Philippe Négrel

Romain Millot

Emmanuelle Petelet-Giraud

Catherine Guerrot

Pb isotope systematics in volcanic river system: Constraints about weathering processes

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Pb isotope systematics in volcanic river system: Constraints about weathering processes

Ph. NEGREL, R. MILLOT, E. PETELET-GIRAUD, C. GUERROT BRGM, Laboratory Division, Orléans, France p.negrel@brgm.fr, r.millot@brgm.fr, e.petelet@brgm.fr,

We present a series of lead isotopes in soils and sediments developed on volcanic rocks forming a small watershed flowing through the Massif Central (France). The Massif Central volcanic province is a widespread area of Tertiary to Recent continental alkaline volcanism comprising alkali basalts and basanites. The Allanche watershed has an area of 160 km2, a maximum altitude in the watershed of 1400 m (a.s.l.) and the relief between the extreme sampling points of 340 m The river is 29 km long from headwaters to the outlet and from its origin in the Cézallier area to its mouth in the Allagnon river (a tributary of the Allier river), the Allanche river flows through the volcanic terrains of the lava plateau (11 to 2.5 Ma Using Pb isotope ratios, major and trace elements (from Négrel and Deschamps, 1996, Aquatic Geochemistry, 2, 1-27) we therefore compare sediments and soils evolution over the Allanche river watershed. K and Ca are considered as mobile reference elements and illustrate the weathering state of soils and sediments relative to parent rocks through a large decrease in K and Ca content when compared to Si; the sediments being less depleted than soils. Lead, with regards to Si shows three behaviour with depleted Si content-same lead content that bedrock, depleted Si content-less lead content and depleted Si content -high lead content that bedrock. The comparison of 1000Pb/K versus Si/K ratio evidenced the evolution line from weathering processes and the lead enrichment from atmospheric deposition as a major contributor to explain the deviation of several points from this line.

Lead isotopes decrease from bedrock to sediments-soils without any clear relationship when compared to lead contents. The use of Pb-isotopic compositions showed that most of the lead budget in sediments and soils result from bedrock weathering with an influence of gasoline additive-lead derived inputs and a lack of lead input from agricultural activities.

  ). Main bedrocks are basanites (nepheline or leucitic basalts), with SiO2 around 41-45%, low Na2O + K2O (<5%), and with modal or normative nepheline or leucite and a ground mass of clinopyroxene and plagioclase. Surrounding rocks are feldspatic basalts with SiO2 close to 46-49%, low Na2O + K2O (<5%). The main phase in these basalts is plagioclase with normative nepheline, hyperstene and olivine. Crustal contamination (e.g. by granite, gneiss or metasedimentary granulite, as stated byDownes, 1987, doi: 10.1144/GSL.SP.1987.030.01.25) has occurred in the differentiated magmas of both series, as witnessed by lead isotopic variations in conjunction with Rb/La ratios and lead contents.


Hue Sophie

Kared Hassen

Mehwish Younas

Mouhammad Shahul

Balbo Michelle

Levy Yves
email: yves.levy@hmn.aphp.fr

Pr Y Lévy

Notch activation on effector T cells increases their sensitivity to Treg cell-mediated suppression through upregulation of TGF-βRII expression

Notch proteins play an important role in embryonic development and cell-fate decisions.

Notch influences also the activation and differentiation of peripheral T cells. Here, we investigated whether Notch signalling modulates the response of effector T cells to Treg cells.

Pre-exposure of CD4 + CD25 -effector T cells to the Notch ligands Delta-4 and Jagged-1, but not Delta-1, increases significantly effector T-cell sensitivity to Treg cell-mediated suppression through upregulation of TGF-βRII expression and increased levels of the phosphorylated form of the Smad 3 protein. This effect is relieved by anti-TGF-β Abs. We demonstrate that HES, the main transcription factor downstream of Notch, induces strong transactivation of TGF-ßRII by binding the TGF-βRII promoter through its DNA-binding domain. Thus, the crosstalk between Notch and the TGF-β pathway leads to potentiation of the suppressive effect of Treg cells.

Introduction

The Notch signalling system is conserved from Drosophila to humans and regulates cell differentiation, proliferation and survival. Notch pathways play an important role in embryonic development, T-cell development and function, and in disease processes including carcinogenesis and autoimmunity. In mammals there are four Notch receptors (1 to 4) and five Notch ligands (Jagged-1, Jagged-2, Delta-like 1 (DL-1), DL-3 and DL-4) [START_REF] Yuan | Functions of notch signaling in the immune system: consensus and controversies[END_REF].

Notch proteins exert their pleiotropic effects through the regulation of expression of various downstream genes, many of which require the interaction of Notch proteins with the DNA binding transcription factor CSL in order to form a short-lived nuclear transcription complex [START_REF] Osborne | Notch signalling during peripheral T-cell activation and differentiation[END_REF]. After engagement with its ligands, successive proteolytic events cause clipping of the Notch protein. The first is mediated by ADAM proteases and the second by the γ-secretase complex, in which presenilins (PS1 and PS2) constitute the active center of the enzyme complex. These proteolytic events ultimately release the intracellular domain of Notch (NICD). The formation of a complex of activated intracellular Notch protein and CSL converts CSL from a transcriptional repressor to a transcriptional transactivator. The genes encoding the HES (hairy and enhancer of split) family of basic helix-loop-helix proteins are Notch targets that are known to be essential for T-cell development and signalling.

In addition to influencing Th1-and Th2-cell differentiation [START_REF] Amsen | The different faces of Notch in T-helper-cell differentiation[END_REF], Notch signalling has also been involved in the differentiation and expansion of regulatory T (Treg) cells [START_REF] Ostroukhova | Treg-mediated immunosuppression involves activation of the Notch-HES1 axis by membrane-bound TGF-beta[END_REF]. Several reports have shown that the presence of Notch ligands, mostly of the Jagged family, can enhance Treg-cell differentiation and function in vitro [START_REF] Vigouroux | Induction of antigen-specific regulatory T cells following overexpression of a Notch ligand by human B lymphocytes[END_REF]. For example, exposure of Treg cells to Jagged-2 expressed by hematopoietic progenitor cells has been shown to modulate peripheral Treg-cell expansion and prevents the development of diabetes in an autoimmune disease mice model [START_REF] Kared | Jagged2-expressing hematopoietic progenitors promote regulatory T cell expansion in the periphery through notch signaling[END_REF]. Transgenic mice over expressing the active intra-cellular form of Notch3 exhibit an increased percentage of Treg cells and are refractory to the induction of experimental autoimmune diabetes when treated with streptozotocin [START_REF] Anastasi | Expression of activated Notch3 in transgenic mice enhances generation of T regulatory cells and protects against experimental autoimmune diabetes[END_REF]. Moreover, there is emerging data suggesting that Notch can crosstalk or cooperate with other signalling pathways and thereby broaden the spectrum of target genes that are influenced by Notch signalling. For example, the interaction of Notch and TGF-β signal pathways plays a role in Treg-cell effector function through a modulation of FoxP3 expression or by facilitating TGFβ mediated suppressive function of Treg cells [START_REF] Samon | Notch1 and TGFbeta1 cooperatively regulate Foxp3 expression and the maintenance of peripheral regulatory T cells[END_REF][START_REF] Asano | Notch1 signaling and regulatory T cell function[END_REF]. For example, Asano et al showed that Treg cells express Notch ligands and also that Notch signaling plays an important role on effector function of TGF-β and its signaling through an interaction of NICD and Smad 3 [START_REF] Asano | Notch1 signaling and regulatory T cell function[END_REF]. Therefore, most of the reports focused on the influence of Notch signalling on Treg cells.

Here, we hypothesized that Notch signalling modulates the response of effector T cells to Treg cell-mediated suppressive effects. We show that Notch activation through its ligands increases exquisitely the sensitivity of effector CD4 + T cells to the suppressive effect of Treg cells even at low frequency. This effect is mediated through an upregulation of TGF-βRII and the phosphorylated form of Smad 3 protein on effectors T cells. We show that HES-1 transactivates the TGF-βRII promoter.

Results

Notch1 signaling is involved in Treg cell-mediated suppression

In order to investigate the involvement of Notch signalling in Treg cell-mediated suppression, we performed co-culture experiments of purified CD4 + CD25 high and autologous CD4 + CD25 - T cells stimulated with anti-CD3 mAbs in the presence or absence of a γ-secretase inhibitor (GSI). While GSI did not affect the proliferation of CD4 + CD25 -T cells cultured alone, addition of this compound at the begining of the coculture relieved the Treg cell-mediated suppression (Fig1A). We found that CD4 + CD25 -T cell proliferation was inhibited (mean +/-SD) by 79.2% +/-12.2 and 60.2 % +/-24.8 when cultured with Treg cells at a ratio (T REG /T EFF ) 1:4 and 1:8, respectively (mean of 4 experiments) (Fig 1B). In the presence of GSI, Treg-cell mediated suppression became 46.9% +/-20.5 (P<0.05 for comparison with coculture performed with GSI vehicle) and 49.3 % +/-21 at the corresponding ratios.

In a second step, the same experiments were performed in the presence of immobilized Notch ligands DL-1, DL-4 and Jagged-1 or IgG control at 5µg/ml. As shown in Fig1C, the presence of Notch ligands potentiates the inhibitory effects of Treg cells on the proliferation of CD4 + CD25 -T cells. Percentages of inhibition in cultures performed at ratio 1:8 (T REG /T EFF ) were 58, 73 and 72% in the presence of DL-1, DL-4 and Jagged-1, respectively as compared to 43% in the presence of Ig control (P<0.05 for DL-4 and Jagged-1 as compared to IgG control). At a higher ratio (1:16) these percentages are 39%, 54% and 64% as compared to 20% in control conditions (P<0.05 for DL4 and Jagged-1 as compared to IgG control).

Altogether, these results suggest that notch ligands potentate Treg cell-mediated suppression. However, they could not discriminate between direct effects of Notch ligands on Treg cells and/or effector T cells.

Notch signalling sensitizes effector T cells to Treg cell-mediated suppression.

In order to discriminate between an effect of Notch signalling on Treg cells or effector T cells, we performed coculture experiments in which either Treg or CD4 + CD25 -T cells were first preincubated or not with immobilized recombinant Notch ligands or IgG control overnight.

