Marie Le Guilly
email: marie.le-guilly@liris.cnrs.fr

Jean-Marc Petit
email: jean-marc.petit@liris.cnrs.fr

Ihab F Ilyas
email: ilyas@uwarteloo.ca

ExplIQuE: Interactive Databases Exploration with SQL

Keywords: SQL, extensions, imprecise queries

To help databases users who have just started learning SQL or are not familiar with their database, we propose ExplIQuE, an exploration interface with query extensions. Its purpose is to assist users to smoothly dive into data exploration, and to be able to express imprecise questions over their data. Indeed, such situations are more and more current with the increasing desire for users to get value out of their data. In this configuration, in addition to classic SQL querying possibilities, ExplIQuE offers the possibility to extend a given SQL query, by suggesting a set of possible selection predicates to add to the query, that aim at dividing the initial answer set to identify interesting exploration zones. In addition, ExplIQuE proposes some indicators to help the user in choosing its desire extension and in understanding her data, as well as interactive visualizations of the result set, in two dimensions revealed by PCA techniques. In this demonstration, we offer the audience the possibility to try the various functionalities of ExplIQuE by trying to express an imprecise question over a scientific database on bacterial colonies, through an iterative process. A video of the proposed demonstration is available at https://youtu.be/oK8xWGCWj_A.

INTRODUCTION

SQL is a language that appeared in the 1960s, and is still widely used nowadays. Over the years, the volume of data that is stored has increased tremendously, creating new needs for users to be able Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. CIKM '19, November 3-7, 2019, Beijing, China © 2019 Association for Computing Machinery. ACM ISBN 978-1-4503-6976-3/19/11. . . $15.00 https://doi.org/10.1145/3357384.3357847 to store, access, and analyze their data. Moreover, with the recent development and popularity of data analysis techniques, combined with an easier access to data collecting and storing devices, more and more users are trying to make sense and get value out of their data.

As a consequence, the structure of databases tends to change, with bigger schemas, larger tables, and of course, more and more tuples. In addition, the use of databases by users is also evolving, because the data they are looking for is less easy to find, because they are less experienced, and because the question they are trying to answer is not always totally clear: we call such questions imprecise queries, that are expressed in natural language but not easily translatable into SQL. This is especially true in exploratory contexts, where it is necessary to deeply understand the data, and to try several different SQL queries, before translating the initial question into SQL and reaching the desired data. Indeed, when a user is confronted to a relational database with a question that is not necessarily clear in the first place, she will go through different phases. First, she can explore and navigate the database to understand its content and structure better. Then, once more familiar with the data structure, she can express a first simple and general query, before refining it over and over, until she answers the initial question at hand. As a result, the desired tuple set is not reached with a single query, but with a sequence of iterative refinement that, one by one, that gets the user closer to what she is looking for, and gradually builds her final answer set.

This query refinement process is not always easy for users, especially when they do not know where to start it from, or when for example their initial query returns an answer set that presents too many tuples. Tools to assist users in such situations are therefore necessary, as the one for interactive query refinement presented in [START_REF] Mishra | Interactive query refinement[END_REF].

More specifically, there is a gap to bridge to help SQL users to get confident about their data and queries, and to tackle real life problems. Such users have often just started to master its basic functionalities, and have been trained with well-defined questions on small datasets. But whenever they are confronted to fuzzy questions on new databases, they can get lost. This situation is common, for examples students confronted to their first internship experiences, or for freshly trained employees in companies that are implementing new data analysis processes to make use of the data they collect daily. Helping these new users to smoothly dive into data exploration is therefore an important challenge that would be beneficial in many domains.

In this paper, we present ExplIQuE, our exploration interface of SQL databases with query extension. The purpose of this interface is to help users in refining an initial SQL query over a database, through an iterative process, by suggesting a set of possible extensions of this query, that consist in possible additional selection conditions for its where clause. These extensions give new queries, that can then be explored and extended, until the desired result is reached. Assistance is provided to understand and select the desired extensions, with metrics and visual representations. These extensions have several objectives:

• Help the user understand her database by paying attention to some interesting attributes they might have overlooked, and observing how the query result is divided and distributed, using 2D visualizations. • Unblock the user when she does not know where to go or where to start, by providing suggestions even to a very general query. • Provide a semantic help where most SQL editors only provide syntactic help that is not helpful to understand the content of the database itself. ExplIQuE is an interface aimed to connect to any database, and to provide, in addition to classic querying options in SQL, extensions to queries, with interactive and intuitive visual support. we offer the audience the possibility to try the various functionalities of Ex-plIQuE by trying to express an imprecise question over a scientific database on bacterial colonies, through an iterative process.

SYSTEM OVERVIEW 2.1 Query extensions

The purpose of extensions is to give suggestions to the user, so that she has directions to refine her initial query Q, by adding additional selection predicates in the Where clause of the query. More specifically, we propose, given a query Q, to compute a k extensions set, such that the extensions in this set do not overlap in terms of results, but also cover all the initial tuples returned by Q. Our solution, to compute such extensions, presently being patented 1 , is based on two machine learning algorithms:

Clustering In order to identify interesting zone of refinement in Q's answer set (denoted by ans (Q, d )), we propose to group similar tuples together using a clustering algorithm. ans (Q, d ) is thus divided into k clusters, and each cluster becomes a possible refinement zone for the user. The intuition between this process is that users are formulating queries in order to reach a specific set of tuples, answering a question, and that the results to be found are not random, but likely to contain tuples that fit together and have similar characteristics. We therefore chose to base the identification of refinement zone of the k-means algorithm with a Euclidean distance (see [START_REF] Lloyd | Least Squares Quantization in PCM[END_REF]). Decision tree The second part of our extension process consists in finding a query to describe each refinement zone identified by clustering. To do so, an additional column label is added to the dataset from ans (Q, d ), and filled with the number of the cluster each tuple has been assigned to. A 1 Patent number FR1757682 from the 14/08/2017 in France binary decision tree is then built to discriminate between the different clusters. For each leaf of the tree, the decision path from the root of the tree to this leaf is computed, and the conjunction of decision clause gives a new selection predicate. In order to get as many extensions as clusters, we offer two possibilities. We either limit the depth of the tree to only obtain k leaves, and therefore each leaf gives exactly one extension. Otherwise, we do not limit the depth of the tree, but all the leaves corresponding to the same cluster are joined by a disjunction, forming together a unique extension. These algorithms require some parameters. Predefined ones are proposed to the user, who can then personalize them using the ExplIQuE's interface. For the clustering, several values of k are used (by default for k = 2 to k = 10), each giving an extension set of different size. These sets are presented to the user, ranked according to well-known clustering score of the silhouette coefficient [START_REF] Peter | Silhouettes: a graphical aid to the interpretation and validation of cluster analysis[END_REF]. Inside a single extension set, extensions are ranked according to the size of their result, by showing the most general extension first (with the most tuples), and the most specific one last.

The algorithm to compute a k extension set given a fixed value of k and a tree with k-leaves is given in algorithm 1. It should be noted that the data is preprocessed and normalized for clustering, and that the projection of the query is removed, to allow other attributes, that the user might have overlooked, to be included in the extensions.

procedure Extension (Q, d, k );

Input : A query Q over R, d a database over R, k the number of extensions Output :S c a set of k extensions of Q if Q = π X (Q ′ ) ; // remove the projection then Q = Q' end wd = ans (Q, d ) ;
// wd: working data lwd = kmeans (wd, k) ; // lwd: labelled wd tree = DecisionTree (lwd, k) conjunctions = дetRules (tree)

S c = {} foreach c in conjunctions do S c = S c ∪ σ c (Q ) end return S c ;
Algorithm 1: Query extension procedure

Data visualization

To understand the extensions, and assist the user in choosing or refining one, ExplIQuE offers several hints. First, for each extension, two scores are displayed, in addition to the ranking among extension sets with the silhouette coefficient.

• The narrowing ratio, which is the percentage of tuples removed from the initial query when adding the extension. • The result set size of the extension, when added to the initial query.

In addition, two visualizations are available to assist the user in choosing an interesting extension. The first is a scatterplot of the results of the query being extended, where the tuples are grouped by extension. The data is projected on two dimensions using principal component analysis (PCA), and each extension is presented using a different color. Moreover, the visualization is interactive, as the user can see the extension corresponding to the datapoints by moving the mouse over the scatterplot. The purpose of this visualization is to show in one glance to the user the size and dispersion of an extension, as well as how separated each extension is with respect to the others. Such a visualization is presented on figure 2.

In the specific case of image databases, where tuples are associated with images, another visualization is possible. The images associated to the results of an extension can be displayed as a mosaic in ExplIQuE. This is useful for the user to easily identify the diversity of data that an extension represents: she might see visually that the extension contains homogeneous images, or on the opposite easily identify outliers. In this demonstration, we propose to use such a database to demonstrate this additional functionality.

Implementation

ExplIQuE's is implemented as a web interface. The backend relies on Flask2 framework, and is therefore implemented using Python 3. The clustering and decision trees algorithms are the ones from the scikit-learn library [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF].

