_{14},\eps_{15},\eps_{16},\eps_{17},\eps_{18},\eps_{19},\eps_{20})=(40,40,40,40,40,40,60,60,60)$. Evaluating the value function in these values and letting $(\gamma,\eps_1,\eps_3,\eps_8,\eps_9,\eps_{10},\eps_{11})\longrightarrow (0,0,0,\infty,\infty,\infty,\infty)$, we obtain this bound. This is, $f$ does not attain its maximum value, but in the feasible region it can get as close as possible.

Case 1: $1-10\eps_{12}^{-1} <\min\{  1-10\eps_{13}^{-1} ,\;1-10\eps_{14}^{-1} ,\;1-10\eps_{15}^{-1} \}$. We refer to the previous inequality as $($C1$)$. Then the $\min$ is attained by the first expression and we optimise on $\eps_{12}$.

We then write the objective as

\[f(\eps_{12}):= \frac{  \eps_{12} -10}{\eps_{12}( \eps_{12}+ \eps_{13}+\eps_{14}+\eps_{15} )}\]

By first order analysis

\[ \partial_{\eps_7}f(\eps_7)= \frac{- \eps_{12}^2 +20\eps_{12} +10(\eps_{13}+\eps_{14}+\eps_{15})}{\eps_{12}^2(\eps_{13}+\eps_{14}+\eps_{15} )^2}=0, \text{ yields, }\eps_{12}^{\pm}= 10 \pm \sqrt{100+10  (\eps_{13}+\eps_{14}+\eps_{15})}
\]

Therefore, as the optimal value is positive we must have $1 -10\eps_{12}^{-1} >0$, which implies that the maximum is attained at

\[\eps_{12}^{\star}(\eps_{13},\eps_{14},\eps_{15}):=10 + \sqrt{100+10  (\eps_{13}+\eps_{14}+\eps_{15})} .\]

and by inspecting the sign of $\partial_{\eps_{12}} f$ we verify $\eps_{12}^\star(\eps_{13},\eps_{14},\eps_{15})$ is indeed a maximum. Thus we obtain that

\begin{align}\label{Eq:c1:1}
f\big(\eps_{12}^\star(\eps_{13},\eps_{14},\eps_{15})\big)=\frac{1}{(\sqrt{10}+\sqrt{10+ \eps_{13}+\eps_{14}+\eps_{15} })^2},
\end{align}

and note
$f\big(\eps_{12}^\star(\eps_{13},\eps_{14},\eps_{15})\big)$ is maximised whenever $(\eps_{13},\eps_{14},\eps_{15})\longrightarrow (0,0,0)$. However this choice does not verify the $($C1$)$. We thus now enforce $($C1$)$. We note that (<ref>) is symmetric in $\eps_{13},\eps_{14},\eps_{15}$ therefore, we set $\eps_{13}=\eps_{14}=\eps_{15}=\eps$.  It then follows from $($C1$)$ that

\[ \eps_{12}^\star(\eps,\eps,\eps)< \eps  \Longleftrightarrow \eps>50. \]

\[f(\eps_{12}^\star(50,50,50))= \frac{1}{5^3 2} \]

Case 2: $1 -5\eps_7^{-1}>1 -5\eps_5^{-1}-5\eps_6^{-1}$. In this case we note the objective is completely symmetric in $\eps_5$ and $\eps_6$, and so are its partial derivatives. Thus we assume, without lost of generality, $\eps_5=\eps_6=2 \lambda$, and optimise for $\lambda$. The analysis is completely analogous to the previous case.

We write the objectives as

\[ f(\lambda):=   \frac{  \lambda  -5}{ \lambda (4\lambda+\eps_7 ) }, \]

and first order analysis yields candidates $\lambda^\pm(\eps_7)= 5 \pm1/2  \sqrt{10^2+5\eps_7}$. Now, $1-5\lambda>0$ yield $\lambda^\star(\eps_7)=\lambda^+(\eps_7)$ and

\[ f(\lambda^\star(\eps_7)):= \frac{ 1}{(10+\sqrt{10^2+  5\eps_7 })^2},\]

thus the maximum value is $1/(2^4 5^2)$ whenever $\eps_7 \longrightarrow 0 $. However this choice does not verify the $($C2$)$. Enforcing it we find that $\eps_7(4 \eps-45)>0$ and thus $f(\lambda^\star(45/4))=2^2/(3^4 5^2)$. The maximum remain the one in Case 1.

$({\rm OPT2})=3(\sqrt{30+(\tilde\eps_1+\tilde\eps_2)}+\sqrt{30})^2$, where

\begin{align*}
({\rm OPT2}):=&\ \inf \bigg\{ \big(3 \tilde \eps_1 + 3 \tilde \eps_2 +2\tilde \eps_3+2\tilde \eps_4 +2\tilde \eps_{5}+3\tilde \eps_{6}\big)  \min \bigg\{   \frac{\tilde \eps_3}{ \tilde \eps_3 -10} ,\;  \frac{\tilde \eps_4}{ \tilde \eps_4 -10},\;  \frac{\tilde \eps_{5}}{ \tilde \eps_{5} -10},\;  \frac{\tilde \eps_{6}}{ \tilde \eps_{6} -10} \bigg\}\bigg\} \\
{\rm s.t.}\;  &  1-10\tilde\eps_i^{-1} \in (0,1] ,\;  \tilde \eps_i \in (0,\infty),\; i\in\{3,4,5,6\}.
\end{align*}

