Pradorn Sureephong

Nopasit Chakpitak

Yacine Ouzrout

Gilles Neubert

Abdelaziz Bouras

Yacine Ouzroute

Knowledge Engineering Technique for Cluster Development

Keywords: Knowledge Engineering, Industry Cluster, CommonKADS, Knowledge Management System

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The knowledge-based economy is affected by the increasing use of information technologies. Most of industries try to use available information to gain competitive advantages [START_REF] Malmberg | How) do (firms in) cluster create knowledge[END_REF]. From the study of ECOTEC in 2005 [2] about the critical success factors in cluster development, first two critical success factors are collaboration in networking partnership and knowledge creation for innovative technology in the cluster which are about 78% and 74% of articles mentioned as success criteria accordingly. This knowledge is created through various forms of local interorganizational collaborative interaction [START_REF] Malmberg | On the roal of global demand in local innovation processes: Rethinking Regional Innovation and Change[END_REF].

Study of Yoong and Molina [START_REF] Young | Knowledge Sharing and Business Clusters[END_REF] assumed that one way of surviving in today's turbulent business environment for business organizations is to form strategic alliances or mergers with other similar or complementary business companies. Thus, grouping as a cluster seems to be the best solution to increase the competitiveness for companies [5][6]. Although, many literatures claimed that knowledge is very important for cluster development but there is no empirical method to initiate or improve knowledge sharing for cluster.

Developing knowledge-based application creates difficulties to knowledge engineers [START_REF] Lodbecke | Co-opetition and Knowledge Transfer[END_REF]. Knowledge-based project cannot be handle by general software engineering methodology. The lifecycle of knowledge based application and software application is different in many aspects. In order to achieve the objective of knowledge engineering, Knowledge-Based Engineering (KBE) application lifecycle [START_REF] Preston | Knowledge Acquisition for Knowledge-based Engineering Systems[END_REF] focuses on these six critical phases as shown in figure 1.

Knowledge Engineering Techniques

Actually, knowledge is not a new idea [START_REF] Davenport | Working Knowledge: How Organizations Manage What They Know[END_REF] [10], philosophers and scholars had been studying it for centuries. There are many knowledge engineering techniques which are used for solving problem. However, we choose well known techniques that widely used in many projects for this study, i.e. MOKA, SPEDE and CommonKADS. These techniques were applied in different projects in various domains. All these methods are based on this KBE application lifecycle [START_REF] Preston | Knowledge Acquisition for Knowledge-based Engineering Systems[END_REF] which focuses on six critical phases as shown in fig. 1. We will analyze their capacity to be implemented in the cluster context.

MOKA (Methodology and tools Oriented to Knowledge based engineering Applications) aims to help the structure side of the capture process (Fig. 1) in KBE application lifecycle. It focuses on two levels of representation -an informal and a formal model. These models provide the means of recording the structure behind the knowledge -including not only things about the product and design process but about the design rationale as well. Informal model is assembled from five categories of knowledge types; described on forms, known as ICARE forms (Illustrations, Constraints, Activities, Rules and Entities) [START_REF] Stroke | MOKA: methodology for knowledge-based engineering applications[END_REF]. SPEDE (Structured Process Elicitation Demonstrations Environment) provides an effective means to capture, validate and communicate vital knowledge to provide business benefit [START_REF] Shadbolt | From knowledge engineering to knowledge management[END_REF]. The concept of this methodology is to develop a guide that suitable for using with a variety of BPR/BPI projects, while at the same creating a means of enhancing process. This was made possible by breaking the activities into generally acknowledged high level BPR/BPI stages and providing a sequence at that level. The concept of this methodology is called Swim Lane. SPEDE provides Knowledge Acquisition (KA) tools to facilitate and assist the process in knowledge context.

CommonKADS (Common Knowledge Acquisition and Design System) is a methodology to support structured knowledge engineering. It provided CommonKADS model suite for creating requirements specification for knowledge system. The organization, task, and agent models analyze the organizational environment and the corresponding critical success factors for a knowledge system. The knowledge and communication models yield the conceptual description of problem-solving functions and data that were handled and delivered by a knowledge system. The design model converts it into a technical specification that is the basics for software system implementation [START_REF] Schreiber | Knowledge Engineering and Management: The CommonKADS Methodology[END_REF].

Knowledge Engineering Technique Selection

In the study, we used knowledge-based engineering lifecycle and their provided tools as our criteria to select knowledge engineering technique for this study. The result of the comparison was shown in Table . 1. MOKA focuses on capturing and formalizing knowledge [START_REF] Stroke | MOKA: methodology for knowledge-based engineering applications[END_REF] to solve the specific engineering problems. On the other hand, we found that both SPEDE and CommonKADS techniques support KBE lifecycle from knowledge identifying to packaging phase. Then, we consider in the detail of each models of SPEDE and CommonKADS. SPEDE provided swim lane flowcharts as tools for each processes. SPEDE technique mainly focused on business process improvement in knowledge context. CommonKADS provided models and templates for each level. These templates support knowledge engineer for knowledge elicitation in different knowledge task, i.e. analytic tasks and synthetic tasks. These help knowledge engineer to be able to apply this technique in different type of knowledge problems. From two criteria, support KBE lifecycle and provided tools, CommonKADS technique is suitable for applying with industry cluster problems.

The concept of proposed methodology is adopted from CommonKADS methodology which was divided into three levels called CommonKADS model suite, i.e. Context Level, Concept Level and Artifact Level. However, managing structured knowledge in the industry cluster is different from single organization in many aspects because of characteristic of the organization. For example, there is no single policy maker in the cluster. So, KE could not utilize Context level's worksheets to assess single company for developing KMS for all companies within the cluster.

Context Level

This level contain organization, task and agent model. The main objectives of this level are giving the scope and clear view of the organization, knowledge intensive tasks and actors who involved in the task. It also provides the impact assessment, changes and consensus for knowledge engineering project. Knowledge Engineer (KE) should start at the most influence association in the cluster. Due to, association always be a group of potential companies in the industry and able to set policy/direction for the industry. KE could utilize Organization Mode (OM) worksheets for interviewing with associations. Then, the outputs from OM are knowledge intensive tasks from broken down process and agents who are related to each task. Then, KE could interview with experts in each task by using TM and AM worksheet. Finally, KE validate the result of each module with association again to assess impact and changes with OTA worksheet.

Concept Level

The second level contain knowledge and communication model. This level is related with capturing and formalizing phases in KBE lifecycle. The main objectives of this to the level are to explicate in detail the types and structures used in performing task and to model the communicative transaction between the agents involved [START_REF] Schreiber | Knowledge Engineering and Management: The CommonKADS Methodology[END_REF]. We acquired Task knowledge from context level which composed of task goal, decomposition and control. Inference and Domain knowledge could be obtained from knowledge elicitation process.

Artifact Level

The last level is the artifact level, contain design model. This level is related to packaging phase in KBE lifecycle or application development. The main objectives of this level are to give the technical system specification, such as architecture, implement platform, software modules, representation constructs, and computational mechanisms needed to implement the functions laid down in the knowledge and communication models [START_REF] Schreiber | Knowledge Engineering and Management: The CommonKADS Methodology[END_REF].

Validation and Results

The initial investigations have been done with 10 firms within the two biggest handicraft associations in Thailand and Northern Thailand. NOrthern Handicraft Manufacturer and EXporter (NOHMEX) association is the biggest handicraft association in Thailand which includes 161 manufacturers and exporters. Another association which is the biggest handicraft association in Chiang Mai is named Chiang Mai Brand. It is a group of qualified manufacturers who have capability to export their products and passed requirements of Thailand's ministry of commerce. Until end of 2006, there are 99 authorized enterprises to use Chiang Mai brand on their products [START_REF] Sureephong | Economic based Knowledge Management System for SMEs Cluster: case study of handicraft cluster in Thailand[END_REF].

At the beginning of this study, CommonKADS was used as a knowledge engineering methodology in the context level (organization model, task model, and agent model) in order to understand organization environment and corresponding critical success factors for knowledge system.

As shown in Fig. 2, Organization Model (OM-1 to OM-5), we found that handicraft cluster has its own vision as "Knowledge sharing hub for handicraft exporter". And, companies defined their problems, such as intellectual property problem, lack of collaboration, CDA development, product innovation, and product exporting. However, this cluster has many opportunities and solutions as well. We used "product exporting" and "product innovation" as our mock-up problems due to these problems is knowledge intensive and feasible in business and technical aspect.

From the Task Model (TM-1 to TM-2), we analyzed the feasibility of each tasks that related to product exporting and product innovation processes. This model makes it possible to rank and prioritize the different knowledge-improvement scenarios.

Agent Model (AM-1) proposed organizational recommendations, improvements, and actions. From the experts' point of view, they proposed actions for solving product exporting and product innovation problems as follow, 1) Develop information system that provides knowledge from experts about product selection, marketing information, or economic data from government organization.

2) Archive past lesson learn or experiences with in electronic forum 3) Create best practice of each task and store in knowledge-based system 4) Increasing the collaboration and information sharing within the cluster. 5) Create tools to support the capability of cluster development agency (CDA) to facilitate the cluster.

Fig. 1 .

 1 Fig. 1. Knowledge based engineering application lifecycle

Fig. 2 .

 2 Fig. 2. A road map for carrying out knowledge oriented organization and task analysis.

Table 1 . Three methods compared with Knowledge Based Engineering Lifecycle

 1

	KBE Lifecycle	MOKA	SPEDE	CommonKADS
	1. Identify	-	Understand the project	Context Level
	2. Justify	-	Understand the project	OTA Model
	3. Capture	Informal Model	Design the process	Concept Level
	4. Formalize	Formal Model	Evaluate the new process	Concept Level
	5. Package	-	Communicate Process	Artifact Level
	6. Activate	-	-	-


Stefan Ankirchner
email: s.ankirchner@uni-jena.de

Thomas Kruse
email: thomas.kruse@uni-due.de

Mikhail Urusov
email: mikhail.urusov@uni-due.de

A functional limit theorem for coin tossing Markov chains

Keywords: one-dimensional Markov process, speed measure, Markov chain approximation, functional limit theorem, sticky Brownian motion, slow reflection, Brownian motion slowed down on the Cantor set 60F17, 60J25, 60J60. Secondary: 60H35, 60J22

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability 1/2, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process can be approximated with such Markov chains arbitrarily well. It applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. We illustrate the theorem with Markov processes exhibiting sticky features, e.g., sticky Brownian motion and a Brownian motion slowed down on the Cantor set.