Data is stored in a relational database that can be either MySQL or Oracle. Optional external files can be stored outside the database in a dedicated folder. The user has access to a web interface that is implemented using the React3 javascript library. A first page allows to connect to the desired database, before accessing to the second page that allows to extend queries over it. A snapshot of this page is presented on figure 2, showing the main functionalities. On the left panel, the user can query the database using SQL, and when needed, ask to extend the current query. If necessary, the schema can be displayed using the link on the top left corner. On the right panel, the extensions are displayed. For a given value of k, a global visualization is displayed, and the corresponding extension is highlighted when the mouse is over a cluster, to link the visualization to the SQL extension it represents. For each extension, additional information such as the reduction ratio is given to the user when she clicks on it. Finally, the extension can be added to the current query in just one click, it can be updated in the query writing zone, and the process can be repeated iteratively until convergence.

As this is an online system, the extensions have to be computed in a reasonable amount of time. This can be challenging on large instances: as a result, we use random sampling, that allows to reach a trade-off between the computing time and the quality of the results. The experimentations on this specific problem are out of the scope of this demonstration paper.

Experimentation results

A preliminary version of ExplIQuE (see [START_REF] Le Guilly | Partitioning queries for data exploration using query extensions[END_REF]) was experimented with a group of 70 computer science students, who had just started their lessons on SQL 4 . They were divided in two groups, one with access to ExplIQuE (group EXT), the other with a similar interface but with only classic SQL querying possibilities (group NoEXT). They were asked to answer a series of ten questions over a database designed for the test:

• The first three questions were precise and easy to translate into SQL. They were used to assess the participant's level in SQL, and to ensure that the performances of both groups were balanced. • The other were imprecise questions that had been designed to be deliberately fuzzy with a more exploratory purpose. As such, they where not easily translatable to SQL, and required some fumbling around and playing with SQL to answer them using only classic SQL tools. The answering time of each participant to each question was monitored, in order to compare the performances of both groups. The results are presented on figure 1, that shows the boxplot of answering time per question for each group, only when student correctly answered the question. On the first three questions, the results are very similar for both groups, which is what was expected: this questions were easy, and did not require the use of extensions. However, on the other ones, impressive differences can be observed, as students who had access to extensions performed much faster (the difference on question 10 is due to the fact that very few students from group NoEXT had the time to answer it correctly). This experiment therefore showed how extensions can help users in better understanding their data, and to write their SQL queries faster. This is what we want to show in this demonstration.

DEMONSTRATION SCENARIO

The audience will have the opportunity to use ExplIQuE's web interface, over a scientific database, in order to answer a list of imprecise questions on the database. The interface is presented on figure 2, showing its main features.

Running database

The database used for the demonstration comes from a study where colonies of different bacteria grow on solid plates 5 . It contains 10000 tuples over 29 columns, that correspond to measures on the development of the bacteria. It is both simple (only one table) and difficult, as the content of its columns is not easy to under at first glance, with some confusing column names. It describes the shape, texture, and color of the colonies. Scientists use it to detect colonies with specific characteristics, for example bacteria belonging to the same species. Each tuple is also associated to an image representing the bacteria. We chose this dataset for the audience to be in the conditions where ExplIQuE can be useful, as they are not likely to have prior knowledge on this subject. Moreover, the 29 columns, with names that are not so expressive except for domain experts, will make the extensions useful to understand what they contain and the type of data they represent.

Audience interaction

The audience will have the opportunity to experience the various features of ExplIQuE. First, as with any DBMS, the audience will be able to browse the database's schema, to understand its structure and content. Moreover, ExplIQuE allows user to evaluate queries like any other DBMS's interface. The audience will then be allowed

Figure 1 :

 1 Figure 1: Boxplot of answering time for students with (EXT) and without extensions (NoEXT)

Figure 2 :

 2 Figure 2: Web interface for ExplIQuE

http://flask.pocoo.org/

https://reactjs.org/

more details at https://marielgy.github.io/sql_experimentation/

The authors would like to thank Christopher Pease from Darlington EURL for releasing the dataset to play with the dataset, by querying it in a traditional setting. Second, the audience will be asked to answer questions, like the following one: What are the bacteria that have a very similar texture, and all have a circular shape ? Using our query extensions, the answer to this question can be found in three iterations, and in less than five minutes. Users will be invited to start with a very general query, and to use the suggestions of the extensions to reach the desired result set. They will also be invited to play with the several visualizations to understand and select the most useful extensions. Finally, we also propose to play with several parameters available to tune the extensions. They will therefore have the possibility to change the number of clusters to produce, and to change how the decision tree is computed.


Keywords: Reflected SDEs, Time-dependent domains, Generalized BSDEs, Viscosity solution of PDEs, Nonlinear Neumann boundary conditions. 2020 MSC: 35G30, 60H10, 60H30

Approximation of reflected SDEs in non-smooth time-dependent domains and application to fully nonlinear PDEs with Neumann boundary condition on time-dependent domains

Manal Jakani

To cite this version:

Manal Jakani. Approximation of reflected SDEs in non-smooth time-dependent domains and application to fully nonlinear PDEs with Neumann boundary condition on time-dependent domains. 2023. hal

Introduction

Let d ≥ 1, T > 0 be fixed, and let D ′ be a bounded open connected subset of R 1+d . We will refer to D = D ′ ∩ [0, T ] × R d , as a time-dependent domain. For t ∈ [0, T ], the time sections of D are defined by D t = {x : (t, x) ∈ D} and are assumed to be convex and increasing in time. We deal with the normally reflected SDE in time-dependent domain of the following form: ∀t ⩽ T ,      X t = x + t 0 b(r, X r )dr + t 0 σ (r, X r )dW r + t 0 ⃗ n(r, X r )d|Λ| r ;

X t ∈ Dt , |Λ| t = t 0 χ {Xr ∈∂ Dr } d|Λ| r < ∞.

(

This class of reflected SDEs has been introduced by Costantini-Gobet-El Karoui [START_REF] Costantini | Boundary sensitivities for diffusion processes in time dependent domains[END_REF] for smooth timedependent domains. Then, in the case of SDEs with oblique reflection in non-smooth time-dependent domains, existence of weak solutions has been established by . This result has been generalized by Lundström-Önskog [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF], the authors proved the existence and uniqueness of a strong solution for obliquely reflected SDEs in non-smooth time-dependent domains. In the previous papers, the results of existence and uniqueness are obtained mainly by solving Skorohod problem. The aim of this work is to provide an approximation scheme for the reflected SDE (1.1) using standard SDEs. This problem has been considered by Lions-Menaldi-Sznitman [START_REF] Lions | Construction de processus de diffusion réfléchis par pénalisation du domaine[END_REF] and by Menaldi [START_REF] Menaldi | Stochastic Variational Inequality for Reflected Diffusion[END_REF] where an approximation is given in convex time-independent domains when b and σ are Lipschitz.

The same domain was considered by Bahlali-Maticiuc-Zalinescu [START_REF] Bahlali | Penalization method for a nonlinear Neumann PDE via weak solution of reflected SDEs[END_REF] when the coefficients are only measurable, the authors established a weak convergence result. In general time-independent domains, Ren-Wu [START_REF] Ren | Penalization of Reflected SDEs and Neumann Problems of HJB Equations[END_REF] extended these results to the case where the domain may have corners.

In this paper, we are able to extend similar results of convergence to the case of not necessarily smooth time-dependent domains whose time sections are increasing with time which constitutes the main result of this paper. As an application, we consider the following generalized backward stochastic differential equations (BSDE for short): ∀t ⩽ s ⩽ T , Y t,x s = h(X t,x T ) + T s f (r, X t,x r ,Y t,x r , Z t,x r )dr + T s ψ(r, X t,x r ,Y t,x r )d|Λ t,x | r -T s Z t,x r dW r ,

This class of BSDEs has been first introduced and studied by Pardoux-Zhang [START_REF] Pardoux | Generalized BSDEs and nonlinear Neumann boundary value problems[END_REF]. Moreover, the authors showed that if (X t,x , Λ t,x ) is the solution of a normally reflected SDE in a smooth timeindependent domain, then the solution of (1.2) provides a probabilistic formula for u which is a solution of a system of PDEs with nonlinear Neumann boundary conditions on smooth time-independent domain. Based on this connection between BSDEs and PDEs, and the approximation by standard SDEs of reflected SDEs in regular convex time-independent domains, many authors studied the convergence in the S-topology of the approximation of the generalized BSDE (1.2) associated with a reflected SDE in a regular convex when the driver function f does not depend on z. Then, they give an approximation (by standard PDEs) for the associated PDE with nonlinear Neumann boundary conditions on smooth time-independent domain (see e.g. [START_REF] Boufoussi | An approximation result for a nonlinear Neumann boundary value problem via BSDEs[END_REF][START_REF] Bahlali | Penalization method for a nonlinear Neumann PDE via weak solution of reflected SDEs[END_REF][START_REF] Bahlali | Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients[END_REF]). A recent paper by Bahlali-Boufoussi-Mouchtabih [START_REF] Bahlali | Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition[END_REF] treats the case where the driver function is allowed to depend on the variable z. Moreover, the authors provide a strong approximation for the generalized BSDE using a sequence of standard BSDEs. Then they use it to obtain, once again, the approximation by standard PDEs for a PDE with nonlinear boundary condition on time-independent regular convex.