Without lost of generality let us assume the min is attained by the first quantity, i.e.  the optimisation problem becomes

\[ \inf   \big(3(\tilde\eps_1+\tilde\eps_2)+ 2\tilde \eps_3+2\tilde \eps_4 +2\tilde \eps_{5}+3\tilde \eps_{6}\big)    \frac{\tilde \eps_3}{ \tilde \eps_3-10} , \; \text{ s.t. } \tilde \eps_8\leq \min\{\tilde\eps_4,\tilde\eps_{5},\tilde\eps_{6}\},\;  1-10\tilde\eps_3^{-1} \in (0,1],\;  \tilde \eps_i \in (0,\infty)\ \forall i.\]

As the value function is increasing in $(\tilde\eps_4,\tilde\eps_{5},\tilde\eps_{6})$, $\tilde \eps_3\leq \min\{\tilde\eps_4,\tilde\eps_{5},\tilde\eps_{6}\}$ implies we must have $\tilde \eps_3=\tilde\eps_4=\tilde\eps_{5}=\tilde\eps_{6}$ a thus we minimise

\[ f(\tilde \eps):=3\frac{3 \tilde \eps^2 +\tilde\eps(\tilde \eps_1+\tilde \eps_2)}{\tilde \eps-10}.\]

First order analysis renders

\[ \partial_{\tilde\eps}f(\tilde\eps) = \frac{9\tilde\eps-180\tilde\eps-30(\tilde\eps_1+\tilde\eps_2)}{\tilde\eps-10}=0, \text{ yields, }\tilde\eps^\pm= 10\pm \frac{1}6\sqrt{60^2+120(\tilde\eps_1+\tilde\eps_2)}.
\]

The minimum occurs at $\tilde\eps^\star=10+ \frac{1}6\sqrt{60^2+120(\tilde\eps_1+\tilde\eps_2)}$, and $f(\tilde\eps^\star)=3(\sqrt{30+(\tilde\eps_1+\tilde\eps_2)}+\sqrt{30})^2$. We conclude the minima occurs when $(\tilde\eps_3,\tilde\eps_4,\tilde\eps_{5},\tilde\eps_{6})=(20,20,20,20)$. Evaluating the value function in these values and letting $(\tilde \eps_1,\tilde \eps_2)\longrightarrow (0,0)$, we obtain this bound. This is, $f$ does not attain its minimum value, but in the feasible region it can get as close as possible.

Let $f(\tilde \eps_1):= \tilde \eps_1^{-1} 2T L_u^2 +\tilde  \eps_1 T L_y^2$ and consider the problem

\begin{align*}
({\rm OPT3})&:= \inf\big\{  f(\tilde \eps_1)\big\}\\
{\rm s.t.}\;  &    (\sqrt{15} + \sqrt{15 + 2\tilde \eps_1})^2 \leq 2^2 5^2, \;    \tilde \eps_1 \in (0,\infty).
\end{align*}

Let $2\alpha:= (10-\sqrt{15})^2-15$ and $\tilde \eps_1^\star:={\rm min} \big \{  \sqrt{2}L_u L_y^{-1}, \alpha\big\}$. Then $({\rm OPT3})=f(\tilde \eps_1^\star)$.

We note that the constraint can be rewritten as $\tilde \eps_1 \leq  \alpha$, and therefore

\begin{align*}
({\rm OPT3})= \inf\{ f(\tilde \eps_1)\},\;  {\rm s.t.}\;        \tilde \eps_1 \in (0,\alpha].
\end{align*}

By first order analysis we find

\[ \partial_{\tilde \eps_1} f(\tilde \eps_1)=-2TL_yL_u^2 \tilde \eps_1^{-2}+TL_y^2=0, \text{ yields } \tilde \eps_1=\sqrt{2}L_u L_y^{-1} .\]

we conclude $\tilde \eps_1^\star=\min\big \{  \sqrt{2}L_u L_y^{-1}, \alpha\big\}.$

Consider the problem

\begin{align*}
({\rm OPT4})&:= \inf\big\{ \max\{a+\eps,b+c\eps^{-1}\}\big\}\\
{\rm s.t.}\;  &     \eps \in (0,\infty).
\end{align*}

$({\rm OPT4})=\min\{ b+c/\eps^{+}, a+\eps^{-}\}$, where $\eps^-\leq 0\leq  \eps^+$ denote the real valued roots of $f(\eps):= \eps^2+(a-b)\eps-c$.

Case 1: $a+\eps\leq b+c \eps^{-1}$. In this case, we must have $f(\eps)<0$, yielding $\eps\in [\eps^-,\eps^+]$. Back in the objective, we obtain

\[ \inf \ \{b+c\eps^{-1}\},\ {\rm s.t. }\ \eps\in [\eps^-,\eps^+] \]

Thus $\eps^\star=\eps^+$, and $({\rm OPT4})=b+c\eps^{-1}$.

Case 2: $a+\eps>b+c /\eps^{+}$. Arguing similarly we obtain, $\eps^\star=\eps^-$, and $({\rm OPT4})=a+\eps^{-}$.

We conclude $({\rm OPT4})=\min\{ b+c/\eps^{+}, a+\eps^{-}\}$.