Notch suppresses the proliferation of CD4 + CD25 -T cells through TGF-β signalling.

Although the mechanisms of Treg-cell suppression are not fully elucidated, the TGF-β pathway plays a key role in the regulation of Treg-cell function. Moreover, previous results have shown an interaction between Notch and TGF-β pathways [START_REF] Samon | Notch1 and TGFbeta1 cooperatively regulate Foxp3 expression and the maintenance of peripheral regulatory T cells[END_REF][START_REF] Asano | Notch1 signaling and regulatory T cell function[END_REF][START_REF] Blokzijl | Cross-talk between the Notch and TGF-beta signaling pathways mediated by interaction of the Notch intracellular domain with Smad3[END_REF]. We surmise that Notch1 affects effector T cell responses in the presence of Treg cells through a TGFβ signalling mechanism. To test this, and to increase the sensitivity of the system, we performed coculture experiments of effector T cells and Treg cells at a low ratio (1:16) in the presence of immobilized Notch ligands DL1, DL4 and Jagged-1 or IgG control (5µg/ml).

Anti-TGF-β (2.5µg/ml) or isotype control mAbs were added at the beginning of the culture.

Coculture experiments performed in the presence of either DL-4 or Jagged-1 led to a mean inhibition of T cell proliferation of 47% and 65%, respectively as compared to 12% when cells were pre-treated with IgG control. As shown in figure 3A, these percentages became 6.3% and 19.3 % in the presence of anti-TGF-β mAbs (P<0.05) while no changes were noted in the presence of isotype control. Finally, anti-TGF-β did not affect coculture performed with Treg cells and effector T cells pre-treated with DL-1 ligands.

Active TGF-β mediates its biological functions by binding to TGF-β type I and type II receptors (TGF-βRII). Therefore, we first investigated the effects of Notch activation on TGF-βRII expression of effector T cells. As shown in figure 3B, CD4 + CD25 -effector T cells stimulated with coated anti-CD3 and Notch ligands (DL-4 or Jagged-1) for 3 hours, exhibited a marked increase in TGF-βRII RNA expression as compared to pre-exposure to IgG control (4.4 and 2.8 fold increase for DL-4 and Jagged-1, respectively). This effect is abrogated in the presence of GSI (P<0.05 for comparison with culture performed without GSI vehicle).

According to the results presented above, pre-exposure of effector T cells to DL-1 alone, or in the presence of GSI, did not significantly modify TGF-βRII RNA expression.

We next assessed TGF-βRII protein expression by flow cytometry. In these experiments, effector CD4 + CD25 -T cells were isolated and incubated for 48 hours in the presence of Notch ligands or IgG control. Flow analysis showed that DL-4 and Jagged-1 increased expression of TGF-βRII (MFI: 166 and 130 respectively) as compared to IgG (MFI: 53) or DL-1 (MFI: 85) (Figure 3B-C). Together these results suggest that engagement of DL-4 and Jagged-1 increases TGF-βRII expression on CD4 + CD25 -T cells.

DL-4 and Jagged-1 sustain TGF-βRII function through HES interaction with the TGF-βRII promoter

Given the above observations showing an upregulation of TGF-βRII following Notch activation, we hypothesized that Jagged-1 and DL-4 led to an activation of TGF-β signalling in effector T cells. CD4 + CD25 -T cells were isolated and cultured for 48 hours in wells coated with Notch ligands or IgG control in the presence of TGF-β (5ng/ml). We then performed immunoblot to assess the expression of pSmad3, a major TGF-β signalling intermediate [START_REF] Li | TGF-beta, T-cell tolerance and immunotherapy of autoimmune diseases and cancer[END_REF].

As shown in figure 4A, DL-4 and Jagged-1 exposure induced pSmad3 expression, while only a weak pSmad3 band was detected in IgG control and DL-1 stimulated cells. Western blot analysis showed that TGF-β induced a significant increase of pSmad3 expression in CD4 + CD25 -T cells after 30 minutes (D0) of stimulation (ratio pSmad/actin: 2.5). This expression decreased dramatically at 48 hours when cells were incubated in the presence of IgG (ratio pSmad3/actin: 0.26) but not in the presence of DL-4 (ratio pSmad3/actin: 2.7) or Jagged-1 (ratio pSmad3/actin: 1.25). By contrast and in accordance with data presented above, DL-1 stimulated CD4 + CD25 -T cells showed very low amount of pSmad3 (ratio pSmad3/actin: 0.26) after 48 hours of TGF-β treatment (Figure 4B). These results suggest that DL-4 and Jagged-1 sustain TGF-βRII function on CD4 + CD25 -T cells.

To explore the mechanism by which Notch may regulate TGF-βRII expression, we have performed transactivation experiments. Since these experiments cannot be performed easily on primary CD4 + T cells, we used rat PC12, a cell line widely used to explore the role of HES-1 as a modulator of cell differentiation and proliferation [START_REF] Castella | HES-1 repression of differentiation and proliferation in PC12 cells: role for the helix 3-helix 4 domain in transcription repression[END_REF]. We analyzed the transactivation of the TGF-βRII promoter by wtHes-1 (pCI-HES-1) or a DN mutant (dnHES-1). In three separate experiments, we found that wtHES-1 induced a strong transactivation of the TGF-ßRII luciferase reporter promoter containing 5' sequences compared with transfection with the empty vector (figure 4C). No increase in luciferase activity was seen when wtHES-1 and dnHES-1 were cotransfected. Interestingly, when transfected alone, the dnHES-1 plasmid decreased luciferase activity below that measured with the empty vector, since it neutralized endogeneous HES-1 activity on the TGF-ßRII reporter promoter. These results demonstrated that HES-1 activates the TGF-ßRII promoter through its DNA-binding activity.

Discussion

Although, Treg cells mediate their suppressive effects through different mechanisms, it is well established that Treg cells produce or express at their surface TGF-β, which participates to their suppressive effects [START_REF] Du | TGF-beta signaling is required for the function of insulin-reactive T regulatory cells[END_REF][START_REF] Gregg | A sudden decline in active membrane-bound TGF-beta impairs both T regulatory cell function and protection against autoimmune diabetes[END_REF][START_REF] Nakamura | Cell contact-dependent immunosuppression by CD4(+)CD25(+) regulatory T cells is mediated by cell surface-bound transforming growth factor beta[END_REF]. In the present study, we focused our interest on the role of Notch pathway on the sensitivity of effectors T cells to Treg cell-mediated suppression. Our results extend the knowledge on the role of Notch in Treg/Effectors T cells cross talk showing that Notch activation may also significantly increase the sensitivity of effector cells to Treg cells even at a low ratio. We demonstrate that Notch ligands DL-4 and Jagged-1, but not DL-1, increase significantly the suppressive effect of Treg cells on effectors T cells through an upregulation of TGF-βRII expression and the phosphorylated form of Smad3 protein. This effect is relieved by anti-TGF-β Abs.

Therefore, the picture of the interaction between Treg cells and effectors T cells in the context of the presence of Notch in the cellular environment becomes more complex. Globally, interactions between Notch and TGF-β pathways impact on the activating or inhibitory functions of Notch. Therefore, in one hand, Notch activation of Treg cells increases TGF-β mediated effector function of these cells and upregulates Notch expression on target cells [START_REF] Ostroukhova | Treg-mediated immunosuppression involves activation of the Notch-HES1 axis by membrane-bound TGF-beta[END_REF].

For example, it has been shown that cells bearing surface TGF-β (but not cells lacking this property) could activate the Notch signalling pathway leading to a tolerance state to respiratory Ags in the airway [START_REF] Ostroukhova | Treg-mediated immunosuppression involves activation of the Notch-HES1 axis by membrane-bound TGF-beta[END_REF]. Notch signalling may also play a role in the differentiation of Treg cells as demonstrated in vitro or in vivo models [START_REF] Vigouroux | Induction of antigen-specific regulatory T cells following overexpression of a Notch ligand by human B lymphocytes[END_REF][START_REF] Kared | Jagged2-expressing hematopoietic progenitors promote regulatory T cell expansion in the periphery through notch signaling[END_REF][START_REF] Anastasi | Expression of activated Notch3 in transgenic mice enhances generation of T regulatory cells and protects against experimental autoimmune diabetes[END_REF] or in the modulation of TGFβ signalling [START_REF] Samon | Notch1 and TGFbeta1 cooperatively regulate Foxp3 expression and the maintenance of peripheral regulatory T cells[END_REF][START_REF] Asano | Notch1 signaling and regulatory T cell function[END_REF]. A recent report suggests that NICD interacts with pSmad3 enhancing its activity and TGF-β mediated effectors function of Treg cells. According to this, it has been recently shown that Notch1 activation enhances TGF-β signalling by increasing Smad translocation into the nucleus and its transactivation effect at promoter sites [START_REF] Asano | Notch1 signaling and regulatory T cell function[END_REF].

On the other hand, and as shown here, Notch activation of effectors T cells increases their sensitivity to TGF-β. Finally, TGF-β is involved also on the generation and maintenance of adaptative Treg cells [START_REF] Samon | Notch1 and TGFbeta1 cooperatively regulate Foxp3 expression and the maintenance of peripheral regulatory T cells[END_REF]. Collectively these data suggest that activated effectors T cells and Treg cells may be maintained at certain equilibrium by a regulatory loop induced by Notch activation.

Earlier studies established an interrelationship between TGF-β signalling and Notch

signalling at the biochemical level. TGF-β signalling leads to a rapid upregulation of HES, a major target of Notch signalling in keratinocytes [START_REF] Zavadil | Integration of TGF-beta/Smad and Jagged1/Notch signalling in epithelial-to-mesenchymal transition[END_REF], in neural stem cells and in myoblasts.

In addition, Smad3 and NICD signalling physically interact and such interaction lead to recruitment of Smad3 to Notch target genes via binding to CSL, a key DNA-binding protein of the Notch pathway. Here, we found that HES, the main transcription factor donwstream of Notch, induced a strong transactivation of the TGF-ßRII by binding the TGF-βRII promoter through its DNA-binding activity. Taken together, this set of findings is consistent with the conclusion that Notch signalling plays an important role in facilitating TGF-β signalling in effector function and, in turn, TGF-β signalling amplifies these effects by upregulating HES.