Using the approximation for the normally reflected diffusion (1.1) that we give in the first part of this paper, we get an approximation for the associated generalized BSDE (1.2) when randomness stems from (1.1), using a sequence of standard BSDEs. As we know that (see Jakani [START_REF] Jakani | System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains[END_REF]), the solution of the generalized BSDE (1.2) provides a solution for the following PDE with nonlinear Neumann boundary condition on the time-dependent domain

D ′ ∩ [0, T ] × R d : ∀i = 1, ..., m,            ∂ t u i (t, x) + L u i (t, x) + f i (t, x, u(t, x), σ ⊤ (t, x)D x u i (t, x)) = 0, (t, x) ∈ D ′ ∩ [0, T ) × R d ; ∂ u i ∂⃗ n (t, x) + ψ i (t, x, u(t, x)) = 0, (t, x) ∈ D ′ \ D ′ ∩ [0, T ) × R d ; u(T, x) = h(x), x ∈ D T , (1.3
) where the operator L is defined by

L = 1 2 Tr(σ σ ⊤ )D 2 xx (.) + b ⊤ D x (.) and at a point (t, x) ∈ ∂ D we set ∂ ∂⃗ n = ⟨⃗ n(t, x), D x (.)⟩.
We then obtain an approximation for the PDE with nonlinear Neumann boundary condition on time-dependent domain (1.3) by a sequence of standard PDEs defined on [0, T ] × R d . The paper is organized as follows: In Section 2, we define the geometry of the domain considered in this paper and we collect some existing results on reflected SDEs in time-dependent domains. Section 3 is devoted to the results of convergence, we propose a sequence of standard SDEs for which we establish a priori estimates and we show that it converges to the solution of the reflected SDE in the time-dependent domain D. Then, in Section 4, we use the approximation of reflected SDEs in a smooth time-dependent domain to provide an approximation for both generalized BSDEs associated with reflected diffusions in time-dependent domains and PDEs with normal derivative and nonlinear Neumann boundary condition defined on time-dependent domains.

Preliminaries and formulation of the problem 2.1 Geometry of the time-dependent domain

Let d ≥ 1, T > 0 be fixed, we follow the notation of [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF] and we let D ′ be a bounded open connected subset of R 1+d . We will refer to D = D ′ ∩ [0, T ] × R d , as a time-dependent domain. Given D and t ∈ [0, T ], we define the time sections of D as D t = {x : (t, x) ∈ D}. We assume that: (2.1)

D t ̸ = ∅,
Our basic assumption is the following:

D t ⊂ D t ′ , whenever t ≤ t ′ , t,t ′ ∈ [0, T ]. (2.2)
Let t ∈ [0, T ], the boundary of D t will be denoted ∂ D t . Then, define N(t, x) the cone of unit inward normal vectors at a boundary point x ∈ ∂ D t which is nonempty thanks to assumption (2.1). Note that under this assumption, the domain may have corners which does not rule out the possibility of several unit inward normal vectors at the same boundary point. Now, let ⟨•, •⟩ denote the standard inner product on R d and |x| = ⟨x, x⟩ 1/2 be the euclidean norm of x ∈ R d . For x ∈ R d and r > 0, let B(x, r) and S(x, r) denote the ball and sphere of radius r, centered at x, respectively, i.e. B(x, r) = y ∈ R d : |x -y| < r and S(x, r) = y ∈ R d : |x -y| = r . We assume that there exists a radius r 0 > 0 such that the exterior sphere condition holds for all time-sections of D. This implies that for any t ∈ [0, T ], we have:

B(x -r 0 ⃗ n(t, x), r 0 ) ⊂ D c t , (2.3)
whenever x ∈ ∂ D t and ⃗ n(t, x) ∈ N(t, x). This is equivalent to say that

⟨y -x,⃗ n(t, x)⟩ + 1 2r 0 |y -x| 2 ⩾ 0, ∀x ∈ ∂ D t , y ∈ D t , (2.4
)

whenever ⃗ n(t, x) ∈ N(t, x) for t ∈ [0, T ].
Unless otherwise stated, we fix ⃗ n(t, x) in N(t, x) and we assume that:

⃗ n ∈ C 1,2 b (R 1+d , {0, 1}), (2.5)
where C 1,2 b (R 1+d , {0, 1}) denotes the space of bounded functions that are continuously differentiable once with respect to the time variable and twice with respect to the space variable and having bounded derivatives. Then, let us recall the temporal variation of the domain d(t, x) := inf

y∈D t |y -x|, ∀t ∈ [0, T ], x ∈ R d that is assumed to satisfy for some p ∈ (1, ∞), d(., x) ∈ W 1,p ([0, T ], [0, ∞)), (2.6)
for all x ∈ R d , where W 1,p ([0, T ], [0, ∞)) denotes the Sobolev space of functions whose first order weak derivatives belong to L p ([0, T ]) with Sobolev norm uniformly bounded in space and such that the first weak derivative ∂ t d(t, x) is jointly measurable in (t, x).

Remark 2.1 Thanks to assumption (2.1), the exterior cone condition assumed in [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF] which is weaker than the uniform sphere condition (2.3) holds, i.e, there exists a constant ρ ∈ (0, 1) such that

∪ 0⩽ξ ⩽ρ B(x -ξ⃗ n(t, x), ξ ρ) ⊂ D c t , ∀t ∈ [0, T ], x ∈ ∂ D t . (2.7)
It follows, that the interior cone condition is satisfied as well:

∪ 0⩽ξ ⩽ρ B(x + ξ⃗ n(t, x), ξ ρ) ⊂ D t , ∀t ∈ [0, T ], x ∈ ∂ D t . (2.8)
Moreover, by Remark 2.2 in [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF], there exists α = 1 -1/p ∈ (0, 1) and K ∈ (0, ∞) such that for all s,t ∈

[0, T ], x ∈ R d , |d(s, x) -d(t, x)| ≤ K|s -t| α . ( 2

Reflected SDEs in time-dependent domain

Let (Ω, F , P) be a fixed probability space on which is defined an n-dimensional Brownian motion W = (W t ) 0⩽t⩽T , where F = (F t ) 0≤t≤T is the completed filtration of (σ (W s , 0 ≤ s ≤ t)) t≤T with all P-null sets of We start by recalling the definition of the solution:

F . Let b : [0, T ] × R d -→ R d and σ : [0, T ] × R d -→ R d × R n be
Definition 2.1 (Lundström-Önskog [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF]) A strong solution of the reflected SDE in D driven by W and with coefficients b and σ , direction of reflection along ⃗ n and initial condition x ∈ D0 is an F t -adapted stochastic process X t which satisfies P-almost surely, whenever t ∈ [0, T ],

     X t = x + t 0 b(r, X r )dr + t 0 σ (r, X r )dW r + t 0 ⃗ n(r, X r )d|Λ| r ; X t ∈ Dt , |Λ| t = t 0 χ {Xr ∈∂ Dr } d|Λ| r < ∞.
(2.10)

We introduce the following assumptions:

(a) The functions b and σ are Lipschitz continuous with respect to x, i.e., there exists a positive constant C such that

|b(t, x) -b(t, x ′ )| + |σ (t, x) -σ (t, x ′ )| ≤ C|x -x ′ |, ∀(t, x, x ′ ) ∈ [0, T ] × R d × R d . (2.11) (b)
The functions b and σ are of linear growth in (t, x), i.e., there exists a positive constant

C such that |b(t, x)| + |σ (t, x)| ≤ C(1 + |x|), ∀(t, x) ∈ [0, T ] × R d . (2.12)
Next, we recall the following result of existence and uniqueness of the solution of the reflected SDE (2.10) under the above geometric setting:

Theorem 2.1 (Lundström-Önskog [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF]) Under assumption (a), the reflected SDE (2.10) has a unique strong solution.

Moreover, it has been shown in [START_REF] Jakani | System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains[END_REF] that the solution satisfies the following properties:

Proposition 2.1 There exists a constant C such that for all x, x ′ ∈ D 0 , E[ sup 0⩽t⩽T | X x t -X x ′ t | 4 + | |Λ x | t -|Λ x ′ | t | 4 ] ≤ C | x -x ′ | 4 . (2.13)
Moreover, for each µ > 0, t ∈ [0, T ], there exists C(µ,t) such that for all x ∈ D 0 ,

E[e µ|Λ x | t ] ≤ C(µ,t).
(2.14)

Approximation of reflected SDEs in time-dependent domains

Let D be a time-dependent domain satisfying (2.1)- (2.6). From now on we assume that the functions b and σ satisfy assumptions (a) and (b). Let x ∈ D0 , we introduce the following penalized SDE: ∀n ≥ 1, ∀t ∈ [0, T ],

X n t = x + t 0 b(r, X n r )dr + t 0 σ (r, X n r )dW r -n t 0 (X n r -π(r, X n r ))dr. (3.1) Note that if X n t / ∈ Dt , the vector -X n t -π(t,X n t ) |X n t -π(t,X n t )| is an element of N(t, π(t, X n t )).
Then the penalized SDE (3.1) can be written as:

X n t = x + t 0 b(r, X n r )dr + t 0 σ (r, X n r )dW r + Λ n t ,
where Λ n and |Λ n | are given by:

Λ n t = t 0 n(X n r -π(r, X n r ))dr,
and

|Λ n | t = t 0 n|X n r -π(r, X n r )|dr = t 0 n|X n r -π(r, X n r )|dr = t 0 nd(r, X n r )dr.