Introduction

Let (ξ k ) k∈N be an iid sequence of random variables, on a probability space with a measure P , satisfying P (ξ 1 = ±1) = 1 2 . Given y ∈ R, h ∈ (0, ∞) and a function a h : R → R, we denote by (X h kh ) k∈N 0 the Markov chain defined by

X h 0 = y and X h (k+1)h = X h kh + a h (X h kh )ξ k+1 , for k ∈ N 0 . (1)
We choose as the Markov chain's index set the set of non-negative multiples of h because we interpret h as the length of a time step. We extend (X h kh ) k∈N 0 to a continuous-time process by linear interpolation, i.e., we set

X h t = X h t/h h + (t/h -t/h )(X h ( t/h +1)h -X h t/h h ), t ∈ [0, ∞). (2)
Let h ∈ (0, ∞) and let (a h ) h∈(0,h) be a family of real functions and (X h ) h∈(0,h) the associated family of extended Markov chains defined as in [START_REF] Ankirchner | Stopping with expectation constraints: 3 points suffice[END_REF]. A fundamental problem of probability theory is to find conditions on (X h ) h∈(0,h) such that the laws of the processes X h , h ∈ (0, h), converge in some sense as h → 0. In this article we provide an asymptotic condition on the family (a h ) h∈(0,h) guaranteeing that the laws of the processes X h , h ∈ (0, h), converge as h → 0 to the law of a one-dimensional regular continuous strong Markov process (in the sense of Section VII.3 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF] or Section V.7 in [START_REF] Rogers | Diffusions, Markov processes, and martingales[END_REF]). In what follows we use the term general diffusions for the latter class of processes. Recall that a general diffusion Y = (Y t ) t∈[0,∞) has a state space that is an open, half-open or closed interval I ⊆ R. We denote by I • = (l, r) the interior of I, where -∞ ≤ l < r ≤ ∞.

Moreover, the law of any general diffusion is uniquely characterized by its speed measure m on I, its scale function and its boundary behavior. Throughout the introduction we assume that Y is in natural scale and that every accessible boundary point is absorbing (see the beginning of Section 1 and Section 6 on how to incorporate diffusions in general scale and with reflecting boundary points). This setting covers, in particular, solutions of driftless SDEs with discontinuous and fast growing diffusion coefficient (see Section 2) and also diffusions with "sticky" behavior that cannot be modeled by SDEs (see Section 7).

Our main result, Theorem 1.1, shows that if a family of functions (a h ) h∈(0,h) satisfies for all y ∈ I • , h ∈ (0, h) the equation 1 2 (y-a h (y),y+a h (y))

(a h (y) -|u -y|) m(du) = h,

with a precision of order o(h) uniformly in y over compact subsets of I • (see Condition (A) below for a precise statement), then the associated family (X h ) h∈(0,h) converges in distribution, as h → 0, to the general diffusion Y with speed measure m. We show that for every general diffusion a family of functions (a h ) h∈(0,h) satisfying (3) exists implying that every general diffusion can be approximated by a Markov chain of the form [START_REF] Amir | Sticky Brownian motion as the strong limit of a sequence of random walks[END_REF]. Equation (3) dictates how to compute the functions (a h ) h∈(0,h) and therefore paves the way to approximate the distribution of a general diffusion numerically (see, e.g., Section 8). The central idea in the derivation of Equation ( 3) is to embed for every h ∈ (0, h) the Markov chain (X h kh ) k∈N 0 into Y with a sequence of stopping times. To explain this idea assume for the moment that the state space is I = R. For every h ∈ (0, h) let τ h 0 = 0 and then recursively define τ h k+1 as the first time Y exits the interval

(Y τ h k - a h (Y τ h k ), Y τ h k + a h (Y τ h k )) after τ h k .
It follows that the discrete-time process (Y τ h k ) k∈N 0 has the same law as the Markov chain (X h kh ) k∈N 0 . Instead of controlling now directly the spacial errors |Y τ h k -Y kh |, we first analyze the temporal errors |τ h k -kh|, k ∈ N 0 . We show that for every y ∈ R, a ∈ [0, ∞) the expected time it takes Y started in y to leave the interval (y -a, y + a) is equal to 1 2 (y-a,y+a) (a -|u -y|) m(du). In particular, if a h satisfies (3) for all y ∈ I, it follows that for all k ∈ N 0 the time lag τ h k+1 -τ h k between two consecutive stopping times is in expectation equal to h. In this case we refer to (X h kh ) k∈N 0 as Embeddable Markov Chain with Expected time Lag h (we write shortly (X h kh ) k∈N 0 ∈ EMCEL(h)).

For some diffusions Y one can construct EMCEL approximations explicitly (see, e.g., Section 7). For cases where (3) cannot be solved in closed form, we perform a perturbation analysis and show that it suffices to find for all h ∈ h, y ∈ I • a number a h (y) satisfying [START_REF] Ankirchner | Numerical approximation of irregular SDEs via Skorokhod embeddings[END_REF] with an error of order o(h) uniformly in y belonging to compact subsets of I • . We prove that for the associated stopping times (τ h k ) k∈N 0 the temporal errors |τ h k -kh|, k ∈ N 0 , converge to 0 as h → 0 in every L α -space, α ∈ [1, ∞). This ultimately implies convergence of (X h ) h∈(0,h) to Y in distribution as h → 0.

To illustrate the benefit of the perturbation analysis, we construct in Section 8 approximations for a Brownian motion slowed down on the Cantor set (see Figure 3). Moreover, we note that our main result, Theorem 1.1, is not only applicable to perturbations of the EMCEL approximation but can also be used to derive new convergence results for other approximation methods such as, e.g., weak Euler schemes (see Corollary 2.3).

The idea to use embeddings in order to prove a functional limit theorem goes back to Skorokhod. In the seminal book [START_REF] Skorokhod | Studies in the theory of random processes[END_REF] scaled random walks are embedded into Brownian motion in order to prove Donsker's invariance principle. In [START_REF] Ankirchner | A functional limit theorem for irregular SDEs[END_REF] we embed Markov chains into the solution process of an SDE and prove a functional limit theorem where the limiting law is that of the SDE. In [START_REF] Skorokhod | Studies in the theory of random processes[END_REF] and [START_REF] Ankirchner | A functional limit theorem for irregular SDEs[END_REF] the approximating Markov chains have to be embeddable with a sequence of stopping times (τ k ) k∈N 0 such that the expected distance between two consecutive stopping times is exactly equal to h, the time discretization parameter. In contrast, in the present article we require that the expected distance between consecutive embedding stopping times is only approximately equal to h. We show that for the convergence of the laws it is sufficient to require that the difference of the expected distance and h is of the order o(h). Moreover, compared to [START_REF] Ankirchner | A functional limit theorem for irregular SDEs[END_REF], we allow for a larger class of limiting distributions. Indeed, our setting includes processes that can not be characterized as the solution of an SDE, e.g. diffusions with sticky points.

There are further articles in the literature using random time grids to approximate a Markov process, under the additional assumption that it solves a one-dimensional SDE. In [START_REF] Etoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF] the authors first fix a finite grid in the state space of the diffusion. Then they construct a Bernoulli random walk on this grid that can be embedded into the diffusion. The authors determine the expected time for attaining one of the neighboring points by solving a PDE.

[17] describes a similar approximation method for the Cox-Ingersoll-Ross (CIR) process. Also here the authors first fix a grid on [0, ∞) and then construct a random walk on the grid that can be embedded into the CIR process. In contrast to [START_REF] Etoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF], the authors in [START_REF] Milstein | Uniform approximation of the Cox-Ingersoll-Ross process via exact simulation at random times[END_REF] compute the distributions of the embedding stopping times (and not only their expected value) by solving a parabolic PDE. In the numerical implementation of the scheme the authors then draw the random time increments from these distributions and thereby obtain a scheme that is exact along a sequence of stopping times. Note that in contrast to [START_REF] Etoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF] and [START_REF] Milstein | Uniform approximation of the Cox-Ingersoll-Ross process via exact simulation at random times[END_REF], in our approach the space grid is not fixed a priori. Instead, we approximately fix the expected time lag between the consecutive embedding stopping times.

Yet a further scheme that uses a random time partition to approximate a diffusion Y with discontinuous coefficients is suggested in [START_REF] Lejay | An exponential timestepping algorithm for diffusion with discontinuous coefficients[END_REF]. In contrast to our approach the distribution of the time increments is fixed there. More precisely, the authors of [START_REF] Lejay | An exponential timestepping algorithm for diffusion with discontinuous coefficients[END_REF] use the fact that the distribution of Y sampled at an independent exponential random time is given by the resolvent of the process. Consequently, if it is possible to generate random variables distributed according to the resolvent kernel, one obtains an exact simulation of Y at an exponentially distributed stopping time. Iterating this procedure and letting the parameter of the exponential distribution go to infinity provides an approximation of Y .

The article is organized as follows. In Section 1 we rigorously formulate and discuss the functional limit theorem. In Section 2 we discuss some of its implications for diffusions that can be described as solution of SDEs. In Sections 3 and 4 we explain, for a given general diffusion, how to embed an approximating coin tossing Markov chain into the diffusion and prove some properties of the embedding stopping times. Section 5 provides the proof of the functional limit theorem, where, in particular, we need the material discussed in Sections 3 and 4. The functional limit theorem is shown under the additional assumption that if a boundary point is attainable, then it is absorbing. In Section 6 we explain how one can extend the functional limit theorem to general diffusions with reflecting boundary points. In the last two sections we illustrate our main result with diffusions exhibiting some stickiness. In Section 7 we construct coin tossing Markov chains approximating sticky Brownian motion, with and without reflection, respectively. In Section 8 we first describe a Brownian motion that is slowed down on the Cantor set, and secondly we explicitly construct coin tossing Markov chains that approximate this process arbitrarily well.

Approximating general diffusions with Markov chains

Let (Ω, F, (F t ) t≥0 , (P y ) y∈I , (Y t ) t≥0 ) be a one-dimensional continuous strong Markov process in the sense of Section VII.3 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]. We refer to this class of processes as general diffusions in the sequel. We assume that the state We further assume that Y is regular. This means that for every y ∈ I • and x ∈ I we have that [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]). Recall that for all a < b in

P y [H x (Y ) < ∞] > 0, where H x (Y ) = inf{t ≥ 0 : Y t = x}.
I • we have 0 < m([a, b]) < ∞. (4)
Finally, we also assume that if a boundary point is accessible, then it is absorbing. We drop this assumption in Section 6, where we extend our approximation method to Markov processes with reflecting boundaries. The extension works for both instantaneous and slow reflection. Let h ∈ (0, ∞) and suppose that for every h ∈ (0, h) we are given a measurable function a h : I → [0, ∞) such that a h (l) = a h (r) = 0 and for all y ∈ I • we have y ± a h (y) ∈ I. We refer to each function a h as a scale factor. We next construct a sequence of Markov chains associated to the family of scale factors (a h ) h∈(0,h) . To this end we fix a starting point y ∈ I • of Y . Let (ξ k ) k∈N be an iid sequence of random variables, on a probability space with a measure P , satisfying P (ξ k = ±1) = 1 2 . We denote by (X h kh ) k∈N 0 the Markov chain defined by

X h 0 = y and X h (k+1)h = X h kh + a h (X h kh )ξ k+1 , for k ∈ N 0 . (5)
We extend (X h kh ) k∈N 0 to a continuous-time process by linear interpolation, i.e., for all t ∈ [0, ∞), we set

X h t = X h t/h h + (t/h -t/h )(X h ( t/h +1)h -X h t/h h ). (6)
To highlight the dependence of X h = (X h t ) t∈[0,∞) on the starting point y ∈ I • we also sometimes write X h,y .