There is clear evidence that stimulation of APC (Dendritic Cells) via Jagged enhances APC function from the point of view of maturation markers, cytokine production and differentiation effects on T cells [START_REF] Weijzen | The Notch ligand Jagged-1 is able to induce maturation of monocyte-derived human dendritic cells[END_REF]. This raises the question of whether DCs are exposed to Notch ligands in the microenvironments where they reside. In skin, keratinocytes constitutively express high levels of Jagged-1 [START_REF] Souza | Canonical and non-canonical Notch ligands[END_REF]. Therefore, to prevent constitutive maturation of DCs, Notch signalling must be tightly regulated. It is conceivable that TGF-β inhibit the DCs maturation in physiological conditions. In the context of inflammation or infection for example [START_REF] Skokos | CD8-DCs induce IL-12-independent Th1 differentiation through Delta 4 Notch-like ligand in response to bacterial LPS[END_REF][START_REF] Mukherjee | Regulation of T cell activation by Notch ligand, DLL4, promotes IL-17 production and Rorc activation[END_REF], upregulation of Notch ligands on DCs, may modulate specific T cell responses in the presence of Treg cells.

As shown by our group and others in various culture systems, Notch ligands and engagement of different Notch family members may lead to distinct functional outcomes either in early or late stages of T cell development [21,[START_REF] Lefort | Short exposure to Notch ligand Delta-4 is sufficient to induce T-cell differentiation program and to increase the T cell potential of primary human CD34+ cells[END_REF]. Indeed, Notch proteins have been described as having a role in directing the differentiation of activated T cells to Th1 or Th2 T-cell lineages.

Interactions between DL-1 and Notch3 were described to influence the differentiation of activated CD4 + T cells, by promoting a Th1 cell phenotype [START_REF] Maekawa | Delta1-Notch3 interactions bias the functional differentiation of activated CD4+ T cells[END_REF][START_REF] Minter | Inhibitors of gamma-secretase block in vivo and in vitro T helper type 1 polarization by preventing Notch upregulation of Tbx21[END_REF] while CD4 + T cells incubated with APCs expressing Jagged-1 adopted a Th2 cell-cell fate, producing IL-4 and IL-5 [START_REF] Amsen | Instruction of distinct CD4 T helper cell fates by different notch ligands on antigen-presenting cells[END_REF]. Taken together, these data implicated Jagged and DL ligands in the development of Th2 and Th1 cell immune responses, respectively. However, the molecular mechanisms by which Delta-like and Jagged-1 drive differently these functions are currently unknown. Our data may contribute to the understanding mechanisms behind this observation since Notch signalling delivered through DL-4 and Jagged-1, but not DL-1, increases the sensitivity of effector T cells to TGF-β through upregulation of TGF-βRII and a sustained expression of phosphorylated smad3 protein expression.

Collectively these data show that cell environment may modulate T cell responses through Notch/Notch ligand interactions by increasing sensitivity of effectors T cells to Treg suppression. From a physiology stand point, our study suggest that, by increasing the susceptibility of effectors T cells to TGF-β, Notch may provide signals which converge to limit tissue/organ damage mediated by inflammation.

Materials and methods

Cell culture

Peripheral blood mononuclear cells were separated by Ficoll-Hypaque centrifugation (Amersham Biosciences) from buffy coats obtained from healthy blood donors (EFS, Créteil). CD4 + CD25 -T cells and CD4 + CD25 high T cells were isolated using a CD4 + T cell enrichment column and a human CD4 + CD25 + Treg isolation kit purchased from Miltenyi Biotec. CD4 + CD25 -cells and CD4 + CD25 + cells were enriched to greater than 95 % purity.

Cells were plated in media at a concentration of 1.25x10 5 to 5x10 5 cells per ml in a 96-well round-bottom plates coated with 1µg/ml of anti-CD3 (UCHT1) and 5µg/ml of Notch Ligand (DL-1-Fc, DL-4-Fc and Jagged-1-Fc are a generous Gift from Dr. Sakano).

Suppression assay

A total of 2.5x10 4 CD4 + CD25 -T cells were cultured with various ratios of CD4 + CD25 high cells and 1µg/ml of anti-CD3 for 5 days, in the presence or absence of 10µm of the gammasecretase inhibitor Compound E. A total of 0.5 Ci [ 3 H] thymidine (Amersham Pharmacia) was added to the wells during the last 16 hours of culture. The cells were then harvested and assessed for [ 3 H] thymidine incorporation using a liquid scintillation counter. Results were expressed as mean cpm of quadriplicate culture wells.

Percentage of inhibition was calculated as follows: 1 -(mean cpm of co-culture wells divided by mean cpm of CD4 + CD25 -cells cultured alone) x 100.

In sensitizing asssays CD4 + CD25 -and CD4 + CD25 high were pre-incubated during 16 hours on coated Notch Ligand. Cells were also collected, washed and incubated alone or with Treg cells and stimulated as described above. Some experiments were performed in the presence of anti-TGF-β or IgG control (2.5 µg/ml).

Real-time PCR

CD4 + CD25 -T cells were cultured for 3 hours in coated wells with anti-CD3 (200ng/ml) and Notch ligands (5µg/ml) or IgG control (5µg/ml). Total RNA was extacted with Trizol (Invitrogen) and purified by chloroform extraction. RNA was then reverse transcribed using 1mM oligodT and the Superscript™ HH Rnase H-reverse transcriptase (Invitrogen Life technologies) according to the manufacturer's instructions. Quantitative PCR was performed in a LightCycler System (Roche diagnostics) using a SYBR Green PCR kit from Roche Diagnostics. The cDNA input for each population was normalized to obtain equivalent signals with Splicing Factor 3A1 (SF3A1) used as housekeeping gene. Primers used were: S14:

Forward: 5' GGCAGACCGAGATGAATCCTCA 3'

Reverse: 5' CAGGTCCAGGGGTCTTGGTCC 3' TGF-βRII Forward: 5' CTGCAAGATACATGGCTCCA 3'

Reverse: 5' CTCGATCTCTCAACACGTTGT 3'

Surface and intracellular staining

For surface staining, the cells were harvested, washed twice and resuspended in 1%FCS buffer. The cells were stained with mAbs for CD4 (Beckman Coulter), TGF-βRII (R&D) as per manufacturer's instructions for 20 min at 4°C. Cells were then analyzed by flow cytometry using FACScalibur™ and CellQuest™ software (Becton Dickinson, San Jose, California, USA).

Immuno-blot analysis

CD4 + CD25 -T cells were stimulated with Notch Ligand during 2 days. 1ng/ml of TGF-β was also added during 30 min to induce Smad3 phosphorylation. Cells were collected, washed and lysed with Tris HCL (20mM) 0,5 % SDS buffer in presence of DNase (benzonase), inhibitors of phosphatase and protease. The samples were centrifuged for 5min at 4°C at 15,000g. The resulting supernatants were boiled for 3min and frozen at 80°C or used immediately. Aliquots of the supernatants were used for protein determination (micro-bicinchoninic acid protein assay; Pierce). The 3X Laemmli buffer with 2-βME and bromophenol blue was added, and cell lysate proteins (10µg/lane) were resolved by on precast polyacrylamide gels (gradient 4-12% gels, Invitrogen). Proteins were then electroblotted onto nitrocellulose filters with 0.45 µm pores, and the filters were blocked by incubation for 1h with 5% non fat milk powder in TBS, 0.1% Tween20. The filters were incubated for 1h at room temperature or overnight at Transfection procedure. Transfections reporter assays were carried out in six-well tissue culture dishes with the indicated plasmids by using the Lipofectamine plus reagent (Life technologies) as indicated by the manufacturer. PC12 cells were seeded the day prior to transfection at a concentration that will give 50% confluency. Transfection was carried out 18 hours in serum free optiMEM, and cells were then incubated another 48 hours in complete medium (RPMI, 10% HS, 5% FCS). For normalization of transfection efficiency, pCMV-ß Gal plasmid diluted one-fourth was added as an internal control.

Statistical analysis

Results were expressed as means ± SE. Statistical significance was determined by two-tailed t test with a significance level of 0.05, using GraphStat software (GraphPad, San Diego,CA).

  Exposure of Treg cells to DL-1, DL-4 or Jagged-1 did not modify the Treg-cell capacity to suppress CD4 + CD25 -proliferation in the presence of anti-CD3 mAbs (Fig 2A). In contrast, initiation of Notch signalling in CD4 + CD25 -following exposure to DL-1, DL-4 or Jagged-1 increased significantly their sensitivity to Treg cell-mediated suppression in 5-days coculture experiments in the presence of anti-CD3 (Fig 2B). As shown in figure 2C, percentages of Treg-cell inhibition of cell proliferation were 64.8, 64.2, 65 % (mean of 4 experiments at a ratio 1:8) in cocultures performed with CD4 + CD25 -T cells pre-exposed to DL-1, DL-4 and Jagged-1, respectively as compared to pre-incubation with IgG control (29.2 %) (P<0.05 for all comparisons to IgG control). These results show that Notch signalling acts on effector T cells and that Notch ligands potentiate Treg cell-mediated suppression.

  4°C with anti-Phospho Smad3 (Cell Signalling); anti-Smad3 or β-actin. Blots were washed three times for 10min each, in TBS, 0.1% Tween20, and were then incubated for 1h with peroxidase-labeled anti-mouse or anti-rabbit Ig. Blots were developed by enhanced chemiluminescence (ECL + from GE Healthcare) using a CCD camera (G:BOX, Syngene)Transient transfection and Luciferase assay.Expression plasmids. The TGF-ßRII promoter-luciferase constructs were described previously, wtHES-1 and dominant negative (DN) HES-1 were provided by R. Kageyama and ß-Galactosidase by I. Dusanter-Fourt (INSERM U 567, Paris). The DN HES-1 has three amino acid mutations only in the basic region (DNA-binding domain), so it cannot bind to the DNA or interact with promoters but can dimerize with endogenous wild-type HES-1 to form a non DNA-binding heterodimer complex.