A priori estimates

Proposition 3.1 Under assumptions (a) and (b), for any q ≥ 1, ∀t ≤ T , we have:

sup n≥1 E sup 0≤t≤T |X n t | 2q + sup 0≤t≤T |Λ n | q t < ∞. (3.2)
PROOF. From Itô's formula, we have: ∀t ∈ [0, T ],

|X n t -P 0 | 2 + 2n t 0 ⟨X n r -P 0 , X n r -π(r, X n r )⟩dr = |x -P 0 | 2 + t 0 |σ (r, X n r )| 2 dr + 2 t 0 ⟨X n r -P 0 , b(r, X n r )⟩dr + 2 t 0 ⟨X n r -P 0 , σ (r, X n r )⟩dW r .
Recall Lemma 2.1 -(d), then there exists P 0 ∈ D 0 such that for some 1 ⩽ γ < ∞, we have:

⟨X n r -P 0 , X n r -π(r, X n r )⟩ ≥ 1 γ |X n r -π(r, X n r )|. (3.3)
This implies that:

2n t 0 ⟨X n r -P 0 , X n r -π(r, X n r )⟩dr ≥ 2n 1 γ t 0 |X n r -π(r, X n r )|dr.
Besides, from assumption (a), we deduce that for some constant M > 0 depending on the Lipschitz constant of b and σ , we have:

|b(r, X n r )| ≤ |b(r, P 0 )| + M|X n r -P 0 |, (3.4) |σ (r, X n r )| ≤ |σ (r, P 0 )| + M|X n r -P 0 |. (3.5)
Therefore,

|X n t -P 0 | 2 +2n 1 γ t 0 |X n r -π(r, X n r )|dr ≤ |x -P 0 | 2 + (M 2 + M + 1) t 0 |X n r -P 0 | 2 dr + t 0 |b(r, P 0 )| 2 dr + t 0 |σ (r, P 0 )| 2 dr + 2 t 0 ⟨X n r -P 0 , σ (r, X n r )⟩dW r .
Then from the boundedness of D0 and thanks to assumption (b), we get:

|X n t -P 0 | 2 +2n 1 γ t 0 |X n r -π(r, X n r )|dr ≤ C D 0 ,σ ,b,x +C t 0 |X n r -P 0 | 2 dr + 2 t 0 ⟨X n r -P 0 , σ (r, X n r )⟩dW r ,
and we have for q ≥ 1,

|X n t -P 0 | 2q + 2n 1 γ t 0 |X n r -π(r, X n r )|dr q ≤ C D 0 ,σ ,b,x +C t 0 |X n r -P 0 | 2q dr +C t 0 ⟨X n r -P 0 , σ (r, X n r )⟩dW r q . (3.6)
First, let us examine the term |X n t -P 0 | 2q . By taking the supremum over [0,t] and the expectation, we get:

E sup 0≤r≤t |X n r -P 0 | 2q ≤ C D 0 ,σ ,b,x +CE r 0 sup 0≤u≤r |X n u -P 0 | 2q du (3.7) +CE sup 0≤r≤t r 0 ⟨X n u -P 0 , σ (u, X n u )⟩dW u q .
From BDG inequality, it follows that:

E sup 0≤r≤t r 0 ⟨X n u -P 0 , σ (u, X n u )⟩dW u q ≤ CE t 0 |X n r -P 0 | 2 |σ (r, X n r )| 2 dr q 2 ≤ CE t 0 |X n r -P 0 | 4 + |σ (r, X n r )| 4 dr q 2 ≤ CE t 0 |X n r -P 0 | 2q |dr + t 0 |σ (r, X n r )| 2q |dr .
Again, using (3.5), we deduce from (3.7) that:

E sup 0≤r≤t |X n r -P 0 | 2q ≤ C D 0 ,σ ,b,x +C t 0 E sup 0≤u≤r |X n u -P 0 | 2q dr.
Finally, we apply Gronwall's lemma and we obtain: ∀q ≥ 1,

E sup 0≤r≤t |X n r -P 0 | 2q ≤ C, ∀t ≤ T, which implies that E sup 0≤t≤T |X n t | 2q ≤ C, ∀n ≥ 1.
Moreover, with the use of the latter estimate and taking into account (3.6), it follows that: ∀q ≥ 1,

E sup 0≤t≤T |Λ n | q t = E sup 0≤t≤T t 0 n|X n r -π(r, X n r )|dr q ≤ 1 2 γC, ∀n ≥ 1.

Uniform control of the distance

In this part, we are interested in the uniform control of d(t, X n t ). Note that assumptions (2.1) and (2.6) do not ensure the smoothness of the boundary. Inspired by [START_REF] Nyström | Reflected BSDE of Wiener-Poisson type in time-dependent domains[END_REF], we use a smooth approximation of D that allows to apply Itô's formula with a function involving the distance. More precisely, we recall the following lemma from the same work:

Lemma 3.1 Let ε > 0, there exists a C ∞ -smooth time-dependent domain D ε ⊂ D ′ satisfying (2.1) and (2.2) such that: h(D t , D ε,t ) < ε, ∀t ∈ [0, T ], (3.8)
where h stands for the Hausdorff distance, which is defined by: h(F, G) = max(sup{d(y, F); y ∈ G}, sup{d(y, G); y ∈ F})

for any two sets F and G of R d .

Thanks to Lemma 3.1, we deduce that the cone of unit inward normal vectors at each boundary point of D ε is reduced to a unique vector that we denote⃗ n ε . Then for t ∈ [0, T ] and x ∈ R d \ Dε,t the projection of x along ⃗ n ε (t, x) will be denoted π ε (t, y). Note that π ε satisfies:

π ε (t, y) = y, ∀y ∈ D ε,t , ∀t ∈ [0, T ],
and

d ε (t, y) = d(y, D ε,t ) = |y -π ε (t, y)|, ∀y ∈ R d , ∀t ∈ [0, T ].
Proposition 3.2 Let (X n ) n⩾1 be the unique solution of the penalized SDE (3.1). Then for any p > 2, there exists c > 0 such that ∀n ⩾ 1, we have:

E sup 0≤t≤T d(t, X n t ) p ⩽ c n p-2 2 , (3.9)
and

E T 0 d(t, X n t ) p dt ⩽ c n p 2 . ( 3.10)
First we shall recall some properties of the projection π ε as stated in the following lemma which is borrowed from [START_REF] Nyström | Reflected BSDE of Wiener-Poisson type in time-dependent domains[END_REF].

Lemma 3.2 Let D ε be a smooth approximation of D satisfying (3.8). Then, there exists a constant c ≥ 0, such that, if ε ∈ (0, 1), y ∈ R d and t ∈ [0, T ], we have:

(i) |π(t, y) -π ε (t, y)| ≤ c min( ε 2 + εd ε (y,t); √ ε(1 + d ε (y,t)); ε 2 + εd(y,t)), (ii) |π(t, y) -π ε (t, y)| ≤ c √ ε d ε (t, y) whenever d ε (t, y) > ε.
PROOF OF PROPOSITION 3.2. We establish a uniform control of d ε (t, X n t ). This will be done using Itô's formula with the function ϕ ε (t, y) := (d ε (t, y)) p = |yπ ε (t, y)| p , ∀p > 2, which is continuously differentiable with respect to y and for which the derivative with respect to the time variable exists. Now, recall Lemma 3.1, then (D ε,t ) t⩾0 is increasing in time and by definition of ϕ ε , we have:

∂ t ϕ ε ≤ 0.
Using Itô's formula and thanks to the previous remark, we get:

∀t ∈ [0, T ], ϕ ε (t, X n t ) ≤ ϕ ε (0, X n 0 ) + t 0 ∂ x ϕ ε (s, X n s )dX n s + 1 2 t 0 ∂ xx ϕ ε (s, X n s )d⟨X n i , X n j ⟩ s ≤ ϕ ε (0, x) + t 0 ⟨∂ x ϕ ε (s, X n s ), b(s, X n s )⟩ds + t 0 ⟨∂ x ϕ ε (s, X n s ), σ (s, X n s )dW s ⟩ -n t 0 ⟨∂ x ϕ ε (s, X n s ), X n s -π(s, X n s )⟩ds + 1 2 t 0 σ ⊤ (s, X n s )∂ xx ϕ ε (s, X n s )σ (s, X n s )ds.
First, note that Lemma 3.2-(i) gives the following boundedness from above of ϕ ε (0, x):

ϕ ε (0, x) = |x -π ε (0, x)| p = |π(0, x) -π ε (0, x)| p ≤ c p ε 2 + εd(x, D 0 ) p = (cε) p .
Next, for the term involving ∂ x ϕ ε , we observe that:

∂ x ϕ ε (t, x) = ∂ x (d ε (t, x) 2 ) p 2 = p 2 ∂ x d ε (t, x) 2 × d ε (t, x) 2 p 2 -1 = p × (x -π ε (t, x)) × d ε (t, x) p-2 .
Then by taking the norm, we get:

|∂ x ϕ ε (t, X n t )| ≤ pd ε (t, X n t ) p-1 .
Note that for any c 1 > 0, there exists C 1 depending on c 1 and p:

pd ε (s, X n s ) p-1 |b(s, X n s )| ≤ c 1 nd ε (s, X n s ) p + C 1 n p-1 |b(s, X n s )| p , (3.11)
Then we obtain the following inequality:

∀c 1 > 0, t 0 ⟨∂ x ϕ ε (s, X n s ), b(s, X n s )⟩ds ≤ p t 0 d ε (s, X n s ) p-1 |b(s, X n s )|ds ≤ c 1 n t 0 d ε (s, X n s ) p ds + C 1 n p-1 t 0 |b(s, X n s )| p ds. (3.12)
Next, note that -x-π ε (t,x) |x-π ε (t,x)| coincides with⃗ n ε (t, x) the unit normal vector pointing toward the interior of D ε,t whenever x ∈ ∂ D ε,t for each t ∈ [0, T ] and null elsewhere. Hence, we can see that,

∂ xx ϕ ε (t, x) = ∂ x -p⃗ n ε (t, x)d ε (t, x) p-1 = -p⃗ n ε (t, x)∂ x (d ε (t, x) 2 ) p-1 2 -p(d ε (t, x)) p-1 ∂ x ⃗ n ε (t, x).
Taking into account the smoothness of D ε and the boundedness of D ′ , the derivative ∂ x ⃗ n ε is bounded. Thus, by taking the norm there exists a constant c > 0 independent of ε such that:

1 2 t 0 (σ ⊤ ∂ xx ϕ ε σ (s, X n s )ds ≤ t 0 cpd ε (s, X n s ) p-1 + cp(p -1)d ε (s, X n s ) p-2 |σ (s, X n s )| 2 ds.
Similarly, we can see that for any c 2 , c 3 > 0 there exist C 2 ,C 3 > 0 depending on c 2 and c 3 respectively and p such that,

cpd ε (s, X n s ) p-1 ≤ c 2 nd ε (s, X n s ) p + C 2 n p-1 |σ (s, X n s )| 2p , and
cp(p -1)d ε (s, X n s ) p-2 |σ (s, X n s )| 2 ≤ c 3 nd ε (s, X n s ) p ds + C 3 n p-2 2 |σ (s, X n s )| p .
Therefore, the following upper bound holds: ∀c 2 , c 3 > 0,

1 2 t 0 (σ ⊤ ∂ xx ϕ ε σ (s, X n s )ds ≤ (c 2 + c 3 )n t 0 d ε (s, X n s ) p ds + C 2 n p-1 t 0 |σ (s, X n s )| 2p ds + C 3 n p-2 2 t 0 |σ (s, X n s )| p ds.
Now, let us examine the term involving the penalization term,

-n t 0 <∂ x ϕ ε (s, X n s ), X n s -π(s, X n s ) > ds = -np t 0 d ε (s, X n s ) p ds -np t 0 d ε (s, X n s ) p-2 ⟨X n s -π ε (s, X n s ), π ε (s, X n s ) -π(s, X n s )⟩χ {dε (s,X n s )>ε} (s, X n s )ds -np t 0 d ε (s, X n s ) p-2 ⟨X n s -π ε (s, X n s ), π ε (s, X n s ) -π(s, X n s )⟩χ {dε (s,X n s )⩽ε} (s, X n s )ds.
On the one hand, Lemma 3.2-(ii) yields:

| -np t 0 d ε (s, X n s ) p-2 ⟨X n s -π ε (s, X n s ), π ε (s, X n s ) -π(s, X n s )⟩χ {dε (s,X n s )>ε} (s, X n s )ds| ≤ npc √ ε t 0 d ε (s, X n s ) p-2 |X n s -π ε (s, X n s )|d ε (s, X n s ) 1 2 χ {dε (s,X n s )>ε} (s, X n s )ds ≤ npc √ ε t 0 d ε (s, X n s ) 2p-1 2 ds.
This implies that, for any c 4 > 0 there exists C 4 > 0 depending on c 4 and p such that:

| -np t 0 d ε (s, X n s ) p-2 ⟨X n s -π(s, X n s ), π ε (s, X n s ) -π(s, X n s )⟩χ {dε (s,X n s )>ε} (s, X n s )ds| ≤ c 4 n t 0 d ε (s, X n s ) p ds +C 4 n 1 2p ε p .
On the other hand, the second term can be dominated as follows:

np t 0 d ε (s, X n s ) p-2 |X n s -π ε (s, X n s )||π ε (s, X n s ) -π(s, X n s )|(1 -χ {dε (s,X n s )⩽ε} (s, X n s ))ds ≤ np √ ε t 0 d ε (s, X n s ) p-1 (1 + d ε (s, X n s ))ds ≤ cnε p-1 2 .
The last line follows from Lemma 3.2-(i). As a conclusion, we have:

ϕ ε (t, X n t ) + (p -c 1 -c 2 -c 3 -c 4 )n t 0 ϕ ε (s, X n s )ds ≤ (c p +C 4 n 1 2p )ε p + cnε p-1 2 + C 1 n p-1 t 0 |b(s, X n s )| p ds + t 0 C 2 n p-1 |σ (s, X n s )| 2p + C 3 n p-2 2
|σ (s, X n s )| p ds

+ t 0 ⟨∂ x ϕ ε (s, X n s ), σ (s, X n s )dW s ⟩. ( 3
E d ε (t, X n t ) p + n t 0 d ε (s, X n s ) p ds ≤ (c p +C 4 n 1 2p )ε p + cnε p-1 2 + C n p-2 2 E t 0 |b(s, X n s )| p + |σ (s, X n s )| p + |σ (s, X n s )| 2p ds . (3.14)
By taking the supremum over [0, T ] and by recalling (3.2) and assumption (b), we conclude that:

sup 0≤t≤T E d ε (t, X n t ) p ⩽ (c p +C 4 n 1 2p )ε p + cnε p-1 2 + C n p-2 2
, ∀p > 2.

Next, using B-D-G inequality there exists c > 0 such that:

E sup 0≤t≤T | t 0 ⟨∂ x ϕ ε (s, X n s ), σ (s, X n s )dW s ⟩| ≤ cE T 0 d ε (s, X n s ) p-1 |σ (s, X n s )|ds ≤ c 5 E T 0 nd ε (s, X n s ) p ds + C 5 n p-1 E T 0 |σ (s, X n s )| p ds , ∀c 5 > 0,
where C 5 > 0 is independent of n. Hence, from (3.13) and (3.14), we conclude that there exists c > 0 such that

E sup 0≤t≤T d ε (t, X n t ) p ⩽ (c p +C 4 n 1 2p )ε p + cnε p-1 2 + C n p-2 2
, ∀p > 2.

Finally, since h(D t , D ε,t ) < ε, we have d(s, X n ) ≤ d ε (s, X n s ) + ε. Then, it suffices to take the limit as ε → 0.

Convergence of the Penalized SDE

First, we show that (X n ) n⩾1 is a Cauchy sequence as stated in the following proposition:

Proposition 3.3 Let (X n ) n⩾1 be
|X n t -X m t | 2 = 2 t 0 ⟨X n s -X m s , b(s, X n s ) -b(s, X m s )⟩ds + 1 2 t 0 |σ (s, X n s ) -σ (s, X m s )| 2 ds -2n t 0 ⟨X n s -X m s , X n s -π(s, X n s )⟩ds + 2m t 0 ⟨X n s -X m s , X m s -π(s, X m s )⟩ds + 2 t 0 ⟨X n s -X m s , (σ (s, X n s ) -σ (s, X m s ))dW s ⟩.
Using the Lipschitz continuity of b and σ , we deduce that:

|X n t -X m t | 2 ≤ c b,σ t 0 |X n s -X m s | 2 ds -2n t 0 ⟨X n s -X m s , X n s -π(s, X n s )⟩ds + 2m t 0 ⟨X n s -X m s , X m s -π(s, X m s )⟩ds + 2 t 0 ⟨X n s -X m s , (σ (s, X n s ) -σ (s, X m s ))dW s ⟩. (3.16) Note that, -⟨X n s -X m s , X n s -π(s, X n s )⟩ = -|X n s -π(s, X n s )| 2 + π(s, X n s ) -π(s, X m s ), - X n s -π(s, X n s ) |X n s -π(s, X n s )| d(s, X n s ) -π(s, X m s ) -X m s , X n s -π(s, X n s ) . (3.17)
By recalling the exterior sphere property (2.4) and using the Lipschitz continuity of π w.r.t. y, we deduce that:

-2n t 0 ⟨X n s -X m s , X n s -π(s, X n s )⟩ds ≤ cn t 0 |π(s, X n s ) -π(s, X m s )| 2 d(s, X n s )ds + 2n t 0 d(s, X n s )d(s, X m s )ds ≤ cn t 0 d(s, X n s )|X n s -X m s | 2 ds + 2n t 0 d(s, X n s )d(s, X m s )ds.
We do likewise with the third term in the right hand side of (3.16). Then, we obtain the following inequality:

|X n t -X m t | 2 ≤ M m,n t + H m,n t + t 0 Ψ m,n s |X n s -X m s | 2 ds
, where M m,n is a local martingale and the processes H m,n and Ψ m,n are defined by:

H m,n t :=2(m + n) t 0 d(s, X n s )d(s, X m s )ds Ψ m,n s :=c b,σ ,r (1 + nd(s, X n s ) + md(s, X m s )
). Then thanks to Stochastic Gronwall's inequality (see e.g. Lemma 2.3 in Ren-Wu [START_REF] Ren | Penalization of Reflected SDEs and Neumann Problems of HJB Equations[END_REF], Theorem 4 in Scheutzow [START_REF] Scheutzow | A stochastic Gronwall lemma[END_REF]), we get:

E sup 0≤t≤T |X n t -X m t |e - c b,σ ,r 2 T 0 (1+nd(s,X n s )+md(s,X m s ))ds ≤ 3cE (n + m) T 0 d(s, X n s )d(s, X m s )ds 1 2 ≤ cE sup 0≤t≤T d(t, X m t ) T 0 nd(s, X n s )ds + sup 0≤t≤T d(t, X n t ) T 0 md(s, X m s )ds 1 2 ≤ c E sup 0≤t≤T d(t, X m t ) + sup 0≤t≤T d(t, X n t ) 1 2 × E n T 0 d(s, X n s )ds + m T 0 d(s, X m s )ds 1 2 .
Recall Proposition 3.1, then from (3.2) we deduce that

E n T 0 d(s, X n s )ds = E |Λ n | T ≤ c. This implies that, E n T 0 d(s, X n s )ds + m T 0 d(s, X m s )ds ≤ c
On the other hand, from (3.9) it follows that for p > 2, there exists c > 0 that may change from line to line such that:

E sup 0≤t≤T d(t, X m t ) + sup 0≤t≤T d(t, X n t ) 1 2 ≤ E sup 0≤t≤T d(t, X m t ) + sup 0≤t≤T d(t, X n t ) p 1 2p ≤ c E sup 0≤t≤T d(t, X m t ) p + sup 0≤t≤T d(t, X n t ) p 1 2p ≤ c 1 m p-2 2 + 1 n p-2 2 1 2p .
Therefore, the following holds: ∀p > 2,

E sup 0≤t≤T |X n t -X m t |e - c b,σ ,r 2 (1+|Λ n | T +|Λ m | T ) ≤ c 1 m p-2 2 + 1 n p-2 2 1 2p -→ m,n→∞ 0.
First, observe that: ∀κ > 0,

sup 0≤t≤T |X n t -X m t | = sup 0≤t≤T |X n t -X m t | χ e - c b,σ ,r 2 (1+|Λ n | T +|Λ m | T ) ⩽κ + χ e - c b,σ ,r 2 (1+|Λ n | T +|Λ m | T ) >κ ≤ sup 0≤t≤T |X n t -X m t |χ 1+|Λ n | T +|Λ m | T ⩾ln ( 1 κ ) + 1 κ sup 0≤t≤T |X n t -X m t |e - c b,σ ,r 2 (1+|Λ n | T +|Λ m | T ) .
By taking the expectation, we obtain: ∀κ > 0,

E sup 0≤t≤T |X n t -X m t | ≤ E sup 0≤t≤T |X n t -X m t | 1 2 P 1 + |Λ n | T + |Λ m | T ⩾ ln 1 κ 1 2 + 1 κ 1 m p-2 2 + 1 n p-2 2 1 2p ≤ E sup 0≤t≤T |X n t -X m t | 1 2 E 1 + |Λ n | T + |Λ m | T ln 1 κ 1 2 + 1 κ 1 m p-2 2 + 1 n p-2 2 1 2p
.

Thanks to the estimate (3.2), there exists a constant c > 0 independent of n and m such that ∀κ > 0,

E sup 0≤t≤T |X n t -X m t | ≤ c 1 ln 1 κ 1 2 + 1 κ 1 m p-2 2 + 1 n p-2 2 1 2p .
It follows that:

lim sup m,n→∞ E sup 0≤t≤T |X n t -X m t | = 0.
Again from the estimate (3.2), we can see that for any p ⩾ 1 the family (|X n t -X m t | p ) m,n is uniformly integrable. Therefore, the following convergence holds: ∀p ⩾ 1,

E sup 0≤t≤T |X n t -X m t | p -→ m,n→∞ 0.
It follows that (X n ) n⩾1 is a Cauchy sequence in the space S p defined by:

S 2 = {(ψ t ) 0⩽t⩽T F t -progressively measurable such that E[ sup 0≤t≤T | ψ t | p ] < ∞}, then (X n ) n⩾1 converges.
To conclude this section, we show that the limit of (X n , Λ n ) n⩾1 is the solution of the reflected SDE (2.10). Proposition 3.4 For any p ⩾ 1, we have:

lim n→∞ E sup 0≤t≤T |X n t -X t | p = 0. (3.18)
PROOF. Let n ⩾ 1, by applying Itô's formula to |X n t -X t | 2 , we get:

|X n t -X t | 2 = 2 t 0 ⟨X n s -X s , b(s, X n s ) -b(s, X s )⟩ds + 1 2 t 0 |σ (s, X n s ) -σ (s, X s )| 2 ds + 2 t 0 ⟨X n s -X s , (σ (s, X n s ) -σ (s, X s ))dW s ⟩ + 2 t 0 (X n s -X s )dΛ n s -2 t 0 (X n s -X s )dΛ s .
The Lipschitz continuity of b and σ , implies that;

|X n t -X t | 2 ≤ c b,σ t 0 |X n s -X s | 2 ds + 2 t 0 ⟨X n s -X s , (σ (s, X n s ) -σ (s, X s ))dW s ⟩ -2n t 0 ⟨X n s -X s , X n s -π(s, X n s )⟩ds + 2 t 0 (X n s -X s )dΛ s . (3.19)
Then, by repeating the same calculus as in the previous proof, we obtain:

-2n t 0 ⟨X n s -X s , X n s -π(s, X n s )⟩ds ≤ cn t 0 |π(s, X n s ) -π(s, X s )| 2 d(s, X n s )ds + 2n t 0 d(s, X n s )d(s, X s )ds ≤ cn t 0 d(s, X n s )|X n s -X s | 2 ds + 2n t 0 d(s, X n s )d(s, X s )ds.
Since (X t ) 0⩽t⩽T is the solution of the reflected SDE in the time-dependent domain (2.10), then X t ∈ D t . This implies that d(t, X t ) = 0 for any 0 ⩽ t ⩽ T . Therefore, the inequality (3.19) becomes:

|X n t -X t | 2 ≤ M n t + c t 0 (1 + nd(s, X n s ))|X n s -X s | 2 ds + 2 t 0 (X n s -X s )dΛ s ,
where M m is a local martingale. Besides, note that

|X n s -X s | ≤ |X n s -π(s, X n s ) + π(s, X n s ) -X s | ≤ d(s, X n s ) + c, since π(s, X n s )
and X s belong to the bounded set D T . It follows that,

|X n t -X t | 2 ≤ M n t + c t 0 (1 + nd(s, X n s ))|X n s -X s | 2 ds + c t 0 (1 + d(s, X n s ))d|Λ s |.
Again thanks to Stochastic Gronwall's inequality (see e.g. Lemma 2.3 in Ren-Wu [START_REF] Ren | Penalization of Reflected SDEs and Neumann Problems of HJB Equations[END_REF], Theorem 4 in Scheutzow [START_REF] Scheutzow | A stochastic Gronwall lemma[END_REF]), we get:

E sup 0≤t≤T |X n t -X t,x t |e - c b,σ ,r 2
T 0 nd(s,X n s )ds ≤ 3cE T 0 d(s, X n s )d|Λ| s 1 2 ≤ cE sup 0≤t≤T d(t, X n t )|Λ| T 1 2 ≤ c E sup 0≤t≤T d(t, X n t ) p 1 p × E |Λ| T q 1 q 1 2 , p > 2, q > 1 and 1 p + 1 q = 1.
Thanks to (2.14) and (3.2), we deduce that the right hand side tends to zero as n → ∞. The remaining of the proof is similar to the proof of Proposition 3.3.

Note that the above convergence implies the convergence of Λ n to Λ as stated in the following corollary: Corollary 3.1 For any p ⩾ 1, we have:

lim n→∞ E sup 0≤t≤T |Λ n t -Λ t | p = 0. (3.20) PROOF. Recall that Λ n t -Λ t = X n t -X t + t 0 (b(r, X r ) -b(r, X n r ))dr + t 0 (σ (r, X r ) -σ (r, X n r ))dW r .
Then using the Lipschitz continuity of b and σ , we get: ∀p ⩾ 1,

|Λ n t -Λ t | p ≤ |X n t -X t | p + t 0 (b(r, X r ) -b(r, X n r ))dr p + t 0 (σ (r, X r ) -σ (r, X n r ))dW r p .
By taking the expectation, together with the use of B-D-G inequality and the convergence (3.18), we obtain the result.