To formulate our main result we need the following condition. Condition (A) For all compact subsets K of I • it holds that

sup y∈K 1 2 (y-a h (y),y+a h (y)) (a h (y) -|u -y|) m(du) -h ∈ o(h), h → 0. (7)
Theorem 1.1. Assume that Condition (A) is satisfied. Then, for any y ∈ I • , the distributions of the processes (X h,y t ) t∈[0,∞) under P converge weakly to the distribution of (Y t ) t∈[0,∞) under P y , as h → 0; i.e., for every bounded and continuous functional 1

F : C([0, ∞), R) → R, it holds that E[F (X h,y )] → E y [F (Y )], h → 0. ( 8
)
1 As usual, we equip C([0, ∞), R) with the topology of uniform convergence on compact intervals, which is generated, e.g., by the metric

d(x, y) = ∞ n=1 2 -n x -y C[0,n] ∧ 1 , x, y ∈ C([0, ∞), R),
(a h (y) -|u -y|) m(du) = I (a h (y) -|u -y|) + m(du).
This yields an alternative representation of Condition (A) which is occasionally used below.

It is important to note that for every speed measure m there exists a family of scale factors such that Condition (A) is satisfied and hence every general diffusion Y can be approximated by Markov chains of the form [START_REF] Bass | A stochastic differential equation with a sticky point[END_REF]. Indeed, for all y ∈ I • , h ∈ (0, h) let a h (l) = a h (r) = 0 and

a h (y) = sup a ≥ 0 : y ± a ∈ I and 1 2 (y-a,y+a) (a -|z -y|) m(dz) ≤ h (9)
and denote by ( X h ) h∈(0,1) the associated family of processes defined in ( 5) and [START_REF] Engelbert | Stochastic differential equations for sticky Brownian motion[END_REF]. Then the proof of Corollary 1.4 below shows that for all compact subsets K of I • there exists h 0 ∈ (0, h) such that for all y ∈ K, h ∈ (0, h 0 ) it holds that

1 2 (y-a h (y),y+ a h (y)) ( a h (y) -|z -y|) m(dz) = h.
In particular, the family ( a h ) h∈(0,1) satisfies Condition (A) and we show in Section 3 below that the Markov chain ( X h kh ) k∈N 0 is embeddable into Y with a sequence of stopping times with expected time lag h. We refer to ( X h t ) t∈[0,∞) , h ∈ (0, h), as EMCEL approximations and write shortly ( X h kh ) k∈N 0 ∈ EMCEL(h). Corollary 1.4. For every y ∈ I • the distributions of the EMCEL approximations ( X h,y t ) t∈[0,∞) under P converge weakly to the distribution of (Y t ) t∈[0,∞) under P y as h → 0. Proof. Let K be a compact subset of I • . Without loss of generality assume that K = [l 0 , r 0 ] with l < l 0 < r 0 < r. Let a 0 = r-r 0 2 ∧ l 0 -l 2 ∧ 1. It follows with dominated convergence that the function

K y → 1 2 I (a 0 -|u -y|) + m(du) ∈ (0, ∞)
is continuous. In particular, it is bounded away from zero, i.e., there exists h 0 ∈ (0, h) such that for all y ∈ K it holds that 1 2 I (a 0 -|u -y|) + m(du) ≥ h 0 . Next observe that for all y ∈ K the function [0, a 0 ] a → 1 2 I (a -|u -y|) + m(du) ∈ [0, ∞) is continuous and strictly increasing. Hence for all y ∈ K, h ∈ (0, h 0 ) the supremum in ( 9) is a maximum and it holds that 1 2 (y-a h (y),y+ a h (y)) ( a h (y) -|u -y|) m(du) = h. In particular, Condition (A) is satisfied and the statement of Corollary 1.4 follows from Theorem 1.1.

Application to SDEs

A particular case of our setting is the case, where Y is a solution to the driftless SDE

dY t = η(Y t ) dW t , (10)
where η :

I • → R is a Borel function satisfying the Engelbert-Schmidt conditions η(x) = 0 ∀x ∈ I • , (11)
η -2 ∈ L 1 loc (I • ) (12)
(L 1 loc (I • ) denotes the set of Borel functions locally integrable on I • ). Under ( 11)-( 12) SDE ( 10) has a unique in law weak solution (see [START_REF] Engelbert | On solutions of one-dimensional stochastic differential equations without drift[END_REF] or Theorem 5.5.7 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]). This means that there exists a pair of processes (Y, W ) on a filtered probability space (Ω, F, (F t ), P ), with (F t ) satisfying the usual conditions, such that W is an (F t )-Brownian motion and (Y, W ) satisfies SDE [START_REF] Gyöngy | A note on Euler's Approximations[END_REF]. The process Y possibly reaches the endpoints l or r in finite time. By convention we force Y to stay in l (resp., r) in this case. This can be enforced in [START_REF] Gyöngy | A note on Euler's Approximations[END_REF] by extending η to I with η(l) = η(r) = 0. In this example Y is a regular continuous strong Markov process with the state space being the interval with the endpoints l and r (whether l and r belong to the state space is determined by the behavior of η near the boundaries). Moreover, Y is in natural scale, and its speed measure on I • is given by the formula

m(dx) = 2 η 2 (x) dx.
In this situation a change of variables shows that it holds for all h ∈ (0, h), y ∈ I • that (y-a h (y),y+a h (y))

(a h (y) -|u -y|) m(du) = 2a h (y) 2 1 -1 1 -|z| η 2 (y + a h (y)z) dz. (13)
Condition (A) hence becomes that for every compact subset

K of I • it holds that lim h→0 sup y∈K a h (y) 2 h 1 -1 1 -|z| η 2 (y + a h (y)z) dz -1 = 0. (14)
Example 2.1 (Brownian motion). In the special case where

Y = W is a Brownian motion (i.e., I = R, η(x) ≡ 1), Condition (A) requires that for all compact sets K ⊂ R it holds that sup y∈K a h (y) 2 h -1 → 0 as h → 0. In particular, Condition (A) is satisfied for the choice a h (y) = √ h, h ∈ (0, ∞), y ∈ R,
and we recover from Theorem 1.1 Donsker's functional limit theorem for the scaled simple random walk.

Moreover, in the case of a Brownian motion it is natural to restrict ourselves to spacehomogeneous (i.e., constant) scale factors a h (y) ≡ a h , h ∈ (0, h), so that Condition (A) takes the form lim h→0 a 2 h h = 1. It is straightforward to show that the latter condition is also necessary for the weak convergence of approximations ( 5)-( 6) driven by spacehomogeneous scale factors to the Brownian motion.

Example 2.2 (Geometric Brownian motion). Let σ > 0 and assume that η satisfies for all x ∈ (0, ∞) that η(x) = σx. Then the solution Y of [START_REF] Gyöngy | A note on Euler's Approximations[END_REF] with positive initial value Y 0 = y ∈ (0, ∞) is a geometric Brownian motion. Its state space is I = (0, ∞) and both boundary points are inaccessible. Note that for all y ∈ (0, ∞), a ∈ (0, y) it holds that

1 -1 1 -|z| η 2 (y + az) dz = 1 (σy) 2 1 -1 1 -|z| (1 + az/y) 2 dz = - 1 (σa) 2 log 1 - a 2 y 2 .
Hence, Condition (A) requires that for all compact sets K ⊂ (0, ∞) it holds that

lim h→0 sup y∈K 1 hσ 2 log 1 - a h (y) 2 y 2 + 1 = 0. ( 15
)
To obtain the EMCEL approximation of Y we solve for all y ∈ (0, ∞), h ∈ (0, ∞) the equation

1 hσ 2 log 1 -a 2 y 2 + 1 = 0 in a and obtain a h (y) = y √ 1 -e -σ 2 h . Note that also the usual choice a h (y) = √ hσy, y ∈ (0, ∞), h ∈ (0, 1/σ 2 )
, which corresponds to the weak Euler scheme for geometric Brownian motion, satisfies [START_REF] Karlin | A second course in stochastic processes[END_REF].

Convergence of the weak Euler scheme

Throughout this subsection we assume that I = R. A common method to approximate solutions of SDEs is the Euler scheme. For equations of the form [START_REF] Gyöngy | A note on Euler's Approximations[END_REF] with initial condition Y 0 = y the Euler scheme (X Eu,h kh ) k∈N 0 with time step h ∈ (0, ∞) is given by

X Eu,h 0 = y and X Eu,h (k+1)h = X Eu,h kh + η(X Eu,h kh )(W (k+1)h -W kh ), for k ∈ N 0 .
Weak Euler schemes are variations of the Euler scheme, where the normal increments W (k+1)h -W kh , k ∈ N 0 , are replaced by an iid sequence of centered random variables with variance h. Therefore, with the choice a h (y) = √ hη(y), h ∈ (0, ∞), y ∈ R, the Markov chain (X h kh ) k∈N 0 defined in (5) represents a weak Euler scheme with Rademacher increments.

In this subsection we show how Theorem 1.1 can be used to derive new convergence results for weak Euler schemes. To this end let the setting of Section 2 be given and let a h (y) = √ hη(y), h ∈ (0, ∞), y ∈ R. Then it follows from ( 14) that Condition (A) is equivalent to assuming that for every compact subset K ⊂ R we have

sup y∈K 1 -1 η 2 (y)(1 -|z|) η 2 (y + √ hη(y)z) dz -1 = sup y∈K 1 -1 η 2 (y) -η 2 (y + √ hη(y)z) η 2 (y + √ hη(y)z) (1 -|z|) dz → 0, ( 16
) as h → 0.
Suppose that η is continuous, let K ⊂ R be compact and let ε > 0. Then η is bounded on K and since every continuous function is uniformly continuous on compact sets, we obtain that there exists h 0 ∈ (0, ∞) such that for all h ∈ (0,

h 0 ], y ∈ K, z ∈ [-1, 1] it holds that |η(y) -η(y + √ hη(y)z)| ≤ ε.
By [START_REF] Hajri | Application of stochastic flows to the sticky Brownian motion equation[END_REF] and the continuity of η the function η 2 is strictly bounded away from 0 on every compact subset of R and hence we obtain that there exists C ∈ [0, ∞) such that for all h ∈ (0, h 0 ] it holds that

sup y∈K 1 -1 η 2 (y) -η 2 (y + √ hη(y)z) η 2 (y + √ hη(y)z) (1 -|z|) dz ≤ Cε.
It follows with ( 16) that Condition (A) is satisfied. Therefore we obtain the following Corollary of Theorem 1.1.