Figure 1 :

 1 Figure 1: Notch signalling is involved in Treg cell-mediated inhibition of anti-CD3stimulated CD4 + CD25 -T-cell proliferation A, B) The indicated ratios of CD4 + CD25 -T cells and CD4 + CD25 high T cells (T REG /T EFF ratio)

Figure 2 :

 2 Figure 2: Notch signalling sensitizes effector T cells to Treg cell-mediated suppressionA) CD4 + CD25 + T cells were cultured for 18 hours in wells coated with anti-CD3 and Notch

Figure 3 :

 3 Figure 3: Notch suppresses the proliferation of CD4 + CD25 -T cells through TGF-β A) CD4 + CD25 -T cells were cultured for 5 days in wells coated with anti-CD3 and Notch

Figure 4 :

 4 Figure 4: TGF-βRII function on effector T cells stimulated with Notch ligands

Acknowledgments

This work was supported by grants from INSERM. K.H. was funded by a fellowship from ANRS.

Conflict of interest

The authors declare no financial or commercial conflict of interest.


A Alfonsi
email: alfonsi@cermics.enpc.fr

B Jourdain
email: jourdain@cermics.enpc.fr.

A Kohatsu-Higa

Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

Keywords: Euler scheme, Wasserstein distance, weak trajectorial error, diffusion bridges AMS (2010): 65C30, 60H35

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with N steps is smaller than O(N -2/3+ε ) where ε is an arbitrary positive constant. This rate is intermediate between the strong error estimation in O(N -1/2 ) obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation O(N -1 ) obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time T . We also check that the supremum over t ∈ [0, T ] of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time t and the Euler scheme at time t behaves like O( log(N )N -1 ).

For σ : R → R and b : R → R, we are interested in the simulation of the stochastic differential equation dX t = σ(X t )dW t + b(X t )dt (0.1)

where X 0 = x 0 ∈ R and W = (W t ) t≥0 is a standard Brownian motion. We make the standard Lipschitz assumptions on the coefficients:

∃K ∈ (0, +∞), ∀x, y ∈ R, |σ(x)σ(y)| + |b(x)b(y)| ≤ K|x -y|.

For T > 0, we are interested in the approximation of X = (X t ) t∈[0,T ] by its Euler scheme X = ( Xt ) t∈[0,T ] with N ≥ 1 time-steps. We consider the regular grid {0 = t 0 < t 1 < t 2 < . . . < t N = T } of the interval [0, T ] with t k = kT N and define inductively X0 = x 0 and Xt

= Xt k + σ( Xt k )(W t -W t k ) + b( Xt k )(t -t k ) for t ∈ [t k , t k+1 ]. (0.2)
It is well known that the order of convergence of the strong error of discretization is N -1/2 . Indeed, we have (see [START_REF] Kanagawa | On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations[END_REF])

∀p ≥ 1, ∃C < +∞, ∀N ≥ 1, E 1/p sup t≤T |X t -Xt | p ≤ C √ N . (0.3)
See Section 1 for a more precise statement. This upper-bound gives the correct order of convergence since according to Remark 3.6 [START_REF] Kurtz | corrections, random evolutions, and simulation schemes for SDEs[END_REF], when σ and b are continuously differentiable, ( √ N (X t -Xt )) t≤T converges in law as N goes to ∞ to some diffusion limit which is non zero as soon as σ is positive and non constant (see also [START_REF] Kurtz | Weak limit theorems for stochastic integrals and stochastic differential equations[END_REF] and [START_REF] Jacod | Asymptotic error distributions for the Euler method for stochastic differential equations[END_REF] where stable convergence is also proved). When σ is constant, then the Euler scheme coincides with the Milstein scheme and the strong order of convergence is N -1 . On the other hand, the order of convergence of the weak error of discretization is always N -1 . For example, according to [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF], when σ and b are C ∞ with bounded derivatives of all orders and f : R → R is C ∞ with polynomial growth together with its derivatives then, for each integer L ≥ 1, the expansion

E[f (X T )] -E[f ( XT )] = L l=1 a l N l + O(N -(L+1) ) (0.4)
in powers of N -1 holds for the weak error. The bound |E[f ( XT )] -E[f (X T )]| ≤ C N holds when σ, b and f are C 4 with the same growth assumptions. When f is only assumed to be measurable and bounded, it is proved in [START_REF] Bally | The law of the Euler scheme for stochastic differential equations (II) : convergence rate of the density[END_REF][START_REF] Bally | The law of the Euler scheme for stochastic differential equations (I) : convergence rate of the distribution function[END_REF] that the expansion (0.4) still holds for L = 1 if b and σ are smooth functions satisfying an hypoellipticity condition. Under uniform ellipticity, [START_REF] Guyon | Euler scheme and tempered distributions[END_REF] even extends this expansion by only assuming that f is a tempered distribution acting on the densities of both X T and XT .

In view of financial applications, the weak error analysis gives the convergence rate to 0 of the discretization bias introduced when replacing X by its Euler scheme X for the computation of the price E[f (X T )] of a vanilla European option with payoff f and maturity T written on X. Let C denote the space C([0, T ], R) of continuous paths endowed with the sup norm. When dealing with exotic options with payoff F : C → R Lipschitz continuous,

E[F (X)] -E[F ( X)] ≤ E|F (X) -F ( X)| ≤ C √ N ,
where the second inequality follows from the strong error estimate. But the first inequality is very rough and prevents from taking advantage of the cancellations in the mean which occur and permit to obtain the upper-bound C N for vanilla options. The weak error analysis has been performed for specific path-dependent payoffs, typically when F (X) = f (X T , Y T ) with Y t a function of (X s ) 0≤s≤t such that ((X t , Y t )) 0≤t≤T is a Markov process. The cases Y t = t 0 X s ds and Y t = max 0≤s≤t X s respectively correspond to Asian [START_REF] Temam | Couverture approchée d'options exotiques. Pricing des options asiatiques[END_REF] and barrier [START_REF] Gobet | Weak approximation of killed diffusion using Euler schemes[END_REF][START_REF] Gobet | Euler Schemes and Half-Space Approximation for the Simulation of Diffusion in a Domain[END_REF][START_REF] Gobet | Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme[END_REF] or lookback options [START_REF] Seumen Tonou | Méthodes numériques probabilistes pour la résolution d'équations du transport et pour l'évaluation d'options exotiques[END_REF]. But no general theory has been developped so far to analyse the weak trajectorial error. The Wasserstein distance between the laws L(X) and L( X) of X and X defined by W 1 (L(X), L( X )) = sup

F :C→R:Lip(F )≤1 |E[F ( X)] -E[F (X)]|,
where Lip(F ) denotes the Lipschitz constant of F is the appropriate measure to deal with the whole class of exotic Lipschitz payoffs. Notice that this distance has already been used in the context of discretization schemes for SDEs : in the multidimensional setting, by a clever rotation of the driving Brownian motion, Cruzeiro, Malliavin and Thalmaier [START_REF] Cruzeiro | Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation[END_REF] construct a modified Milstein scheme which does not involve the simulation of iterated Brownian integrals and with order of convergence N -1 for the Wasserstein distance. A simpler scheme with the same convergence properties is exhibited in [START_REF] Jourdain | High order discretization schemes for stochastic volatility models[END_REF] for usual stochastic volatility models.

The weak and strong error estimations recalled above imply that

∃c, C < +∞, ∀N ≥ 1, c N ≤ W 1 (L(X), L( X)) ≤ C √ N . (0.5)
A very nice feature of the Wasserstein distance is its primal representation in the Kantorovitch duality theory. This representation is obtained by choosing p = 1, E = C and (µ, ν) = (L(X), L( X)) in the general definition

W p (µ, ν) = inf π∈Π(µ,ν) E×E
|x -y| p π(dx, dy)

1/p (0.6)
where p ∈ [1, +∞), (E, | |) is a normed vector space, µ and ν are two probability measures on E endowed with its Borel sigma-field and the infimum is computed on the set Π(µ, ν) of probability measures on E × E with respective marginals µ and ν (see for instance Remark 6.5 p95 [START_REF] Villani | Optimal transport. Old and new[END_REF]).

When one is able to exhibit some coupling (Y, Ȳ ) with Y L = X and Ȳ L = X, then the law of (Y, Ȳ ) belongs to Π(L(X), L( X)) and necessarily W p (L(X), L( X )) ≤ E 1/p sup t∈[0,T ] |Y t -Ȳt | p . For the obvious coupling (Y, Ȳ ) = (X, X) obtained by choosing the same driving Brownian motion for the diffusion and its Euler scheme, one recovers the upper-bound in (0.5) from the strong error analysis. The main result of the present paper is the construction of a better coupling which leads to the upper-bound

∀p ≥ 1, ∀ε > 0, ∃C < +∞, ∀N ≥ 1, W p (L(X), L( X)) ≤ C N 2 3 -ε
proved in Section 3 under additional regularity assumptions on the coefficients and uniform ellipticity. To construct this coupling, we first obtain in Section 2 a time-uniform estimation of the Wasserstein distance between the respective laws L(X t ) and L( Xt ) of X t and Xt :

∀p ≥ 1, ∃C < +∞, ∀N ≥ 1, sup t∈[0,T ] W p (L(X t ), L( Xt )) ≤ C log(N ) N .
Before, in Section 1, we recall well-known results concerning the moments and the dependence on the initial condition of the solution to the SDE (0.1) and its Euler scheme. Also, we explicit the dependence of the strong error estimations

E[sup s≤t | Xs -X s |] with respect to t ∈ [0, T ],
which will play a key role in our analysis.

1 Basic estimates on the SDE and its Euler scheme

We recall some well-known results concerning the flow defined by (0.1) (see e.g. Karatzas and Shreve [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], p 306) and its Euler approximation.

Proposition 1.1 Let us denote by (X x t ) t∈[0,T ] the solution of (0.1), starting from x ∈ R. One has that for any p ≥ 1, the existence of a positive constant C ≡ C(p, T ) such that:

∀x ∈ R, E sup t∈[0,T ] |X x t | p ≤ C(1 + |x|) p (1.1) ∀x ∈ R, ∀s ≤ t ≤ T, E sup u∈[s,t] |X x u -X x s | p ≤ C(1 + |x|) p (t -s) p 2 (1.2) ∀x, y ∈ R, E sup t∈[0,T ] |X x t -X y t | p ≤ C|y -x| p (1.3) Proposition 1.2 Let ( Xx t ) t∈[0,T ] denote the Euler scheme (0.2) starting from x. For any p ∈ [1, ∞), there exists a positive constant C ≡ C(p, T ) such that ∀N ≥ 1, ∀x ∈ R, E sup t∈[0,T ] | Xx t | p ≤ C(1 + |x|) p (1.4) ∀N ≥ 1, ∀x ∈ R, ∀t ∈ [0, T ], E sup r∈[0,t] | Xx r -X x r | p ≤ Ct p 2 (1 + |x|) p N p 2
.