Remark 3.1 It should be pointed out that the smoothness of the direction of reflection (2.5) and the assumption on the boundary of the domain (2.6) are not required to obtain the convergence of the approximation (3.1). However, in case these conditions are not satisfied, the existence and uniqueness of a solution of the reflected SDE (2.10) as well as its properties are not ensured within the geometric setting of Lundström-Önskog [START_REF] Lundström | Stochastic and partial differential equations on nonsmooth timedependent domains[END_REF]. In another framework, when the SDE (2.10) is reflected in D ⊂ R d+1 which is assumed to be at least H 2 -time-dependent domain, the cone of unit inward normal vectors is reduced to a unique element and the time sections of D satisfy the uniform exterior and interior sphere condition. Therefore, the existence and uniqueness of the solution of the reflected SDE (2.10) within this geometric setting as well as the estimate (2.14) follow without the need for the assumptions (2.5) and (2.6) (see Theorem 3.2, Theorem 3.4 and Proposition 3.5 in Costantini-Gobet-El Karoui [START_REF] Costantini | Boundary sensitivities for diffusion processes in time dependent domains[END_REF]). Moreover, with this type of regularity, the control of the distance d(t, X n t ) is obtained without resorting to the approximation of the domain (3.1).

Remark 3.2 In the case of time-independent domains, the estimates (3.2) coincide with the estimates obtained for the approximation of reflected SDEs in regular convex domain given in Bahlali-Maticiuc-Zalinescu [START_REF] Bahlali | Penalization method for a nonlinear Neumann PDE via weak solution of reflected SDEs[END_REF] and Menaldi [START_REF] Menaldi | Stochastic Variational Inequality for Reflected Diffusion[END_REF]. We also note that the convergence results (3.18) and (3.20) are similar to the results of convergence obtained in time-independent domains: for convex domain (Menaldi [10] and Slominski [START_REF] Slominski | Weak and strong approximations of reflected diffusions via penalization methods[END_REF]) and for non-convex domains (see e.g. Lions-Sznitmann [START_REF] Lions | Stochastic differential equations with reflecting boundary conditions[END_REF] and Ren-Wu [START_REF] Ren | Penalization of Reflected SDEs and Neumann Problems of HJB Equations[END_REF]).

Application to generalized BSDEs and PDEs with boundary condition on non-smooth time-dependent domains

Let D be a time-dependent domain satisfying (2.1)-(2.6). In this section, we give an application of the results obtained in the previous section by providing an approximation of generalized BSDEs when the underlying process is a reflected diffusion in a time-dependent domain. This approximation consists of a sequence of standard BSDEs associated with standard diffusions. Therefore, by considering the associated PDEs we also get an approximation of PDEs with boundary conditions on a time-dependent domain using a sequence of standard PDEs defined on R d . However, this requires an additional smoothness condition on the domain. From now on, we assume that the cone of unit inward normal vectors at each boundary point is reduced to a unique element ⃗ n that is assumed to satisfy (2.5).

Remark 4.1 By assuming that ⃗ n is unique at each boundary point, we note that Lemma 2.2 in Boufoussi-Casteren [START_REF] Boufoussi | An approximation result for a nonlinear Neumann boundary value problem via BSDEs[END_REF] can be generalized to the case φ ∈ C 1,2 b (R 1+d ). The result is based essentially on the convergences (3.18) and (3.20). In fact we have: ∀p ⩾ 1,

lim n→∞ E sup 0≤t≤T t 0 ⟨φ (r, X r ), dΛ r ⟩ - t 0 ⟨φ (r, X n r ), dΛ n r ⟩ p = 0.
Therefore, by taking φ :=⃗ n(., .), we get:

lim n→∞ E sup 0≤t≤T t 0 ⟨⃗ n(r, X r ), dΛ r ⟩ - t 0 ⟨⃗ n(r, X n r ), dΛ n r ⟩ p = lim n→∞ E sup 0≤t≤T |Λ n | t -|Λ| t p = 0.
Now, let (t, x) ∈ D be fixed, we define (X t,x s , Λ t,x s ) t⩽s⩽T as the unique solution of: ∀t ⩽ s ⩽ T ,      X t,x s = x + s t b(r, X t,x r )dr + s t σ (r, X t,x r )dW r + s t ⃗ n(r, X t,x r )d|Λ t,x | r ;

X t,x s ∈ Ds , |Λ t,x | s = s 0 χ {X t,x r ∈∂ Dr } d|Λ t,x | r . (4.1)
Then, let m ⩾ 1 and let us introduce the following functions:

f : (t, x, y, z) ∈ [0, T ] × R d × R m × R m×n -→ f (t, x,⃗ y, z) = ( f i (t, x,⃗ y, z i )) i=1,...,m ∈ R m , ψ : (t, x, y) ∈ [0, T ] × R d × R m -→ ψ(t, x,⃗ y) = (ψ i (t, x,⃗ y)) i=1,...,m ∈ R m , h : x ∈ R d -→ h(x) = (h i (x)) i=1,...,m ∈ R m .
From now on, we make the following assumptions:

(H 0 )
The function h is continuous and is of polynomial growth.

(H 1 ) (i) (t, x) -→ f (t, x,⃗ y, z) and (t, x) -→ ψ(t, x,⃗ y) are uniformly continuous with respect to (⃗ y, z) and ⃗ y respectively.

(ii) (t, x) -→ f (t, x, 0, 0) and (t, x) -→ ψ(t, x, 0) are of polynomial growth.

(iii) f and ψ are Lipschitz continuous with respect to (⃗ y, z) and ⃗ y respectively.

(iii) ∃β < 0 such that ⟨y -ȳ, ψ(t, x, y) -ψ(t, x, ȳ)⟩ ≤ β | y -ȳ | 2 .

Associated BSDEs

Now, let (X t,x s , Λ t,x s ) t⩽s⩽T be the unique solution of the reflected SDE (4.1), then we introduce the associated multidimensional generalized BSDE: ∀t ⩽ s ⩽ T ,

Y t,x s = h(X t,x T ) + T s f (r, X t,x r ,Y t,x r , Z t,x r )dr + T s ψ(r, X t,x r ,Y t,x r )d|Λ t,x | r - T s Z t,x r dW r , (4.2)
which has a unique solution thanks to Theorem 1.6 in Pardoux-Zhang [START_REF] Pardoux | Generalized BSDEs and nonlinear Neumann boundary value problems[END_REF] that we denote (Y t,x , Z t,x ). Moreover, the following estimates hold:

E sup t⩽s⩽T | Y t,x s | 2 + T t | Y t,x r | 2 d|Λ t,x | r + T t ||Z t,x r || 2 dr < ∞. (4.3)
Next, let n ⩾ 1 and let X t,x,n be the unique solution of the penalized SDE:

∀t ⩽ s ⩽ T ,          X t,x,n s = x + s t b(r, X t,x,n r )dr + s t σ (r, X t,x,n r )dW r + Λ t,x,n s ; Λ t,x,n r = -n s t (X t,x,n r -π(r, X t,x,n r ))dr, |Λ t,x,n | s = s 0 n|X t,x,n s -π(s, X t,x,n s )|dr. (4.4)
Then, consider the associated sequence of standard BSDEs:

∀n ⩾ 1, ∀t ⩽ s ⩽ T , Y t,x,n s = h(X t,x,n T ) + T s f (r, X t,x,n r ,Y t,x,n r , Z t,x,n r )dr + T s ψ(r, X t,x,n r ,Y t,x,n r )d|Λ t,x,n | r - T s Z t,x,n r dW r . (4.5)
We first give the estimates for the penalized BSDE:

Proposition 4.1 Let n ⩾ 1 and (Y t,x,n , Z t,x,n ) be the solution of the the penalized BSDE (4.5). Then, we have the following estimates: ∀n ⩾ 1,

E sup t⩽s⩽T | Y t,x,n s | 2 + T t | Y t,x,n r | 2 d|Λ t,x,n | r + T t ||Z t,x,n r || 2 dr < c, (4.6)
where c is a constant independent of n.