Corollary 2.3. Assume the setting of Section 2 with I = R and that η is continuous. Let a h : R → R satisfy a h (y) = √ hη(y) for all h ∈ (0, ∞), y ∈ R. Then for all y ∈ R the distributions of the processes (X h,y t ) t∈[0,∞) under P converge weakly to the distribution of (Y t ) t∈[0,∞) under P y , as h → 0.

Remark 2.4. Corollary 2.3 complements convergence results for the Euler scheme for example obtained in [START_REF] Yan | The Euler scheme with irregular coefficients[END_REF] and [START_REF] Gyöngy | A note on Euler's Approximations[END_REF]. Theorem 2.2 in [START_REF] Yan | The Euler scheme with irregular coefficients[END_REF] shows weak convergence of the Euler scheme if η has at most linear growth and is discontinuous on a set of Lebesgue measure zero. Theorem 2.3 in [START_REF] Gyöngy | A note on Euler's Approximations[END_REF] establishes almost sure convergence of the Euler scheme if η is locally Lipschitz continuous. Moreover, [START_REF] Gyöngy | A note on Euler's Approximations[END_REF] allows for a multidimensional setting and a drift coefficient. In contrast, Corollary 2.3 above applies to the weak Euler scheme and does not require linear growth or local Lipschitz continuity of η.

Remark 2.5. As stated in Corollary 1.4, EMCEL approximations can be constructed for every general diffusion. In particular, they can be used in cases where η is not continuous and where (weak) Euler schemes do not converge (see, e.g., Section 5.4 in [START_REF] Ankirchner | Numerical approximation of irregular SDEs via Skorokhod embeddings[END_REF]). In Sections 7 and 8 we consider further irregular examples.

Embedding the chains into the Markov process

In this section we construct the embedding stopping times. To this end, we need some auxiliary results. Throughout the section we assume the setting of Section 1.

We introduce the function q :

I • × I → [0, ∞] defined by q(y, x) = 1 2 m({y})|x -y| + x y m((y, u))du, (17)
where for u < y we set m((y, u)) := -m((u, y)). Notice that, for y ∈ I • , the function q(y, •) is decreasing on [l, y] and increasing on [y, r]. Recall the Feller test for explosions: for any

y ∈ I • , l is accessible (i.e., l ∈ I) ⇐⇒ q(y, l) < ∞, (18)
r is accessible (i.e., r ∈ I) ⇐⇒ q(y, r) < ∞

(see, e.g., Lemma 1.1 in [START_REF] Ankirchner | Stopping with expectation constraints: 3 points suffice[END_REF]). Consequently, q is finite on I • × I. Notice that for all y, z ∈ I • and x ∈ I we have q(z, x) = q(y, x) -q(y, z) -

∂ 0 q ∂x (y, z)(x -z), (20)
where

∂ 0 q ∂x (y, x) = 1 2 ( ∂ + q ∂x + ∂ -q ∂x )(y, x). Lemma 3.1. Let a < b in I and y ∈ (a, b). Then E y H a,b = E y q(y, Y H a,b ). (21)
Proof. Recall that the speed measure m satisfies

E y H a,b = (a,b) (b -x ∨ y)(x ∧ y -a) b -a m(dx),
(see e.g. Section VII.3 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]). This implies

E y H a,b = 1 b -a (a,y) (b -y)(x -a)m(dx) + (y,b) (b -x)(y -a)m(dx) + (b -y)(y -a)m({y}) = b -y b -a (a,y) (x -a)m(dx) + (y -a) m({y}) 2 + y -a b -a (y,b) (b -x)m(dx) + (b -y) m({y}) 2 = b -y b -a q(y, a) + y -a b -a q(y, b) = E y q(y, Y H a,b ).
Lemma 3.2. Let y ∈ I • and a ∈ (0, ∞) such that y ± a ∈ I. Then it holds q(y, y + a) + q(y, y -a) = (y-a,y+a)

(a -|u -y|) m(du).

Proof. It follows from the definition of the function q that q(y, y + a) + q(y, y -a) = m({y})a + Substituting the latter formulas in the former one yield the result.

The following result is an immediate consequence of Lemma 3.1 and Lemma 3.2

Corollary 3.3. Let y ∈ I • and a ∈ (0, ∞) such that y ± a ∈ I. Then E y [H y-a,y+a ] = 1 2 (y-a,y+a) (a -|u -y|) m(du). (22)
We need to introduce an auxiliary subset of I • . To this end, if l > -∞, we define, for all h ∈ (0, h),

l h := l + inf a ∈ 0, r -l 2 : a < ∞ and 1 2 (l,l+2a) (a -|u -(l + a)|)m(du) ≥ h ,
where we use the convention inf ∅ = ∞. If l = -∞, we set l h = -∞. Similarly, if r < ∞, then we define, for all h ∈ (0, h),

r h := r -inf a ∈ 0, r -l 2 : a < ∞ and 1 2 (r-2a,r) (a -|u -(r -a)|)m(du) ≥ h . If r = ∞, we set r h = ∞.
Equations ( 18) and ( 19), together with Lemma 3.2 below, imply that l is inaccessible if and only if l h = l for all h ∈ (0, h). Similary, r is inaccessible if and only if r h = r for all h ∈ (0, h).

The auxiliary subset is defined by

I h = (l h , r h ) ∪ {y ∈ I • : y ± a h (y) ∈ I • } .
Now we have everything at hand to start constructing a sequence of embedding stopping times. Suppose Y starts at a point y ∈ I • and fix h ∈ (0, h). Set τ h 0 = 0. Let σ h 1 = H y-a h (y),y+a h (y) . Note that by Corollary 3.3 we have

E y [σ h 1 ] = 1 2 (y-a h (y),y+a h (y)) (a h (y) -|u -y|) m(du).
We now define τ h 1 by distinguishing two cases. Case 1: y ∈ I h (i.e., y ∈ (l h , r h ) or y ± a h (y) ∈ I • ). In this case we set τ h 1 = σ h 1 . Case 2: y / ∈ I h (i.e., y / ∈ (l h , r h ) and (y + a h (y) = r or y -a h (y) = l)). In this case we deterministically extend σ h 1 so as to make it have expectation h. Observe that by the definition of l h and r h we have in this case E y [σ h 1 ] ≤ h. Moreover, we can assume in this case that it must hold that P y (Y σ N 1 ∈ {l, r}) = 1 2 (only in the case max{|l|, |r|} < ∞, y = l+r 2 and a h (y) = r-l 2 this probability is 1, but we exclude this case by considering a sufficiently small h, so that a h ( l+r 2 ) < r-l 2 ; notice that Condition (A) implies that, for any y ∈ I • , lim h→0 a h (y) = 0). We define τ h 1 by

τ h 1 = σ h 1 + 2 h -E[σ h 1 ] 1 {l,r} (Y σ h 1 ).
Observe that the definition implies E y [τ h 1 ] = h and that the three random variables Y τ h 1 , Y σ h 1 and X h,y h have all the same law. We can proceed in a similar way to define the subsequent stopping times. Let k ∈ N. Suppose that we have already constructed τ h k . We first define

σ h k+1 = inf{t ≥ τ h k : |Y t -Y τ h k | = a h (Y τ h k )}. On the event {Y τ h k ∈ I h } we set τ h k+1 = σ h k+1 . On the event {Y τ h k / ∈ I h } we extend σ h k+1 as follows. Note that Y τ h k takes only finitely many values. Let v ∈ I \ (l h , r h ) be a possible value of Y τ h k such that v -a h (v) = l or v + a h (v) = r. Consider the event A = {Y τ h k = v}. Observe that c := E y [σ h k+1 -τ h k |A] ≤ h.
We extend σ h k+1 on the event A by setting

τ h k+1 = σ h k+1 + 2 (h -c) 1 {l,r} (Y σ h k+1 ) (23)
(notice that P y (Y σ h k+1 ∈ {l, r}|A) = 1 2 ). This implies that E y [τ h k+1 -τ h k |F τ h k ] = h on the event {Y τ h k / ∈ I h }.
Moreover, the processes (Y τ h j ) j∈{0,...,k+1} and (X h,y jh ) j∈{0,...,k+1} have the same law. To sum up, we have the following. Proposition 3.4. For all h ∈ (0, h) and y ∈ I • the sequence of stopping times

(τ h k ) k∈N 0 satisfies 1. Law Py Y τ h k ; k ∈ N 0 = Law P X h,y kh ; k ∈ N 0 .
2. For all k ∈ N 0 we have

E y (τ h k+1 -τ h k |F τ h k ) = 1 2 (Y τ h k -a h (Y τ h k ),Y τ h k +a h (Y τ h k )) (a h (Y τ h k ) -|u -Y τ h k |) m(du), if Y τ h k ∈ I h , h, if Y τ h k / ∈ I h .

Higher moment estimates for exit times

In this section we provide some moment estimates for the exit times of Y from intervals. We use the estimates in the next section to prove convergence in probability of sup k∈{1,..., T /h } τ h k -kh to zero, where (τ h k ) k∈N 0 is the sequence of embedding stopping times from Section 3. This is a crucial ingredient in the prove our main result, Theorem 1.1.

We need the following lemma that identifies a local martingale associated to Y .

Lemma 4.1. Let y ∈ I • . Then the process q(y, Y t ) -(t ∧ H l,r ), t ∈ [0, ∞), is a P y -local martingale.
Proof. Let a, b ∈ I with a < y < b. We first show that q(y, Y t∧H a,b ) -t ∧ H a,b , t ∈ [0, ∞), is a P y -martingale. For this purpose observe that for all t ∈ [0, ∞) it holds

E y q(y, Y H a,b ) -H a,b |F t = (q(y, Y H a,b )-H a,b )1 {H a,b ≤t} +E y q(y, Y H a,b ) -H a,b |F t 1 {H a,b >t} . ( 24
)
On the event {H a,b > t} we have

q(y, Y H a,b ) -H a,b = q(y, Y H a,b ) • θ t -H a,b • θ t -t
, where θ t denotes the shift operator for Y (see Chapter III in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]). The Markov property and [START_REF] Skorokhod | Studies in the theory of random processes[END_REF] imply that on the event {H a,b > t} we have P y -a.s.