(1.5)

The moment bound (1.4) for the Euler scheme holds in fact as soon as the drift and the diffusion coefficients have a sublinear growth. The strong convergence order is established in Kanagawa [START_REF] Kanagawa | On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations[END_REF] for Lipschitz and bounded coefficients. In fact, it is straightforward to extend Kanagawa's proof to merely Lipschitz coefficients by using the estimates (1.1) and (1.4) and obtain

∀N ≥ 1, ∀x ∈ R, ∀t ∈ [0, T ], E sup r∈[0,t] | Xx r -X x r | p ≤ C(1 + |x|) p N p 2
.

(1.6)

The estimate (1.5) precises the dependence on t. This slight improvement will in fact play a crucial role to construct the coupling between the diffusion and the Euler scheme. We prove it for the sake of completeness even though the arguments are really standard.

Proof of (1.5). Let τ s = sup{t i , t i ≤ s} denote the last discretization time before s. We have Xx

t -X x t = t 0 b( Xx τs ) -b(X x s )ds + t 0 σ( Xx τs ) -σ(X x s )dW s
. By Jensen's and Burkholder-Davis-Gundy inequalities,

E sup r∈[0,t] | Xx r -X x r | p ≤ 2 p E t 0 |b( Xx τs ) -b(X x s )|ds p + C p E t 0 (σ( Xx τs ) -σ(X x s )) 2 ds p 2 ≤ 2 p t p-1 t 0 E |b( Xx τs ) -b(X x s )| p ds + C p t p 2 -1 t 0 E |σ( Xx τs ) -σ(X x s )| p ds Denoting by Lip(σ) the finite Lipschitz constant of σ, we have |σ( Xx τs )-σ(X x s )| ≤ Lip(σ)(| Xx τs - X x τs | + |X x τs -X x s |). Thus, (1.2) and (1.6) yield E[|σ( Xx τs ) -σ(X x s )| p ] ≤ C(1+|x|) p N p 2
, and the same bound holds for b replacing σ. Since t p ≤ T p/2 t p/2 , we easily conclude.

The Wasserstein distance between the marginal laws

In this section, we are interested in finding an upper bound for the Wasserstein distance between the marginal laws of the SDE (0.1) and its Euler scheme. It is well known that the optimal coupling between two one-dimensional random variables is obtained by the inverse transform sampling. Thus, let F t and Ft denote the respective cumulative distribution functions of X t and Xt . The p-Wasserstein distance between the time-marginals of the solution to the SDE and its Euler scheme is given by (see Theorem 3.1.2 in [START_REF] Rachev | Mass Transportation problems[END_REF]):

W p (L(X t ), L( Xt )) = 1 0 |F -1 t (u) -F -1 t (u)| p du 1/p . ( 2.1)
Let us state now the main result of this Section. We set:

C k b = {f : R → R k times continuously differentiable s.t. f (i) ∞ < ∞, 0 ≤ i ≤ k}.
Hypothesis 2.1 Let a = σ 2 . We assume that

∃a > 0, ∀x ∈ R, a(x) ≥ a (uniform ellipticity), a ∈ C 2 b and a ′′ is globally γ-Hölder continuous with γ > 0, b ∈ C 2 b .
Since σ is Lipschitz continuous, under Hypothesis 2.1, we have either σ ≡ √ a or σ ≡ -√ a.

From now on, we assume without loss of generality that σ ≡ √ a which is a C 2 b function bounded from below by the positive constant σ = √ a.

Theorem 2.2 Under Hypothesis 2.1, we have for any p ≥ 1,

∀N ≥ 1, sup t∈[0,T ] W p (L(X t ), L( Xt )) ≤ C log(N ) N ,
where C is a positive constant that only depends on p, T , a and (

a (i) ∞ , b (i) ∞ , 0 ≤ i ≤ 2)
and does not depend on the initial condition x ∈ R.

Remark 2.3 When p = 1, the slightly better bound sup t∈[0,T ] W 1 (L(X t ), L( Xt )) ≤ C N holds if σ is uniformly elliptic, according to [START_REF] Sbai | Modélisation de la dépendance et simulation de processus en finance[END_REF] chapter 3. This is proved in a multidimensional setting for C ∞ coefficients σ and b with bounded derivatives by extending the results of [START_REF] Guyon | Euler scheme and tempered distributions[END_REF] but can also be derived from a result of Gobet and Labart [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] only supposing that b, σ ∈ C 3 b . Let p t (x, y) and pt (x, y) denote respectively the density of X 0,x t and X0,x t . Then, Theorem 2.3 in [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] gives:

∀(t, x, y) ∈ (0, T ] × R 2 , |p t (x, y) -pt (x, y)| ≤ T K(T ) N t exp - c|x -y| 2 t .
As remarked in [START_REF] Sbai | Modélisation de la dépendance et simulation de processus en finance[END_REF] chapter 3, for f : R → R a Lipschitz continuous function with Lipschitz constant not greater than one, one deduces that

|E[f (X t )] -E[f ( Xt )]| = R (f (y) -f (x))(p t (x, y) -pt (x, y))dy ≤ K(T )T N t R |y -x| exp - c|x -y| 2 t dy = K(T )T cN ,
which gives sup t≤T W 1 (L(X t ), L( Xt )) ≤ CT N by the dual formulation of the 1-Wasserstein distance.

Our approach consists in controlling the time evolution of the Wasserstein distance. To do so, we need to compute the evolution of both F -1 t (u) and F -1 t (u). In the two next propositions, we derive partial differential equations satisfied by these functions by integrating in space the Fokker-Planck equations and then applying the implicit function theorem. Proposition 2.4 Assume that Hypothesis 2.1 holds.Then for any t ∈ (0, T ], the cumulative distribution function x → F t (x) is invertible with inverse denoted by F -1 t (u). Moreover, the function (t, u) → F -1 t (u) is C 1,2 on (0, T ] × (0, 1)and satisfies

∂ t F -1 t (u) = - 1 2 ∂ u a(F -1 t (u)) ∂ u F -1 t (u) + b(F -1 t (u)). ( 2
∂ t F -1 t (u) = - 1 2 ∂ u α t (u) ∂ u F -1 t (u) + β t (u). (2.3)
where α t (u) = E[a( Xt k )| Xt = F -1 t (u)] and β t (u) = E[b( Xt k )| Xt = F -1 t (u)].
∂ t W p p (L(X t ), L( Xt )) ≤ C W p p (L(X t ), L( Xt )) + 1 0 |F -1 t (u) -F -1 t (u)| p-1 |b( F -1 t (u)) -β t (u)|du + 1 0 |F -1 t (u) -F -1 t (u)| p-2 a( F -1 t (u)) -α t (u) 2 du , (2.4)
where C is a positive constant that only depends on p, a, a ′ ∞ and b ′ ∞ .

The last ingredient of the proof of Theorem 2.2 is the next Lemma, the proof of which is also postponed in Appendix A.

Lemma 2.7 Let τ t = sup{t i , t i ≤ t} denote the last discretization time before t. Under Hypothesis 2.1, we have for all p ≥ 1 :

∃C < +∞, ∀N ≥ 1, ∀t ∈ [0, T ], E E W t -W τt | Xt p ≤ C 1 N ∨ (N 2 t) p/2 .
Proof of Theorem 2.2. Since W p (L(X t ), L( Xt )) ≤ W p ′ (L(X t ), L( Xt )) for p ≤ p ′ , it is enough to prove the estimation for p ≥ 2. Therefore we suppose without loss of generality that p ≥ 2. Let ψ p (t) = W 2 p (L(X t ), L( Xt )) and

for any integer k ≥ 1, h k (x) = k -2/p h(kx) where h(x) = x 2/p if x ≥ 1, 1 + 2 p (x -1)
otherwise. Since h k is C 1 and non-decreasing, Lemma 2.6 and Hölder's inequality imply that

h k ψ p/2 p (t) = h k W p p (L(X 0 ), L( X0 )) + t 0 h ′ k ψ p/2 p (s) ∂ s W p p (L(X s ), L( Xs ))ds ≤ h k (0) + C t 0 h ′ k ψ p/2 p (s) ψ p/2 p (s) + ψ (p-1)/2 p (s) 1 0 |b( F -1 s (u)) -β s (u)| p du 1/p + ψ (p-2)/2 p (s) 1 0 |a( F -1 s (u)) -α s (u)| p du 2/p
ds.

Since for fixed x ≥ 0, the sequence (h ′ k (x)) k is non-decreasing and converges to 2 p x 2 p -1 as k → ∞, one may take the limit in this inequality thanks to the monotone convergence theorem and remark that the image of the Lebesgue measure on [0, 1] by F -1 s is the distribution of Xs to deduce

ψ p (t) ≤ 2C p t 0 ψ p (s) + ψ 1/2 p (s)E 1/p |b( Xs ) -E(b( Xτs )| Xs )| p + E 2/p |a( Xs ) -E(a( Xτs )| Xs )| p ds. (2.5)
One has

a( Xτs ) -a( Xs ) = a ′ ( Xs )σ( Xs )(W τs -W s ) -a ′ ( Xs ) (σ( Xτs ) -σ( Xs ))(W s -W τs ) + b( Xτs )(s -τ s ) + ( Xτs -Xs ) 1 0 a ′ (v Xτs + (1 -v) Xs ) -a ′ ( Xs )dv.
Using Jensen's inequality, the boundedness assumptions on a, b and their derivatives and Lemma 2.7, one gets

E |a( Xs ) -E(a( Xτs )| Xs )| p ≤ CE |σa ′ ( Xs )| p |E(W s -W τs )| Xs )| p + CE (s -τ s ) p + |(σ( Xτs ) -σ( Xs ))(W s -W τs )| p + | Xτs -Xs | 2p ≤ C N p/2 ∨ (N p s p/2 )
.