PROOF. From Itô's formula, we have:

|Y t,x,n s | 2 + T s ||Z t,x,n r || 2 dr = |h(X t,x,n T )| 2 + 2 T s ⟨Y t,x,n r , f (r, X t,x,n r ,Y t,x,n r , Z t,x,n r )⟩dr +2 T s ⟨Y t,x,n r , ψ(r, X t,x,n r ,Y t,x,n r )⟩d|Λ t,x,n | r -2 T s ⟨Y t,x,n r , Z t,x,n r dW r ⟩.
Besides, using assumptions (H 1 )(ii) -(iv), we have:

⟨Y t,x,n r , f (r, X t,x,n r ,Y t,x,n r )⟩ ≤ |Y t,x,n r || f (r, X t,x,n r ,Y t,x,n r )| ≤ |Y t,x,n r |{c(1 + |X t,x,n r | + |Y t,x,n r | + |Z t,x,n r |)} ≤ c c 1 {(1 + |X t,x,n r | 2 + |Y t,x,n r | 2 } + c 1 ||Z t,x,n r || 2 (4.7)
and

⟨Y t,x,n r , ψ(r, X t,x,n r ,Y t,x,n r )⟩ ≤ ⟨Y t,x,n r , ψ(r, X t,x,n r ,Y t,x,n r ) -ψ(r, X t,x,n r , 0)⟩ + ⟨Y t,x,n r , ψ(r, X t,x,n r , 0)⟩ ≤ β |Y t,x,n r | 2 + c 2 |Y t,x,n r | 2 + c c 1 {1 + |X t,x,n r | 2 )}.
Note that the constants c 1 and c 2 can be chosen such that (-βc 2 ) > 0 and (1c 1 ) > 0. Therefore, the equation (4.7) yields,

|Y t,x,n s | 2 + (-β -c 2 ) T s |Y t,x,n r | 2 d|Λ t,x,n | r + (1 -c 1 ) T s ||Z t,x,n r || 2 dr ≤ c{1 + |X t,x,n T | 2 + T s |X t,x,n r | 2 dr + T s |X t,x,n r | 2 d|Λ t,x,n | r } + c T s |Y t,x,n r | 2 dr -2 T s ⟨Y t,x,n r , Z t,x,n r dW r ⟩.
Next, using the estimates (3.2)and by taking the expectation, we get:

E |Y t,x,n s | 2 + (-β -c 2 ) T s |Y t,x,n r | 2 d|Λ t,x,n | r + (1 -c 1 ) T s ||Z t,x,n r || 2 dr ≤ c{1 + E sup t≤r≤T |X t,x,n r | 2 + |Λ n | q [t,s] + T s |Y t,x,n r | 2 dr }.
Then, thanks to Gronwall's lemma, we obtain:

sup t≤s≤T E |Y t,x,n s | 2 + T s |Y t,x,n r | 2 d|Λ t,x,n | r + T s ||Z t,x,n r || 2 dr ≤ c.
Finally, it suffices to apply BDG inequality to conclude.

The convergence of (Y t,x,n , Z t,x,n ) n⩾1 is stated in the following proposition: Proposition 4.2 Let (Y t,x,n , Z t,x,n ) n⩾1 and (Y t,x , Z t,x ) be the unique solutions of the sequence of BS-DEs (4.5) and the generalized BSDE (4.2) respectively. Then, the following convergences hold true: [START_REF] Bahlali | Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition[END_REF] when (X t,x s , Λ t,x s ) t⩽s⩽T is the solution of the reflected SDE (4.1) in a regular convex time-independent domain. The proof is very technical and relies essentially on the results of convergence and properties of the penalized SDE (4.4) established in [START_REF] Bahlali | Penalization method for a nonlinear Neumann PDE via weak solution of reflected SDEs[END_REF][START_REF] Boufoussi | An approximation result for a nonlinear Neumann boundary value problem via BSDEs[END_REF][START_REF] Slominski | Weak and strong approximations of reflected diffusions via penalization methods[END_REF] and the properties of the reflected SDE in a regular convex time-independent domain that can be found in [START_REF] Lions | Stochastic differential equations with reflecting boundary conditions[END_REF][START_REF] Tanaka | Stochastic differential equations with reflecting boundary condition in convex regions[END_REF][START_REF] Menaldi | Stochastic Variational Inequality for Reflected Diffusion[END_REF]. The properties of the solution (X t,x s , Λ t,x s ) t⩽s⩽T and the results of convergence obtained in Section 2 and Section 3 are the generalizations of these results within our geometric setting. Therefore, the proof of Proposition 4.2 can be obtained by mimicking the proof of Theorem 6 in Bahlali-Boufoussi-Mouchtabih [START_REF] Bahlali | Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition[END_REF], we omit any further details.

lim n→∞ E sup t≤s≤T |Y t,

Associated PDEs

D o = D ′ ∩ [0, T ) × R d , ∂ D = D ′ \ D ′ ∩ [0, T ) × R d .
Next, let us consider the following system of PDEs: ∀i = 1, ..., m,

          
∂ t u i (t, x) + L u i (t, x) + f i (t, x, u(t, x), σ ⊤ (t, x)D x u i (t, x)) = 0, (t, x) ∈ D o ;

∂ u i ∂⃗ n (t, x) + ψ i (t, x, u(t, x)) = 0, (t, x) ∈ ∂ D; u(T, x) = h(x), x ∈ D T , (4.9)
where the operator L is defined by L = It follows from Theorem 2 in Jakani [START_REF] Jakani | System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains[END_REF], that u is a viscosity solution of the system of PDEs with boundary condition of Neumann type on time-dependent domain (4.9) in the sense of Definition 2.2 in Jakani [START_REF] Jakani | System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains[END_REF].

Finally, let (Y t,x,n , Z t,x,n ) n⩾1 be the unique solution of (4.5), it is well known that the sequence of deterministic functions (u n ) n⩾1 given by u n (t, x) = Y t,x,n t for any n ⩾ 1 is a viscosity solution of the following PDEs system: ∀x ∈ R d , ∀0 ⩽ t < T , ∀i = 1, . . . , m,

       ∂ t u n i (t, x) + L u n i (t, x) + f i (t,
x, u n (t, x), σ ⊤ (t, x)D x u n i (t, x))

-nψ i (t, x, u n (t, x))⟨⃗ n(t, x), xπ(t, x)⟩ = 0, u n (T, x) = h(x). As an application of the approximation provided for generalized BSDEs (4.5), we obtain an approximation for the system of PDEs (4.9):

  and that D t is open and convex for every t ∈ [0, T ].

. 13 )

 13 It suffices to choose c 1 , c 2 , c 3 and c 4 positive such that pc 1c 2c 3c 4 > 1. Then, by taking the expectation, there exists C > 0 depending on c 1 , c 2 , c 3 , c 4 and p such that:

  Now, we are in a position to get an approximation of PDEs with nonlinear Neumann boundary condition on time-dependent domains. Let us set D = D ′ ∩ [0, T ] × R d . Then, following the notation of Lundström-Önskog [9], let us recall the spaces:

1 2

 1 Tr(σ σ ⊤ )D 2 xx (.) + b ⊤ D x (.) and at a point (t, x) ∈ ∂ D we set ∂ ∂⃗ n = ⟨⃗ n(t, x), D x (.)⟩. Let u : D → R m be the deterministic function defined by Y t,x the solution of the multidimensional generalized BSDE (4.2) as follows: u i (t, x) := Y t,x,i t , ∀i = 1, . . . , m. (4.10)

Proposition 4 . 3

 43 The following convergence holds: ∀(t, x) ∈ D, ∀i = 1, . .

  Now, recall that for any t ∈ [0, T ], D t is convex. Then for each t ∈ [0, T ], we have for all y ∈ R d \ Dt at least one projection of y onto ∂ D t along N(t, •) denoted π(t, y) which satisfies: d(t, y) = d(y, D t ) = |yπ(t, y)| and π(t, y) = y, ∀y ∈ Dt .

	Let D be a time-dependent domain satisfying (2.1) and (2.2), then we have the following lemma
	which is an adaptation of Lemma 3.1 in [11] to our case:
	Lemma 2.1 For any t ∈ [0, T ], and y, y ′ ∈ R d , we have:
	(a) ⟨y ′ -y, y -π(t, y)⟩ ≤ 0, y ′ ∈ Dt ;
	(b) ⟨y ′ -y, y -π(t, y)⟩ ≤ ⟨y ′ -π(t, y ′ ), y -π(t, y)⟩, y ′ ∈ Dt ;
	(c) |π(t, y) -π(t, y ′ )| ≤ |y -y ′ |.
	Furthermore, there exists P 0 ∈ D 0 and 1 ≤ γ < ∞ depending on d(P 0 , ∂ D 0 ) such that:
	(d) ⟨y -P 0 , y -π(t, y)⟩ ≥ 1 γ |y -π(t, y)|, for any y ∈ R d and t ∈ [0, T ].
			.9)
	Recall the function l(r) = sup s,t∈[0,T ] x∈ Ds sup	inf y∈ Dt	|x -y| introduced in [5], then by Remark 2.4 in [9], the
	|s-t|⩽r
	condition (2.9) is equivalent to l(r) ⩽ Kr α .

  the solution of the penalized SDE (3.1). Then, for any p ⩾ 1,

	lim m,n→∞ E sup 0≤t≤T	|X n t -X m t | p = 0.	(3.15)
	PROOF. Let m, n ⩾ 1 and 0 ⩽ t ⩽ T , by applying Itô's formula to |X n s -X m s | 2 , we get:

  Remark 4.2 The convergence (4.8) was established in Bahlali-Boufoussi-Mouchtabih

	s	x,n	-Y t,x s | 2 +	t	T	||Z t,x,n r	-Z t,x r || 2 dr = 0.	(4.8)

  . , m,

	lim n→∞	u n i (t, x) = u i (t, x).	(4.12)

PROOF. Let (t, x) ∈ D, then from the definition of (u n i ) i=1,...,m and (u i ) i=1,...,m , it follows that: ∀i = 1, . . . , m,

lim n→∞ |u n i (t, x)u i (t, x)| 2 = lim n→∞ |Y t,x,n,i t -Y t,x,i t | 2 ≤ lim n→∞ E sup t≤s≤T |Y t,x,n,i s -Y t,x,i s | 2 = 0.