E y q(y, Y H a,b ) -H a,b |F t =E z [q(y, Y H a,b ) -H a,b ] z=Yt -t =E z [q(y, Y H a,b ) -q(z, Y H a,b )] z=Yt -t. (25)
Formula ( 20) yields for all 25) implies that on the event {H a,b > t} we have P y -a.s.

z ∈ I • q(y, Y H a,b ) -q(z, Y H a,b ) = q(y, z) + ∂ 0 q ∂x (y, z)(Y H a,b -z). Since E z [Y H a,b -z] = 0 for all z ∈ I • , equation (
E y q(y, Y H a,b ) -H a,b |F t = q(y, Y t ) -t.
Together with (24) this yields for all t ∈ [0, ∞)

E y q(y, Y H a,b ) -H a,b |F t = q(y, Y t∧H a,b ) -t ∧ H a,b , which shows that q(y, Y t∧H a,b ) -t ∧ H a,b , t ∈ [0, ∞), is a P y -martingale.
The statement of the lemma follows via a localization argument. If l / ∈ I, then choose a decreasing sequence (l n ) n∈N ⊆ I with l 1 < y and lim n→∞ l n = l. If l ∈ I, set l n = l for all n ∈ N. Similarly, if r / ∈ I, then choose an increasing sequence (r n ) n∈N ⊆ I with r 1 > y and lim n→∞ r n = r, and if r ∈ I, then set r n = r for all n ∈ N. The sequence of stopping times inf{t ≥ 0 : X t / ∈ [l n , r n ]}, n ∈ N, is then a localizing sequence for the process q(y, Y t ) -(t ∧ H l,r ), t ∈ [0, ∞).

The next result provides conditions guaranteeing that moments of a stopping time τ can be bounded against an integral with respect to the distribution of Y τ . Theorem 4.2. Let α ∈ [1, ∞) and let y ∈ I. Let τ be a stopping time such that τ ≤ H l,r (Y ), P y -a.s. and the process (q α (y, Y τ ∧t )) t∈[0,∞) is of class (D) under P y . Then τ < ∞, P y -a.s. and it holds that

E y [τ α ] ≤ α α E y [q α (y, Y τ )]. (26)
Proof. If y ∈ I \ I • , then τ = 0 and (26) is satisfied. For the remainder of the proof we assume that y ∈ I • . We first show by contradiction that τ < ∞ P y -a.s. So assume that P y (τ = ∞) > 0. Since on {τ = ∞} we necessarily have τ = H l,r (Y ), we obtain

lim sup t→∞ q α (y, Y t∧τ ) = ∞ P y -a.s. on {τ = ∞}. For any n ∈ N let σ n = inf{t ∈ [0, ∞) : q α (y, Y τ ∧t ) ≥ n}.
Then we obtain that lim n→∞ q α (y, Y σn∧τ ) = ∞ P y -a.s. on {τ = ∞}, which contradicts the uniform integrability of {q α (y, Y σn∧τ )} n∈N . Thus, we proved that τ < ∞ P y -a.s. and, in particular, that Y τ on the right-hand side of (26) is well-defined.

According to Lemma 4.1 the process N t := q(y, Y t ) -(t ∧ H l,r (Y )), t ≥ 0, is a P y -local martingale. The product formula yields for all t ∈ [0, H l,r (Y )]

t α-1 q(y, Y t ) = t α-1 N t + t α = (α -1) t 0 s α-2 N s ds + t 0 s α-1 dN s + t α = (α -1) t 0 s α-2 q(y, Y s )ds + t 0 s α-1 dN s + 1 α t α ≥ t 0 s α-1 dN s + 1 α t α . ( 27
)
Note that ( t 0 s α-1 dN s ) t≥0 is a local martingale and let (τ n ) n∈N be a localizing sequence for it. Set τ n := n ∧ τ n for all n ∈ N. In particular, it holds E y [τ α n ] < ∞ for all n ∈ N. With inequality (27) and Hölder's inequality we obtain for all n ∈ N that

E y [(τ n ∧ τ ) α ] ≤ αE y [(τ n ∧ τ ) α-1 q(y, Y τn∧τ )] ≤ α(E y [(τ n ∧ τ ) α ]) α-1 α (E y [q α (y, Y τn∧τ )]) 1 α .

This implies for all

n ∈ N E y [(τ n ∧ τ ) α ] ≤ α α E y [q α (y, Y τn∧τ )]. (28)
By monotone convergence the left-hand side converges to E y [τ α ] as n → ∞. Since the process (q α (y, Y τ ∧t )) t∈[0,∞) is of class (D), it follows that the family (q α (y, Y τn∧τ )) n∈N is uniformly integrable. Vitali's convergence theorem implies that E y [q α (y, Y τn∧τ )] → E y [q α (y, Y τ )] as n → ∞. Therefore we obtain

E y [τ α ] ≤ α α E y [q α (y, Y τ )]
which is precisely (29). Then it holds that

E y [(H y-a,y+a (Y )) α ] ≤ α α 2 (q α (y, y -a) + q α (y, y + a)) . (29)
Proof. Clearly, H y-a,y+a (Y ) ≤ H l,r (Y ), P y -a.s. Moreover, under P y , the process (q α (y, Y H y-a,y+a (Y )∧t )) t∈[0,∞) is bounded (see [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF] and ( 19)) and hence of class (D). Inequality (29) then follows from Theorem 4.2.

Proof of Theorem 1.1

In this section we prove Theorem 1.1 and show that under Condition (A) the processes (X h ) h∈(0,h) converge in distribution to Y . We use the embedding stopping times (τ h k ) k∈N 0 constructed in Section 3 and control the temporal errors |τ h k -kh|, h ∈ (0, h), k ∈ N 0 . To this end, for every h ∈ (0, h) we apply the Doob decomposition to the process (τ h k -kh) k∈N 0 and write τ h k -kh = M h k + A h k , k ∈ N 0 , for a martingale M h and a predictable process A h . Condition (A) guarantees that A h converges to 0 as h → 0 (see Proposition 5.3 below). We show that also the martingale part M h can be nicely controlled.

For all h ∈ (0, h) let (τ h k ) k∈N be the sequence of embedding stopping times defined in Section 3. Then we have the following result about the time lags

ρ h k = τ h k -τ h k-1 , h ∈ (0, h), k ∈ N, between consecutive embedding stopping times. Lemma 5.1. Let α ∈ [1, ∞), h ∈ (0, h) and y ∈ I • . Then it holds that sup k∈N ρ h k L α (Py) ≤ 2h + 2 1-1/α α sup z∈I E z [H z-a h (z),z+a h (z) (Y )] . (30)
Proof. By construction of the sequence (τ h k ) k∈N 0 (cf. the end of Section 3, in particular, Equation ( 23)) it holds for all k ∈ N that

ρ h k = τ h k -τ h k-1 ≤ inf t ≥ τ h k-1 : |Y t -Y τ h k-1 | = a h (Y τ h k-1 ) + 2h.
This and the strong Markov property of Y imply for all k ∈ N that

E y τ h k -τ h k-1 α ≤ E y E z H z-a h (z),z+a h (z) (Y ) + 2h α z=Y τ h k-1 ≤ sup z∈I E z H z-a h (z),z+a h (z) (Y ) + 2h α . (31)
Notice that

a α + b α ≤ (a + b) α for a, b ∈ [0, ∞), (32)
because the function x → x α , x ∈ [0, ∞), is convex, increasing and starts in zero. It follows from the triangle inequality, Corollary 4.4 and (32) that for all z ∈ I it holds

H z-a h (z),z+a h (z) (Y ) + 2h L α (Pz) ≤ E z [H z-a h (z),z+a h (z) (Y ) α ] 1 α + 2h ≤ 2 -1/α α (q α (z, z -a h (z)) + q α (z, z + a h (z))) 1 α + 2h ≤ 2 -1/α α (q(z, z -a h (z)) + q(z, z + a h (z))) + 2h = 2 1-1/α αE z [H z-a h (z),z+a h (z) (Y )] + 2h
Combining this with (31) completes the proof.

Below, for a random variable ξ, it is convenient to use the notation

ξ L α (Py) = (E y |ξ| α ) 1/α
for all α ∈ (0, ∞), even though it is not a norm for α ∈ (0, 1). Notice that ξ L α (Py) ≤ ξ L β (Py) for 0 < α < β by the Jensen inequality.

Proposition 5.2. Let α ∈ (0, ∞). Then there exists a constant C(α) ∈ (0, ∞) such that, for all T ∈ (0, ∞), y ∈ I • and h ∈ (0, h), it holds that

sup k∈{1,..., T /h } τ h k - k n=1 E[ρ h n |F τ h n-1 ] L α (Py) ≤ C(α) √ T √ h + sup z∈I E z [H z-a h (z),z+a h (z) (Y )] √ h . (33)
Proof. Without loss of generality we consider α ∈ [2, ∞). Throughout the proof we fix T ∈ (0, ∞), y ∈ I • , h ∈ (0, h) and let N = T /h . For all k ∈ {0, . . . , N } we define

G k = F τ h k and M k = τ h k -k n=1 E[ρ h n |G n-1 ]. Notice that (M k ) k∈{0,...,N } is a (G k ) k∈{0,.
..,N }martingale. The Burkholder-Davis-Gundy inequality ensures that there exists a constant C(α) ∈ (0, ∞) (only depending on α) such that

E y sup k∈{1,...,N } τ h k - k n=1 E[ρ h n |G n-1 ] α ≤ C(α)E y   N k=1 (M k -M k-1 ) 2 α 2   = C(α)E y   N k=1 ρ h k -E[ρ h k |G k-1 ] 2 α 2   .
This, together with Jensen's inequality, proves that

E y sup k∈{1,...,N } τ h k - k n=1 E[ρ h k |G k-1 ] α ≤ C(α)N α 2 -1 N k=1 E y ρ h k -E[ρ h k |G k-1 ] α ≤ C(α)2 α N α 2 -1 N k=1 E y ρ h k α ≤ C(α)2 α N α 2 sup k∈N E y ρ h k α ≤ C(α)2 α T α 2 sup k∈N ρ h k L α (Py) √ h α . ( 34
)
Then Lemma 5.1 proves (33).