The same bound holds with a replaced by b. With (2.5) and Young's inequality, one deduces

ψ p (t) ≤ C t 0 ψ p (s) + ψ 1/2 p (s) √ N ∨ (N √ s) + 1 N ∨ (N 2 s) ds ≤ C t 0 ψ p (s) + 1 N ∨ (N 2 s) ds.
One concludes by Gronwall's lemma.

Remark 2.8 When a(x) ≡ a is constant, the term E 2/p |a( Xs ) -E(a( Xτs )| Xs )| p in (2.5)
vanishes and the above reasoning ensures that ψp (t) defined as sup s∈[0,T ] ψ p (s) satisfies

ψp (t) ≤ C t 0 ψp (s)ds + C ψ1/2 p (t) t 0 1 √ N ∨ (N √ s) ds ≤ C t 0 ψp (s)ds + 1 2 ψp (t) + C 2 (T + 1) 2 2N .
By Gronwall's lemma, we recover the estimation sup t∈[0,T ] W p (L(X t ), L( Xt )) ≤ C N which is also a consequence of the strong order of convergence of the Euler scheme when the diffusion coefficient is constant.

The Wasserstein distance between the pathwise laws

We now state the main result of the paper.

Hypothesis 3.1 We assume that a ∈ C 4 b , b ∈ C 3 b , and ∃a > 0, ∀x ∈ R, a(x) ≥ a (uniform ellipticity).
Clearly, Hypothesis 3.1 implies Hypothesis 2.1.

Theorem 3.2 Under Hypothesis 3.1, we have:

∀p ≥ 1, ∀ε > 0, ∃C < +∞, ∀N ≥ 1, W p (L(X), L( X)) ≤ C N 2 3 -ε .
Before proving the theorem, let us state some of its consequences for the pricing of lookback options. It is well-known that if (U k ) 0≤k≤N -1 are independent random variables uniformly distributed on [0, 1] and independent from the Brownian increments

(W t k+1 -W t k ) 0≤k≤N -1 then X def = 1 2 max 0≤k≤N -1 Xt k + Xt k+1 + ( Xt k+1 -Xt k ) 2 -2σ 2 ( Xt k )t 1 ln(U k ) is such that X0 , Xt 1 , . . . , XT , X L = ( X0 , Xt 1 , . . . , XT , max t∈[0,T ] Xt ). Corollary 3.3 If f : R 2 → R is Lipschitz continuous, then, under Hypothesis 3.1, ∀ε > 0, ∃C < +∞, ∀N ≥ 1, E f X T , max t∈[0,T ] X t -E[f ( XT , X)] ≤ C N 2 3 -ε . (3.1)
To our knowledge, this result appears to be new. Of course, when f is also differentiable with respect to its second variable, one has

E f X T , max t∈[0,T ] X t = E [f (X T , x 0 )] + +∞ x 0 E ∂ 2 f (X T , x)1 {max t∈[0,T ] Xt≥x} dx.
One could contemplate combining the weak error analysis for the first term in the right-handside with Theorem 2.3 [START_REF] Gobet | Euler Schemes and Half-Space Approximation for the Simulation of Diffusion in a Domain[END_REF] devoted to barrier options to obtain the order N -1 instead on N -2/3+ε in (3.1). In Theorem 2.3 [START_REF] Gobet | Euler Schemes and Half-Space Approximation for the Simulation of Diffusion in a Domain[END_REF], Gobet assumes C 5 b regularity and uniform ellipticity on the coefficients σ and b and it is not clear whether the estimation is preserved by integration over [x 0 , +∞). More importantly a structure condition on the payoff function implying that

∂ 2 f (x, x) = 0 for all x ≥ x 0 is needed.
Proof of Theorem 3.2. We first deduce from Theorem 2.2 some bound on the Wasserstein distance between the finite dimensional marginals of the diffusion X and its Euler scheme X on a coarse time-grid. For m ∈ {1, . . . , N -1}, we set n = ⌊N/m⌋ and define

s l = lmT N
, for l ∈ {0, . . . , n -1}, and s n = T.

Combining the next proposition, the proof of which is postponed in Appendix B with Theorem 2.2, one obtains that

W p (L(X s 1 , . . . , X sn ), L( Xs 1 , . . . , Xsn )) ≤ C √ log N m (3.2)
where the constant C does not depend on (m, N ).

W p (L( Xx t ), L(X x t )).
There is a probability measure π(dx 1 , . . . , dx n , dx 1 , . . . , dx n ) in Π(L(X s 1 , . . . , X sn ), L( Xs 1 , . . . , Xsn )) which attains the Wasserstein distance in the left-hand-side of (3.2) (see for instance Theorem 3.3.11 [START_REF] Rachev | Mass Transportation problems[END_REF]). Let π(x 1 , . . . , x n , dx 1 , . . . , dx n ) denote a regular conditional probability of (x 1 , . . . , xn ) given (x 1 , . . . , x n ) when R 2n is endowed with π and ( Ȳs 1 , . . . , Ȳsn ) be distributed according to π(X s 1 , . . . , X sn , dx 1 , . . . , dx n ). The vector (X s 1 , . . . , X sn , Ȳs 1 , . . . , Ȳsn ) is distributed according to π so that ( Ȳs 1 , . . . , Ȳsn )

L = ( Xs 1 , . . . , Xsn ) and E 1/p max 1≤l≤n |X s l -Ȳs l | p ≤ C √ log N m . (3.3)
Let p t (x, y) denote the transition density of the SDE (0.1) and ℓ t (x, y) = log(p t (x, y)). According to Appendix C devoted to diffusion bridges, the processes

W l t = t s l dW s -σ(X s )∂ x ℓ s l+1 -s (X s , X s l+1 )ds , t ∈ [s l , s l+1 ) 0≤l≤n-1
are independent Brownian motions independent from (X s 1 , . . . , X sn ). We suppose from now on that the vector ( Ȳs 1 , . . . , Ȳsn ) has been generated independently from these processes and so will be all the random variables and processes needed in the remaining of the proof (see in particular the construction of β below). Moreover

Z x,y t = x + t s l σ(Z x,y s )dW l s + t s l [b(Z x,y s ) + σ 2 (Z x,y s )∂ x ℓ s l+1 -s (Z x,y s , y)]ds, t ∈ [s l , s l+1 ) Z x,y s l+1 = y (3.4
) is distributed according to the conditional law of (X t ) t∈[s l ,s l+1 ] given (X s l , X s l+1 ) = (x, y) and for each l ∈ {0, . . . , n -1}, one has (Z

Xs l ,Xs l+1 t ) t∈[s l ,s l+1 ] = (X t ) t∈[s l ,s l+1 ] .
In order to construct a good coupling between L(X) and L( X), a natural idea would be to extend ( Ȳs 1 , . . . , Ȳsn ) to a process ( Ȳt ) t∈[0,T ] with law L( X) by defining for each l ∈ {0, . . . , n -1}, ( Ȳt ) t∈[s l ,s l+1 ] as an Euler scheme bridge driven by W l and starting from Ȳs l and ending at Ȳs l+1 . Unfortunately, even if the Euler scheme bridge is deduced by a simple transformation of the Brownian bridge on a single time-step, it becomes a complicated process when the difference between the starting and ending times is larger than T N because of the lack of Markov property. We are finally going to choose the difference s l+1s l of order T N 1/3 and therefore much larger than the time-step T N . In addition, it is not clear how to compare the paths of the diffusion bridge and the Euler scheme bridge driven by the same Brownian motion. That is why we are going to introduce some new process ( χt ) t∈[0,T ] such that the comparison will be performed at the diffusion bridge level, which is not so easy yet.

To construct χ, we are going to exhibit a Brownian motion (β t ) t∈[0,T ] such that ( Ȳs 1 , . . . , Ȳsn ) are the values on the coarse time-grid of the Euler scheme with time-step T N driven by β. The extension ( Ȳt ) t∈[0,T ] with law L( X) is then simply defined as the whole Euler scheme driven by

β : Ȳt = Ȳt k + σ( Ȳt k )(β t -β t k ) + b( Ȳt k )(t -t k ), t ∈ [t k , t k+1 ], 0 ≤ k ≤ N -1.
The construction of β is postponed at the end of the present proof. One then defines

χ t = Ȳs l + t s l σ(χ s )dβ s + t s l b(χ s )ds, t ∈ [s l , s l+1 ), 0 ≤ l ≤ n -1
Notice that the process χ = (χ t ) t∈[0,T ] which evolves according to the SDE (0.1) with β replacing W on each time-interval [s l , s l+1 ) is càdlàg : discontinuities may arise at the points {s l+1 , 0 ≤ l ≤ n -1}. We denote by χ s l+1 -its left-hand limit at time s l+1 and set χ T = χ sn-. The strong error estimation (1.5) will permit to estimate the difference between the processes Ȳ and χ. Of course, there is no hope for the processes χ and X to be close. Nethertheless, the process χ obtained by setting ∀l ∈ {0, . . . , n -1}, ∀t ∈ [s l , s l+1 ), χt = Z χs l ,χ s l+1 - t and χT = χ T where Z x,y is defined in (3.4) is such that L( χ) = L(χ) by Propositions C.1 and C.3. On each coarse time-interval [s l , s l+1 ) the diffusion bridges associated with X and χ are driven by the same Brownian motion W l . Moreover the differences |X s l -Y s l | between the starting points and

|X s l+1 -χ s l+1 -| ≤ |X s l+1 -Y s l+1 | + |Y s l+1 -χ s l+1 -|
between the ending points is controlled by (3.3) and the above mentionned strong error estimation. That is why one may expect to obtain a good estimation of the difference between the processes X and χ. By the triangle inequality and since L( X) = L( Ȳ ) and

L( χ) = L(χ), W p (L( X), L(X)) ≤ W p (L( X), L(χ)) + W p (L(χ), L(X)) ≤ E 1/p sup t∈[0,T ] | Ȳt -χ t | p + E 1/p sup t∈[0,T ] |X t -χt | p , (3.5)
where, for the definition of W p (L( X), L(χ)) and W p (L(χ), L(X)), the space of càdlàg samplepaths from [0, T ] to R is endowed with the supremum norm. Let us first estimate the first term in the right-hand-side. Let q ≥ 1. From (1.5), we get