Proposition 5.3. Let α ∈ (0, ∞). Then, for all T ∈ (0, ∞), y ∈ I • and h ∈ (0, h), it holds that

sup k∈{1,..., T /h } k n=1 E[ρ h n |F τ h n-1 ] -kh L α (Py) ≤ T h sup z∈I h E z [H z-a h (z),z+a h (z) (Y )] -h . ( 35
)
Proof. Without loss of generality we consider α ∈ [1, ∞). Throughout the proof we fix T ∈ (0, ∞), y ∈ I • , h ∈ (0, h) and let N = T /h . For all k ∈ {0, . . . , N } we define

G k = F τ h k . The triangle inequality ensures that sup k∈{1,...,N } k n=1 E[ρ h n |G n-1 ] -kh L α (Py) = sup k∈{1,...,N } k n=1 E[ρ h n |G n-1 ] -h L α (Py) ≤ N n=1 (E[ρ h n |G n-1 ] -h) L α (Py) . (36)
By Proposition 3.4, on the event

{Y τ h n-1 ∈ I h } we have E y [ρ h n |G n-1 ] -h = 1 2 (Y τ h n-1 -a h (Y τ h n-1 ),Y τ h n-1 +a h (Y τ h n-1
))

(a h (Y τ h n-1 ) -|u -Y τ h n-1 |) m(du) -h ≤ sup z∈I h 1 2 (z-a h (z),z+a h (z)) (a h (z) -|u -z|) m(du) -h = sup z∈I h E z [H z-a h (z),z+a h (z) (Y )] -h . (37)
On the event

{Y τ h n-1 / ∈ I h } we have |E y [ρ h n |G n-1 ] -h| = 0. Therefore, sup k∈{1,...,N } k n=1 E[ρ h n |G n-1 ] -kh L α (Py) ≤ N sup z∈I h E z [H z-a h (z),z+a h (z) (Y )] -h ≤ T h sup z∈I h E z [H z-a h (z),z+a h (z) (Y )] -h . (38)
This proves (35).

By combining the two preceding theorems we obtain a result about uniform in k convergence of the embedding stopping times (τ h k ) in spaces L α (P y ), as h → 0. To this end, we impose a slightly stronger condition than Condition (A), namely,

sup y∈I h 1 2 (y-a h (y),y+a h (y)) (a h (y) -|u -y|) m(du) -h ∈ o(h), h → 0. ( 39
)
Corollary 5.4. Assume (39). Let α ∈ (0, ∞), T ∈ (0, ∞) and y ∈ I • . Then it holds

sup k∈{1,..., T /h } τ h k -kh L α (Py) ----→ 0, h → 0.
Proof. The proof is an application of Propositions 5.2 and 5.3. The fact that the righthand side of (35) converges to zero as h → 0 is a direct consequence of Corollary 3.3 and (39). Furthermore, Corollary 3.3 and (39) imply

sup z∈I h E z [H z-a h (z),z+a h (z) (Y )] √ h → 0, h → 0.
The remaining property

sup z∈I\I h E z [H z-a h (z),z+a h (z) (Y )] √ h → 0, h → 0 (cf. ( 33
)), follows from the fact that, by the definition of I h , we have

E z [H z-a h (z),z+a h (z) (Y )] ≤ h, z ∈ I \ I h .
This concludes the proof.

Proof of Theorem 1.1. For any h ∈ (0, h), we define the continuous-time process

Y h = (Y h t ) t∈[0,∞) by linear interpolation of (Y τ h k ) k∈N 0 . More precisely, we set Y h t = Y τ h t/h + (t/h -t/h ) Y τ h t/h +1 -Y τ h t/h , t ∈ [0, ∞).
Notice that Y h kh = Y τ h k , k ∈ N 0 , and Proposition 3.4 easily extends to

Law Py Y h t ; t ∈ [0, ∞) = Law P X h,y t ; t ∈ [0, ∞) .
Therefore, in order to prove Theorem 1.1, it is sufficient to show that the processes ) in probability P y uniformly on compact intervals, i.e., that, for all T ∈ (0, ∞), it holds

Y h = (Y h t ) t∈[0,∞) converge to the process Y = (Y t ) t∈[0,∞
Y h -Y C[0,T ] Py -→ 0, h → 0, where • C[0,T ] denotes the sup norm in C([0, T ], R).
In what follows, we use the notation

Y h ucp(Py) ----→ Y, h → 0, (40)
for this mode of convergence.

1. In the first step, we prove (40) under assumption (39), which is stronger than Condition (A). To this end, fix T ∈ (0, ∞) and ε > 0. Take an arbitrary T ∈ (0, ∞), T > T , and choose δ ∈ 0, T -T 2 such that P y (A(δ)) > 1 -ε 2 , where

A(δ) = {|Y t -Y s | < ε 2 for all s, t ∈ [0, T ] such that |t -s| < 3δ}. Corollary 5.4 implies sup k∈{1,..., T /h } τ h k -kh Py -→ 0, h → 0,
hence, there exists γ ∈ (0, h) such that P y (C(h, δ)) > 1 -ε 2 whenever h ∈ (0, γ), where

C(h, δ) = sup k∈{1,..., T /h } τ h k -kh < δ . A somewhat tedious check shows that, if h ∈ (0, δ), Y h -Y C[0,T ] < ε on A(δ) ∩ C(h, δ).
Thus, we get

P y ( Y h -Y C[0,T ] > ε) < ε whenever h ∈ (0, γ ∧ δ).
This completes the proof of the first step.

2. We now prove (40) under Condition (A). Consider strictly monotone sequences {l n } n∈N and {r n } n∈N with l n l and r n r. We define compact subintervals K n of I • by setting K n = [l n , r n ], n ∈ N, and modified scale factors a n h : I → [0, ∞) by setting

a n h (y) = a h (y), y ∈ K n , a h (y), y ∈ I \ K n , n ∈ N, h ∈ (0, h),
where the scale factors a h , h ∈ (0, h), are the ones from the EMCEL algorithm (recall ( 9)). Let ( τ n,h k ) k∈N 0 be the associated sequences of the embedding stopping times and Y n,h = ( Y n,h t ) t∈[0,∞) the analogues of the process Y h = (Y h t ) t∈[0,∞) for the modified scale factors a n h , n ∈ N, h ∈ (0, h). Since the scale factors (a h ) h∈(0,h) satisfy Condition (A), the modified scale factors ( a n h ) h∈(0,h) satisfy (39) for each n ∈ N. By the first step of the proof,

Y n,h ucp(Py) ----→ Y, h → 0, (41)
for any fixed n ∈ N. Fix T ∈ (0, ∞) and ε > 0. For any n ∈ N, we define the events

A n = {H l,r (Y ) ≤ T + 2 and ∃ t ∈ [H ln,rn (Y ), H l,r (Y )] such that |Y t -Y H l,r (Y ) | > ε}, B n = {H l,r (Y ) > T + 2 and H ln,rn (Y ) ≤ T + 1}.
Notice that the expression Y H l,r (Y ) in the above formula for A n is well-defined and finite. Indeed, this is the position of Y at an accessible boundary (because H l,r (Y ) ≤ T + 2 < ∞), while an infinite boundary cannot be accessible (because Y is in natural scale). As H ln,rn (Y ) H l,r (Y ) P y -a.s., as n → ∞, and Y is continuous, we can choose a sufficiently big n 0 ∈ N such that P y (A n 0 ) < ε 3 and P y (B n 0 ) < ε 3 . We also take an arbitrary T ∈ (T, T + 1). Corollary 5.4 applied to the modified scale factors ( a n 0 h ) h∈(0,h) , which satisfy (39), yields that there exists γ > 0 such that, for any h ∈ (0, γ), we have

P y τ n 0 ,h T /h ≤ T + 1 > 1 - ε 3 .
For h ∈ (0, h), we define the event

C h = τ n 0 ,h T /h ≤ T + 1 ∩ (A n 0 ∪ B n 0 ) c
(the notation D c means the complement of an event D). Notice that P y (C h ) > 1 -ε whenever h ∈ (0, γ). Furthermore, on C h we have either

H l,r (Y ) > T + 2, H ln 0 ,rn 0 (Y ) > T + 1, hence Y h t = Y n 0 ,h t , t ∈ [0, T ],
or

H l,r (Y ) ≤ T + 2, |Y t -Y H l,r (Y ) | ≤ ε for t ∈ [H ln 0 ,rn 0 (Y ), H l,r (Y )], hence Y h t -Y t ≤ ε whenever Y h t = Y n 0 ,h t , t ∈ [0, T ].
Together with (41), this proves

Y h -Y C[0,T ] Py -→ 0 as h → 0. As T ∈ (0, ∞)
is arbitrary, we obtain (40). This concludes the proof.

Reflecting boundaries

Throughout the preceding sections we assume that if a boundary point is accessible, then it is absorbing. In this section we explain how one can drop this assumption, i.e. how one can extend our functional limit theorem, Theorem 1.1, to Markov processes with reflecting boundaries.

The idea is to reduce the reflecting case to the inaccessible or absorbing case. Indeed, for every Markov process Z with reflecting boundaries one can find a Markov process Y on an extended state space and a Lipschitz function f such that Y has inaccessible or absorbing boundaries and

Z d = f (Y ).
We illustrate the reduction for a Markov process Z in natural scale with state space I Z = [l, ∞), where l > -∞ is a reflecting boundary. We denote by m Z the speed measure of Z. Since l is non-absorbing, it must hold that m Z ({l}) < ∞. Notice that m Z ({l}) = 0 corresponds to instantaneous reflection, while m Z ({l}) ∈ (0, ∞) to slow reflection.

To proceed with the construction, we first remark that it holds

m Z ((l, l + 1)) < ∞. (42)
Indeed, in terms of the Feller boundary classification (see Table 15.6.2 in [START_REF] Karlin | A second course in stochastic processes[END_REF]), as the accessible boundary point l is reflecting, it can only be regular, which implies (42). Now let Y be a Markov process in natural scale with state space

I Y = R and speed measure m Y satisfying m Y (A) = m Z (A), for all A ∈ B(R), A ⊆ (l, ∞), m Y (A) = m Z (2l -A), for all A ∈ B(R), A ⊆ (-∞, l), m Y ({l}) = 2m Z ({l}),
which is a valid speed measure on I Y = R (i.e., (4) holds) due to (42). Then |Y -l| has the same distribution as Z (see Proposition VII.3.10 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]).

Let (a h ) h∈(0,h) satisfy the assumptions of Theorem 1.1. Then (X h ) h∈(0,h) converges in distribution to Y as h → 0. This implies that the processes (|X h -l|) h∈(0,h) converge in distribution to Z as h → 0.

In a similar way, a Markov process Z on a bounded interval I Z with endpoints l and r (l < r), where l ∈ I Z is reflecting and r is inaccessible (resp., absorbing), can be reduced to a Markov process Y with state space I Y , which is the interval with endpoints 2l -r and r, where both these endpoints are inaccessible (resp., absorbing).

A Markov process Z with two reflecting boundaries can be reduced to a Markov process with state space R. To explain this, suppose for simplicity that the state space of Z is [0, 1]. Define Y as the Markov process on R with speed measure m Y satisfying

m Y (A) = m Y (-A) = m Z (A), for all A ∈ B (0, 1) , m Y (A + 2k) = m Z (A), for all A ∈ B (-1, 0) ∪ (0, 1) and k ∈ Z, m Y ({2k}) = 2m Z ({0}), for all k ∈ Z, m Y ({2k + 1}) = 2m Z ({1}), for all k ∈ Z.
Let f : R → [0, 1] be the periodic function with period 2 satisfying f

(x) = |x|, x ∈ [-1, 1].
Then the process f (Y ) has the same distribution as Z (cf. Proposition VII.3.10 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]).

Examples with sticky points

In this section we apply Theorem 1.1 to sticky Brownian motions on R and on [0, ∞), where the sticky point is zero. Note that recent years have witnessed a renewed interest in the sticky Brownian motion and related processes, see [START_REF] Karatzas | On the one-sided Tanaka equation with drift[END_REF], [START_REF] Bass | A stochastic differential equation with a sticky point[END_REF], [START_REF] Engelbert | Stochastic differential equations for sticky Brownian motion[END_REF], [START_REF] Hajri | Application of stochastic flows to the sticky Brownian motion equation[END_REF] and references therein.

The literature on approximations of diffusions with atoms in the speed measure is scarce. We remark that [START_REF] Amir | Sticky Brownian motion as the strong limit of a sequence of random walks[END_REF] provides a sequence of random walks that converges in distribution to the Brownian motion on R sticky at zero. The random walks considered there are forced to stay in zero for some time whenever they visit zero. In contrast to our approach, the approximating processes are not Markov chains. [START_REF] Fushiya | Weak convergence theorem of a nonnegative random walk to sticky reflected brownian motion[END_REF] constructs Markov chains that converge in distribution to the Brownian motion on [0, ∞) with slow reflection at 0. The approximating Markov chains considered there exhibit sticky behavior in zero in the sense that once the Markov chains reach zero they stay there with positive probability also in the next time period.

Brownian motion on R with sticky point 0

Brownian motion on R sticky at 0 is a Markov process Y in natural scale with state space I = R and speed measure

m(dx) = 2 σ 2 λ(dx) + 2 θ δ 0 (dx), (43)
where σ, θ ∈ (0, ∞) and λ(dx) denotes the Lebesgue measure. Such a process Y behaves like σ times a Brownian motion outside zero, but spends a positive amount of time at zero having no intervals of zeros. Notice that the bigger θ is, the less time Y spends at zero; θ = ∞ corresponds to a standard Brownian motion (times σ).

It is instructive to compute the function q(y, x), y, x ∈ R, of ( 17)

q(y, x) =        (x-y) 2 σ 2 + 2 x - θ if y > 0, x 2 σ 2 + |x| θ if y = 0, (x-y) 2 σ 2 + 2 x + θ if y < 0 (x + = max{x, 0}, x -= -min{x, 0}
) and to observe that, for any y ∈ R, the function q(y, •) has a kink at zero. We now determine, for every h ∈ (0, ∞), a function a h : R → (0, ∞) such that the associated Markov chain ( X h hk ) k∈N 0 , defined in ( 5), belongs to EMCEL(h). Indeed, one can explicitly determine for all y ∈ R the real number a h (y) satisfying

1 2 (y-a h (y),y+ a h (y)) ( a h (y) -|u -y|) m(du) = h. (44)
We state the closed-form representations of a h in the next Lemma.

Lemma 7.1. For all h ∈ (0, ∞) and y ∈ R let

a h (y) =      σ √ h if |y| ≥ σ √ h, σ h + |y| θ + σ 2θ 2 -σ 2θ if |y| < σ √ h.
Then Equation (44) is satisfied for all h ∈ (0, ∞) and y ∈ R.

Proof. Throughout the proof fix h ∈ (0, ∞) and y ∈ R. For every a ∈ [0, ∞) it holds that

1 2 (y-a,y+a) (a -|u -y|) m(du) = a 2 σ 2 + a -|y| θ 1 (y-a,y+a) (0) = a 2 σ 2 + a -|y| θ 1 [0,a) (|y|
F (x) = 1 M M i=1 1 (-∞,x] (X h,i 1 )
Figure 1: Left: Two trajectories of the approximation of the sticky Brownian motion. The black line depicts one realization of ( X h t ) t∈[0,1] with h = 10 -3 , σ = 1 and θ = 0.5. The gray line shows one realization of ( X h t ) t∈[0,1] with h = 10 -3 , σ = 1 and θ = 1. Both trajectories are generated by the same sample of random increments (ξ k ) k∈N . Observe that the smaller the value of θ is, the more the process sticks to 0. Right: Empirical distribution function of the approximation of the sticky Brownian motion. The figure depicts the function F : R → [0, 1],

F (x) = 1 M M i=1 1 (-∞,x] ( X h,i 1 )
, where ( X h,i 1 ) i∈{1,...,M } are M = 10 6 independent realizations of X h 1 with h = 10 -3 , σ = 1 and θ = 1. Observe that a jump at 0 becomes apparent. This reflects the fact that the (weak) limit Y 1 of X h 1 is with positive probability equal to 0.

7.2 Brownian motion on [0, ∞) with slow reflection at 0

In this section we consider a Brownian motion on [0, ∞) with slow reflection at 0. Let σ, θ ∈ (0, ∞). According to Theorem IV.7.2 in [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF] the stochastic differential equation

dZ t = θ1 {Zt=0} dt + σ1 {Zt>0} dW t , Z 0 = 0, (46)
possesses a weak solution that is unique in law. However, it is worth noting that neither existence of a strong solution nor pathwise uniqueness hold for (46) (see [START_REF] Engelbert | Stochastic differential equations for sticky Brownian motion[END_REF] and references therein). The next result shows that Z is a regular diffusion on [0, ∞) and identifies the associated speed measure.

Lemma 7.2. The solution Z of (46) is a regular continuous strong Markov process in natural scale with state space I Z = [0, ∞) and with speed measure

m Z (dz) = 2 σ 2 λ(dz) + 1 θ δ 0 (dz). (47)
Proof. Strong Markov property of Z is implied by the uniqueness in law for (46). Clearly, Z is regular with state space I Z = [0, ∞) and in natural scale. By Itô's formula, we have

f (Z t ) = f (Z 0 ) + t 0 θf (0)1 {Zs=0} + σ 2 2 f (Z s )1 {Zs>0} ds + t 0 σf (Z s )1 {Zs>0} dW s for C 2 functions f : [0, ∞) → R.
Therefore, the generator A of Z takes the form

Af (z) = θf (0) if z = 0, σ 2 2 f (z) if z > 0 (48)
for f ∈ C 2 0 ([0, ∞)) (this means that the function itself and its first and second derivative vanish at infinity) satisfying the boundary condition

θf (0) = σ 2 2 f (0).
By Theorem VII.3.12 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF], we have Af (z) = d dm Z f (z) in the interior of the state space, i.e., for z > 0, while, by Proposition VII.3.13 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF], it holds f (0) = m({0})Af (0) on the boundary. Together with (48), this implies (47) and concludes the proof. 43)). In particular, Z can be approximated by (| X h |) h∈(0,∞) , where each X h is the EMCEL(h) constructed in Section 7.1.

Warren [START_REF] Warren | Branching processes, the Ray-Knight theorem, and sticky Brownian motion[END_REF] determines for all t ∈ (0, ∞) the conditional law of Z t given the driving Brownian motion W . As a consequence, we obtain for all t ∈ (0, ∞) closed form representations of the cumulative distribution function and the expected value of Z t . The precise formulas are provided in Lemma 7.3 below, where we, without loss of generality, consider σ = 1. The notations P 0 for the probability measure and E 0 for the corresponding expectation operator emphasize that the formulas are given for the case Z 0 = 0. We use these formulas to analyze the empirical rate of convergence of EMCEL approximations. The results are presented in Figure 2. Lemma 7.3. Let Z be a solution of (46) with σ = 1. For every t ∈ (0, ∞) the cumulative distribution function

F (•; t) : [0, ∞) → [0, 1] of Z t satisfies F Z (z; t) := P 0 [Z t ≤ z] = 2Φ z √ t -1 + 2e 2θ(z+θt) Φ -2θ √ t - z √ t , z ∈ [0, ∞), (
) where Φ(x) = x -∞ 1 √ 49
2π e -y 2 /2 dy is the cumulative distribution function of the standard normal distribution. Moreover, it holds that

E 0 [Z t ] = 2t π - 1 2θ + e 2θ 2 t θ Φ(-2θ √ t) = E 0 [|W t |] - 1 2θ + e 2θ 2 t θ Φ(-2θ √ t). (50)
-log2(h) ). Then it follows from Theorem 1 in [START_REF] Warren | Branching processes, the Ray-Knight theorem, and sticky Brownian motion[END_REF] that for all z ∈ [0, ∞) it holds

-log 2 (h), log 2 1 M M i=1 1 [0,0.1] (| X h,i 1 |) -F Z (0.1; 1) , h = 2 -6 , . . . ,
(h), log 2 1 M M i=1 | X h,i 1 | -E 0 [Z 1 ] , h = 2
P 0 [Z t ≤ z] = E 0 e -2θ(Wt+sup s∈[0,t] (-Ws)-z) + = E 0 1 [0,z) (|W t |) + e -2θ(|Wt|-z) 1 [z,∞) (|W t |) = 2Φ z √ t -1 + 2 √ 2π ∞ z/ √ t e -2θ( √ ty-z) e -y 2 2 dy = 2Φ z √ t -1 + 2 √ 2π e 2θ(z+θt) ∞ z/ √ t e -(y+2θ √ t) 2 2 dy = 2Φ z √ t -1 + 2e 2θ(z+θt) Φ -2θ √ t - z √ t .
This proves (49). Moreover, this implies that the density function of Z t starting in 0 satisfies

F Z (z; t) = 2 √ t Φ z √ t + 4θe 2θ(z+θt) Φ -2θ √ t - z √ t - 2 √ t e 2θ(z+θt) Φ -2θ √ t - z √ t (51) for all z ∈ (0, ∞). Observe that for all z ∈ (0, ∞) it holds e 2θ(z+θt) Φ -2θ √ t - z √ t = 1 √ 2π e 2θ(z+θt) e - 2θ √ t+ z √ t 2 2 = 1 √ 2π e -z 2 2t = Φ z √ t . (52)
This implies for all z ∈ (0, ∞) that

F Z (z; t) = 4θe 2θ(z+θt) Φ -2θ √ t - z √ t . (53)
This and Fubini's theorem prove that

E 0 [Z t ] = ∞ 0 zF Z (z; t) dz = 4θe 2θ 2 t √ 2π ∞ 0 ze 2θz -2θ √ t-z √ t -∞ e -y 2 2 dy dz = 4θe 2θ 2 t √ 2π -2θ √ t -∞ e -y 2 2 -y √ t-2θt 0 ze 2θz dz dy = e 2θ 2 t θ √ 2π -2θ √ t -∞ e -y 2 2 1 -e -2θ(y √ t+2θt) (2θ(y √ t + 2θt) + 1) dy = e 2θ 2 t θ Φ(-2θ √ t) - 1 θ √ 2π -2θ √ t -∞ e -(y+2θ √ t) 2 2 (2θ √ t(y + 2θ √ t) + 1) dy = e 2θ 2 t θ Φ(-2θ √ t) - 1 θ √ 2π 0 -∞ e -y 2 2 (2θ √ ty + 1) dy = e 2θ 2 t θ Φ(-2θ √ t) - 1 2θ - 2 √ t √ 2π 0 -∞ ye -y 2 2 dy = e 2θ 2 t θ Φ(-2θ √ t) - 1 2θ + 2t π . (54)
This completes the proof.