E sup t∈[s l ,s l+1 ) | Ȳt -χ t | pq Ȳs l ≤ C m pq 2 (1 + | Ȳs l |) pq N pq
where the constant C does not depend on (N, m). We deduce that

E sup t∈[0,T ] | Ȳt -χ t | pq = E max 0≤l≤n-1 sup t∈[s l ,s l+1 ) | Ȳt -χ t | pq ≤ n-1 l=0 E E sup t∈[s l ,s l+1 ) | Ȳt -χ t | pq Ȳs l ≤ C m pq 2 N pq n-1 l=0 E (1 + | Ȳs l |) pq ≤ C m pq 2 -1 N pq-1 ,
where we used (1.4) for the last inequality. As a consequence,

E 1/p sup t∈[0,T ] | Ȳt -χ t | p ≤ E 1/pq sup t∈[0,T ] | Ȳt -χ t | pq ≤ C m 1 2 -1 pq N 1-1 pq . (3.6)
Let us now estimate the second term in the right-hand-side of (3.5). By Proposition C.3 and since for l ∈ {0, . . . , n -1},

χ s l = Ȳs l , sup t≤T |X t -χt | = max 0≤l≤n-1 sup t∈[s l ,s l+1 ) |Z Xs l ,Xs l+1 t -Z χs l ,χ s l+1 - t | ≤ C max 0≤l≤n-1 |X s l -Ȳs l |∨|X s l+1 -χ s l+1 -|.
Since, by the triangle inequality and the continuity of Ȳ ,

|X s l+1 -χ s l+1 -| ≤ |X s l+1 -Ȳs l+1 | + | Ȳs l+1 -χ s l+1 -| ≤ |X s l+1 -Ȳs l+1 | + sup t∈[0,T ] | Ȳt -χ t |, one deduces that sup t≤T |X t -χt | ≤ C max 1≤l≤n |X s l -Ȳs l | + sup t∈[0,T ] | Ȳt -χ t | .
Combined with (3.3) and (3.6), this implies

E 1/p sup t≤T |X t -χt | p ≤ CE 1/p max 1≤l≤n |X s l -Ȳs l | p +CE 1/p sup t∈[0,T ] | Ȳt -χ t | p ≤ C √ log N m + m 1 2 -1 pq N 1-1 pq .
Plugging this inequality together with (3.6) in (3.5), we deduce that

W p (L(X), L( X)) ≤ C √ log N m + m 1 2 -1 pq N 1-1 pq
and conclude by choosing m = ⌊N 2 3 ⌋ and q ≥ 1 3pε . To end the proof, we still have to construct the Brownian motion β. We first reconstruct on the fine time grid (t k ) 1≤k≤N an Euler scheme ( Ȳt k , 0 ≤ k ≤ N ) interpolating the values on the coarse grid (s l ) 1≤l≤n . Let us denote by p(x, y) the density of the law N (x + b(x)T /N, σ(x) 2 T /N ) of the Euler scheme starting from x after one time step T /N . Thanks to the ellipticity assumption, we have p(x, y) > 0 for any x, y ∈ R. Conditionally on ( Ȳs 1 , . . . , Ȳsn ), we generate independent random vectors ( Ȳs l-1 +t 1 , . . . , Ȳs l-1 +t m-1 ) 1≤l≤n-1 and ( Ȳs n-1 +t 1 , . . . , Ȳt N-1 ) with respective densities p( Ȳs l-1 , x 1 )p(x 1 , x 2 ) . . . p(x n-1 , Ȳs l ) R n-1 p( Ȳs l-1 , y 1 )p(y 1 , y 2 ) . . . p(y n-1 , Ȳs l )dy 1 . . . dy n-1 and p( Ȳs n-1 , x 1 )p(x 1 , x 2 ) . . . p(x N -1-m(n-1) , Ȳsn )

R N-1-m(n-1) p( Ȳs n-1 , y 1 )p(y 1 , y 2 ) . . . p(y N -1-m(n-1) , Ȳsn )dy 1 . . . dy N -1-m(n-1)
and get immediately ( Ȳt k ) 0≤k≤n L = ( Xt k ) 0≤k≤n . Then, thanks to the ellipticity condition,

1 σ( Ȳt k-1 ) ( Ȳt k -Ȳt k-1 -b( Ȳt k-1 )) 1≤k≤N
are independent centered Gaussian variables with variance T /N . By using independent Brownian bridges, we can then construct a Brownian motion (β t ) t∈[0,T ] such that

β t k -β t k-1 = 1 σ( Ȳt k-1 ) ( Ȳt k -Ȳt k-1 -b( Ȳt k-1 )),
which ends the construction.

Conclusion

In this paper, we prove that the order of convergence of the Wasserstein distance W p on the space of continuous paths between the laws of a uniformly elliptic one-dimensional diffusion and its Euler scheme with N -steps is not worse that N -2/3+ε . In view of a possible extension to multidimensional settings, two main difficulties have to be overcomed. First, we took advantage of the optimality of the inverse transform coupling in dimension one to obtain a uniform bound on the Wasserstein distance between the marginal laws with optimal rate N -1 up to a logarithmic factor. In dimension d > 1, the optimal coupling between two probability measures on R d is not available, which makes the estimation of the Wasserstein distance between the marginal laws much more complicated even if, for W 1 , the order N -1 may be deduced from the results of [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] (see Remark 2.3). In the second place, one has to generalize the estimation on diffusion bridges given by Proposition C.3 which we deduce from the Lamperti transform in dimension d = 1.

In the perspective of the multi-level Monte Carlo method introduced by Giles [8], coupling with order of convergence N -2/3+ε the Euler schemes with N and 2N steps would also be of great interest for variance reduction, especially in multidimensional situations where the Milstein scheme is not feasible (see [START_REF] Jourdain | High order discretization schemes for stochastic volatility models[END_REF] for the implementation of this idea in the example of a discretization scheme devoted to usual stochastic volatility models). But this does not seem obvious from our non constructive coupling between the Euler scheme and its diffusion limit.

For both the derivation of the order of convergence of the Wasserstein distance on the path space and the explicitation of the coupling, the limiting step in our approach is Proposition 3.4. In this proposition, we bound the dual formulation of the Wasserstein distance between n-dimensional marginals by the Wasserstein distance between one-dimensional marginals multiplied by n.

Even if the order of convergence of the Wasserstein distance on the path space obtained in the present paper may not be optimal, it provides the first significant step from the order N 1/2 obtained with the trivial coupling where the diffusion and the Euler scheme are driven by the same Brownian motion.

A Proofs of Section 2

Proof of Proposition 2.4. According to [START_REF] Friedman | Stochastic differential equations and applications[END_REF], Theorems 5.4 and 4.7, for any t ∈ (0, T ], X t admits a density p t (x) w.r.t. the Lebesgue measure on the real line, the function (t, x) → p t (x) is C 1,2 on (0, T ] × R and, on this set, it is a classical solution of the Fokker-Planck equation

∂ t p t (x) = 1 2 ∂ xx (a(x)p t (x)) -∂ x (b(x)p t (x)). (A.1)
Moreover, the following Gaussian bounds hold

∃C > 0, ∀t ∈ (0, T ], ∀x ∈ R, |p t (x)| + √ t|∂ x p t (x)| ≤ C √ t e -(x-x 0 ) 2 Ct (A.2)
The partial derivatives ∂ x F t (x) = p t (x) and ∂ xx F t (x) = ∂ x p t (x) exist and are continuous on (0, T ] × R. For 0 < s < t ≤ T and y ≤ x, integrating (A.1) over [s, t] × [y, x], then letting y → -∞ thanks to (A.2), one obtains

F t (x) -F s (x) = t s 1 2 ∂ x (a(x)p r (x)) -b(x)p r (x)dr
. By continuity of the integrand w.r.t. (r, x) one deduces that the partial derivative ∂ t F t (x) exists and is continuous on (0, T ] × R. So, (t, x) → F t (x) is C 1,2 on (0, T ] × R and solves

∂ t F t (x) = 1 2 ∂ x (a(x)∂ x F t (x)) -b(x)∂ x F t (x). (A.3)
According to [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF], the density is also bounded from below by some Gaussian kernel : ∃c >

0, ∀(t, x) ∈ (0, T ] × R, |p t (x)| ≥ c √ t e -(x-x 0 ) 2 ct
. This enables us to apply the implicit function theorem to (t, x, u) → F t (x)u to deduce that the inverse

u → F -1 t (u) of x → F t (x) is C 1,2 in
the variables (t, u) ∈ (0, T ] × (0, 1) and solves

∂ t F -1 t (u) = - ∂ t F t ∂ x F t (F -1 t (u)) = - 1 2 ∂ x (a(x)∂ x F t (x))| x=F -1 t (u) ∂ u F -1 t (u) + b(F -1 t (u)) = - 1 2 ∂ u a(F -1 t (u)) ∂ u F -1 t (u) + b(F -1 t (u))
where we used (A.3) for the second equality and

∂ u F -1 t (u) = 1 ∂xFt(F -1 t (u))
for both the second and the third equalities. Proof of Proposition 2.5.

For t ∈ (0, t 1 ], Xt admits the gaussian density with mean x 0 + b(x 0 )t and variance a(x 0 )t. By induction on k and independence of W t -W t k and Xt k in (0.2), one checks that for k ∈ {1, . . . , n-1}, Xt k admits a density pt k (x) and that for t ∈ (t k , t k+1 ], ( Xt k , Xt ) admits the density

ρ(t k , t, y, x) = pt k (y) e - (x-y-b(y)(t-t k )) 2 2a(y)(t-t k ) 2πa(y)(t -t k ) . The marginal density pt (x) = R pt k (y) e - (x-y-b(y)(t-t k )) 2 2a(y)(t-t k ) √ 2πa(y)(t-t k )
dy of Xt is continuous on (t k , t k+1 ] × R by Lebesgue's theorem and positive.

Let

N (x) = x -∞ e -y 2
2 dy √ 2π denote the cumulative distribution function of the standard Gaussian law and k ∈ {0, . . . , N -1}. Again by the independence structure in (0.2), for (t, x)

∈ (t k , t k+1 ]× R, Ft (x) = E N x-Xt k -b( Xt k )(t-t k ) √ a( Xt k )(t-t k )
. One has

∂ t N x -y -b(y)(t -t k ) a(y)(t -t k ) = - x -y -b(y)(t -t k ) 2 2πa(y)(t -t k ) 3 + b(y) 2πa(y)(t -t k ) e - (x-y-b(y)(t-t k )) 2 2a(y)(t-t k )
.