8 Brownian motion slowed down on the Cantor set

In this section we apply our results to construct a family of Markov chains (X h ) h∈(0,1) that converge in distribution to the general diffusion Y on R with speed measure m(dx) = m C (dx)+2 dx, where m C is the Cantor distribution. Such a process Y can be understood as a Brownian motion slowed down on the Cantor set.

For later reference we briefly recall a way to construct the Cantor distribution. To this end let C be the collection of all subsets of [0, 1] and let Ψ : C → C be the map given by Let (X h ) h∈(0,1) be the family of Markov chains defined in (5) and (6) (with scale factor a given by the solution of (58)). Then for all y ∈ R the distributions of (X h,y t ) t∈[0,∞) , h ∈ (0, 1), under P converge weakly to the distribution of (Y t ) t∈[0,∞) under P y , as h → 0.

Ψ(A) = A 3 ∪ A + 2 3 , A ⊆ [0, 1]. (55
Proof. First observe that for all y ∈ R the mapping

[0, ∞) a → 1 2 3 2 n(h) y+a y-a 1 C n(h) (u)(a -|u -y|)du + a 2 ∈ [0, ∞) (59)
is continuous and strictly increasing. This ensures existence of a unique solution a h (y) ∈ [0, ∞) of (58). It follows from

a 2 h (y) ≤ 1 2 3 2 n(h) y+a h (y)
y-a h (y)

1 C n(h) (u)(a h (y) -|u -y|)du + a 2 h (y) = h (60)
that a h (y) ≤ √ h for all h ∈ (0, 1), y ∈ R. Moreover, it follows that for all h ∈ (0, 1) and y ∈ R it holds

Next, observe that Lemma 3.2, definition (17) of q and the fact that m C and m n(h) do not possess atoms ensure that it holds for all h ∈ (0, 1) and y ∈ R that (64)

Hence, Condition (A) is satisfied and weak convergence of X h to Y follows from Theorem 1.1.

Proposition 8.1 provides the way to simulate approximations of the Brownian trajectories slowed down on the Cantor set (see Figure 3). ∈ R, where for all y ∈ R the real number a n(h) (y) is the solution of (58) with h = 10 -4 and n(h) = 4. The gray bars depict the 16 subintervals of the set C n(h) . We see that the black line interpolates smoothly between 1 and considerably smaller values around the gray bars. Right: A realization of an approximation of the Brownian motion slowed down on the Cantor set (CBM) and of a Brownian motion (BM). The black line depicts a trajectory of (X h t ) t∈[0,1] defined in ( 5) and ( 6) with scale factor a given by the solution of (58) with h = 10 -4 and n(h) = 4. The gray line shows a realization of (X h t ) t∈[0,1] defined in ( 5) and ( 6) with scale factor a h given by a h ≡ √ h and h = 10 -4 . Both trajectories are generated from the same sample of random increments (ξ k ) k∈N . The horizontal light gray bars show again the 16 subintervals of the set C n(h) . We see that on the set C n(h) the CBM is indeed slowed down (compared with the BM).

  space is an open, half-open or closed interval I ⊆ R. We denote by I • = (l, r) the interior of I, where -∞ ≤ l < r ≤ ∞, and we set I = [l, r]. Recall that by the definition we have P y [Y 0 = y] = 1 for all y ∈ I.

  y, u)) du + y y-a m((u, y)) du. Using Fubini's theorem we compute y+a y m((y, u)) du = y+a y (y,u) m(dz) du = (y,y+a) (y + a -z) m(dz) and y y-a m((u, y)) du = y y-a (u,y) m(dz) du = (y-a,y) (z -y + a) m(dz).

Remark 4 . 3 .Corollary 4 . 4 .

 4344 Under the assumption of Theorem 4.2 we have equality in (26) for α = 1, i.e., E y [τ ] = E y [q(y, Y τ )]. To obtain the reverse inequality use that equality holds in (27) for α = 1, localization and Fatou's lemma. From Theorem 4.2 we obtain the following moment estimate for first exit times. Let α ∈ [1, ∞), let y ∈ I and let a ∈ [0, ∞) be such that [y-a, y+a] ⊆ I.

  It follows from Section 6 that Z d = |Y |, where Y is a diffusion in natural scale with state space I Y = R and speed measure m Y (dz) = 2 σ 2 λ(dz)+ 2 θ δ 0 (dz), i.e., Y is the process studied in Section 7.1 (cf. (

Figure 2 :

 2 Figure 2: The parameter values are σ = 1 and θ = 1/2 for both plots. Left: Empirical rate of convergence of the distribution function. The circles depict the five data points -log 2 (h), log 2

Proposition 8 . 1 . 1 C

 811 )Next, we define recursively a sequence(C n ) n∈N 0 of subsets of [0, 1]. Let C 0 = [0, 1] and for n ∈ N let C n = Ψ(C n-1 ). (56)The Cantor set is defined asC = ∩ n∈N C n .We define for all n ∈ N the probability measure m n on (R, B(R)) by m n (dx) = converges in distribution to a probability measure m C on (R, B(R)) and that for all n ∈ N it holds supx∈R |m C ((-∞, x]) -m n ((-∞, x])| ≤ 2 -(n-1) .(57)Moreover, it holds that m C (C) = 1 (in particular, m C is concentrated on [0, 1]), µ L (C) = 0 and, for all x ∈ R, m C ({x}) = 0, i.e., m C is a singular-continuous measure. Let m be the measure on R given by m(dx) = m C (dx)+2 dx and let Y be the associated diffusion. Let n : (0, 1) → N be a function satisfying lim h→0 2 n(h) √ h = ∞. Then there exists for all h ∈ (0, 1), y ∈ R a unique solution a h (y) ∈ (0, n(h) (u)(a h (y) -|u -y|)du + a 2 h (y) = h.(58)

1 2 (

 2 y-a h (y),y+a h (y)) (a h (y) -|u -y|) (m n(h) + 2)(du) y+a h (y) y-a h (y) 1 C n(h) (u)(a h (y) -|u -y|) du + a 2 h (y) = h.

  (y-a h (y),y+a h (y))(a h (y) -|u -y|) (m C -m n(h) )(du) = y+a h (y) y m C ((y, u)) -m n(h) ((y, u)) du + y y-a h (y) m C ((u, y)) -m n(h) ((u, y)) du = y+a h (y) y m C ((-∞, u]) -m n(h) ((-∞, u]) du -y y-a h (y) m C ((-∞, u]) -m n(h) ((-∞, u]) du.

1 2sup y∈R 1 2sup y∈R 1 2

 111 57) and the fact that a h (y) ≤ √ h show that, for all h ∈ (0, 1) and y ∈ R, we have1 2 (y-a h (y),y+a h (y)) (a h (y) -|u -y|) (m C -m n(h) )(du) ≤ a h (y) sup u∈R m C ((-∞, u]) -m n(h) ((-∞, u]) ≤ √ h2 -(n(h)-1) .(63)Combining (61) and (63) and using the assumptionlim h→0 2 n(h) √ h = ∞ shows that sup y∈R (y-a h (y),y+a h (y)) (a h (y) -|u -y|) m(du) -h ≤ (y-a h (y),y+a h (y)) (a h (y) -|u -y|) (m C -m n(h) )(du) + (y-a h (y),y+a h (y)) (a h (y) -|u -y|) (m n(h) + 2)(du) -h = sup y∈R 1 2 (y-a h (y),y+a h (y)) (a h (y) -|u -y|) (m C -m n(h) )(du) ∈ o(h).

Figure 3 :

 3 Figure 3: Left: Normalized scale factor for the approximation of the Brownian motion slowed down on the Cantor set. The black line depicts the function R y → a n(h) (y) √ h

  If there is no ambiguity, we simply write H x in place of H x (Y ). Moreover, for a < b in I we denote by H a,b = H a,b (Y ) the first exit time of Y from (a, b), i.e. H a,b = H a ∧ H b . Without loss of generality we suppose that the diffusion Y is in natural scale. If Y is not in natural scale, then there exists a strictly increasing continuous function s : I → R, the so-called scale function, such that s(Y t ), t ≥ 0, is in natural scale. Let m be the speed measure of the Markov process Y (see VII.3.7 and VII.3.10 in

  ).Since a h satisfies Equation (44) exactly, it follows that Condition (A) is satisfied. Theorem 1.1 implies that the processes (X h ) converge in distribution to Y as h → 0. Figure1depicts two realizations of a Brownian motion on R sticky at 0 with σ = 1 and different values for θ as well as the empirical distribution function of X h 1 with h = 10 -3 .

	0.6											1
										θ=0.5
	0.4									θ=1		0.9
												0.8
	0.2
												0.7
	0
												0.6
	-0.2
												0.5
	-0.4
												0.4
	-0.6
												0.3
	-0.8
												0.2
	-1											0.1
	-1.2											0
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	-5	-4	-3	-2	-1	0	1	2	3	4	5
						Time
	Assume first that |y| ≥ σ Next assume that |y| < σ √ √ h. Then it holds that |y| ≥ a h (y) and hence (44) is satisfied. h. In this case it holds that a h (y) > |y|. Moreover it holds
	that This proves (44) in the case |y| < σ a h (y) 2 σ 2 + √ h. The proof is thus completed. a h (y) -|y| = h. θ				(45)

  2-10 , where F Z (0.1; 1) = P 0 [Z 1 ≤ 0.1] ≈ 0.5741 (see Lemma 7.3) and ( X h,i 1 ) i∈{1,...,M } are M = 10 8 independent realizations of X h 1 . The straight line is the linear best fit. Its slope is approximately -0.62. Right: Empirical rate of convergence of the expected value. The circles depict the five data points -log 2

  -6 , . . . , 2-10 , whereE 0 [Z 1 ] ≈ 0.3210 (see Lemma 7.3) and ( X h,i 1 ) i∈{1,...,M } are M = 10 8 independent realizations of X h 1 .The straight line is the linear best fit. Its slope is approximately -0.59.

Proof. Fix t ∈ (0, ∞) throughout the proof. Lévy's distributional theorem implies

W t + sup s∈[0,t] (-W s ) d = |W t | (see Theorem VI.2.3 in

[START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]

2 n 1 Cn (x) dx. Note that m n is absolutely continuous with respect to the Lebesgue measure µ L . It follows from the proof of Theorem 3.1 in[START_REF] Schmidt | The cantor set in probability theory[END_REF] that the sequence (m n ) n∈N