By the growth assumption on σ and b, one easily checks that ∀k ∈ {0, . . . , N }, E( X2 t k ) < +∞. With the uniform ellipticity assumption and Lebesgue's theorem, one deduces that Ft (x) is differentiable w.r.t. t with partial derivative

∂ t Ft (x) = -E   x -Xt k -b( Xt k )(t -t k ) 2 2πa( Xt k )(t -t k ) 3 + b( Xt k ) 2πa( Xt k )(t -t k ) e - (x-Xt k -b( Xt k )(t-t k )) 2 2a( Xt k )(t-t k )   (A.4) continuous in (t, x) ∈ (t k , t k+1 ] × R.
In the same way, one checks smoothness of Ft (x) in the spatial variable x and obtains that this function is

C 1,2 on (t k , t k+1 ] × R. When k ≥ 1, E     b( Xt k ) e - (x-Xt k -b( Xt k )(t-t k )) 2 2a( Xt k )(t-t k ) 2πa( Xt k )(t -t k )     = R b(y)ρ(t k , t, y, x)dy = E[b( Xt k )| Xt = x]p t (x).
For k = 0, even if ( X0 , Xt ) has no density, the equality between the opposite sides of this equation remains true.

Combining Lebesgue's theorem and a similar reasoning, one checks that

-E   x -Xt k -b( Xt k )(t -t k ) 2πa( Xt k )(t -t k ) 3 e - (x-Xt k -b( Xt k )(t-t k )) 2 2a( Xt k )(t-t k )   = ∂ x E     a( Xt k ) e - (x-Xt k -b( Xt k )(t-t k )) 2 2a( Xt k )(t-t k ) 2πa( Xt k )(t -t k )     = ∂ x E(a( Xt k )| Xt = x)p t (x) .
With (A.4), one deduces that

∂ t Ft (x) = 1 2 ∂ x E[a( Xt k )| Xt = x]∂ x Ft (x) -E[b( Xt k )| Xt = x]∂ x Ft (x). (A.5)
One checks that the function (t, u) → F -1 t (u) is smooth and satisfies the partial differential equation (2.3) by arguments similar to the ones given at the end of the proof of Proposition 2.4.

Remark A.1 In the same way, for k ∈ {0, . . . , N -1}, one could prove that on (t k , t k+1 ] × R, (t, x) → pt (x) is C 1,2 and satisfies the partial differential

∂ t pt (x) = 1 2 ∂ xx E[a( Xt k )| Xt = x]p t (x) -∂ x E[b( Xt k )| Xt = x]p t (x) .
obtained by spatial derivation of (A.5). This is related to [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô's differential[END_REF].

Proof of Lemma 2.6.

By the continuity of the paths of X and X and the finiteness of E sup t≤T (|X t | p+1 + | Xt | p+1 ) , one easily checks that t → W p p (L(X t ), L( Xt )) is continuous. Let k ∈ {0, . . . , N -1} and s, t ∈ (t k , t k+1 ] with s ≤ t. Combining Propositions 2.4 and 2.5 with a spatial integration by parts, one obtains for ε ∈ (0, 1/2)

1-ε ε |F -1 t (u) -F -1 t (u)| p du = 1-ε ε |F -1 s (u) -F -1 s (u)| p du + p t s 1-ε ε |F -1 r (u) -F -1 r (u)| p-2 (F -1 r (u) -F -1 r (u))(b(F -1 r (u)) -β r (u))dudr + p(p -1) 2 t s 1-ε ε |F -1 r (u) -F -1 r (u)| p-2 (∂ u F -1 r (u) -∂ u F -1 r (u)) a(F -1 r (u)) ∂ u F -1 r (u) - α r (u) ∂ u F -1 r (u) dudr + p 2 t s |F -1 r (1 -ε) -F -1 r (1 -ε)| p-2 (F -1 r (1 -ε) -F -1 r (1 -ε)) α r (1 -ε) ∂ u F -1 r (1 -ε) - a(F -1 r (1 -ε)) ∂ u F -1 r (1 -ε) dr - p 2 t s |F -1 r (ε) -F -1 r (ε)| p-2 (F -1 r (ε) -F -1 r (ε)) α r (ε) ∂ u F -1 r (ε) - a(F -1 r (ε)) ∂ u F -1 r (ε) dr (A.6)
We are now going to take the limit as ε → 0. We will check at the end of the proof that lim

u→0 + or 1 - sup r∈[s,t] a(F -1 t (u)) ∂ u F -1 t (u) |F -1 t (u) -F -1 t (u)| p-1 + sup r∈[s,t] α t (u) ∂ u F -1 t (u) |F -1 t (u) -F -1 t (u)| p-1 = 0.
(A.7) which enables us to get rid of the two last boundary terms.

Combining Young's inequality with the uniform ellipticity assumption and the positivity of

∂ u F -1 t (u) and ∂ u F -1 t (u), one obtains (∂ u F -1 r (u) -∂ u F -1 r (u)) a(F -1 r (u)) ∂ u F -1 r (u) - α r (u) ∂ u F -1 r (u) = a(F -1 r (u)) -α r (u) ∂ u F -1 r (u) -∂ u F -1 r (u) ∂ u F -1 r (u) ∨ ∂ u F -1 r (u) -a(F -1 r (u)) ((∂ u F -1 r (u) -∂ u F -1 r (u)) + ) 2 ∂ u F -1 r (u)∂ u F -1 r (u) -α r (u) ((∂ u F -1 r (u) -∂ u F -1 r (u)) + ) 2 ∂ u F -1 r (u)∂ u F -1 r (u) ≤ 1 4a a(F -1 r (u)) -α r (u) 2 + a (∂ u F -1 r (u) -∂ u F -1 r (u)) 2 (∂ u F -1 r (u) ∨ ∂ u F -1 r (u)) 2 -(a(F -1 r (u)) ∧ α r (u)) (∂ u F -1 r (u) -∂ u F -1 r (u)) 2 ∂ u F -1 r (u)∂ u F -1 r (u) ≤ 1 4a a(F -1 r (u)) -α r (u) 2 .
Hence, up to the factor p(p-1)

2

, the third term of the right-hand-side of (A.6) is equal to

t s 1-ε ε |F -1 r (u) -F -1 r (u)| p-2 (∂ u F -1 r (u) -∂ u F -1 r (u)) a(F -1 r (u)) ∂ u F -1 r (u) - α r (u) ∂ u F -1 r (u) - a(F -1 r (u)) -α r (u) 2 4a dudr + 1 4a t s 1-ε ε |F -1 r (u) -F -1 r (u)| p-2 a(F -1 r (u)) -α r (u) 2 dudr.
where the integrand in the first integral is non positive. Since

t s 1 0 |F -1 r (u) -F -1 r (u)| p-2 |F -1 r (u) -F -1 r (u)||b(F -1 r (u)) -β r (u)| + (a(F -1 r (u)) -α r (u)) 2 dudr ≤ 2 b ∞ t s W p-1 p (L(X r ), L( Xr ))dr + 4 a 2 ∞ t s W p-2 p
(L(X r ), L( Xr ))dr < +∞, one can take the limit ε → 0 in (A.6) using Lebesgue's theorem for the second term of the righthand-side and combining Lebesgue's theorem with monotone convergence for the third term to obtain

W p p (L(X t ), L( Xt )) = W p p (L(X s ), L( Xs )) + p t s 1 0 |F -1 r (u) -F -1 r (u)| p-2 (F -1 r (u) -F -1 r (u))(b(F -1 r (u)) -β r (u))dudr + p(p -1) 2 t s 1 0 |F -1 r (u) -F -1 r (u)| p-2 (∂ u F -1 r (u) -∂ u F -1 r (u)) a(F -1 r (u)) ∂ u F -1 r (u) - α r (u) ∂ u F -1 r (u)
dudr.

(A.8)

The last term which belongs to [-∞, +∞) is finite since so are all the other terms. We deduce integrability of (r, u)

→ |F -1 r (u) -F -1 r (u)| p-2 (∂ u F -1 r (u) -∂ u F -1 r (u)) a(F -1 r (u)) ∂ u F -1 r (u) - α r (u) ∂ u F -1 r (u)
on [s, t] × (0, 1). Similar arguments show that the integrability property and (A.8) remain true for s = t k . By summation, they remain true for 0 ≤ s ≤ t ≤ T . So integrability holds on [0, T ] for the distribution derivative

∂ t W p p (L(X t ), L( Xt )) = p 1 0 |F -1 t (u) -F -1 t (u)| p-2 (F -1 t (u) -F -1 t (u))(b(F -1 t (u)) -β t (u))du + p(p -1) 2 1 0 |F -1 t (u) -F -1 t (u)| p-2 (∂ u F -1 t (u) -∂ u F -1 t (u)) a(F -1 t (u)) ∂ u F -1 t (u) - α t (u) ∂ u F -1 t (u) du ≤ p 1 0 |F -1 t (u) -F -1 t (u)| p-2 (F -1 t (u) -F -1 t (u))(b(F -1 t (u)) -β t (u)) + (p -1) a(F -1 t (u)) -α t (u) 2 8a du.
Equation (2.4) follows by remarking that

a(F -1 r (u)) -α r (u) 2 ≤ 2 a ′ 2 ∞ |F -1 t (u) -F -1 t (u)| 2 + a( F -1 t (u)) -α t (u)
2 and using a similar idea for |b(F -1 t (u))β t (u)|.

To prove (A.7) for 0 < s ≤ t ≤ T , we use the Aronson estimates recalled in the proof of Proposition 2.4 for X t and deduced from Theorem 2.1 [START_REF] Lemaire | On some non asymptotic bounds for the Euler scheme[END_REF] for the Euler scheme.

c √ r exp - (x -x 0 ) 2 cr ≤ p r (x) ∧ pr (x) ≤ p r (x) ∨ pr (x) ≤ C √ r exp - (x -x 0 ) 2 Cr , (A.9) Setting K 1 = c √ t , c 1 = cs/2, K 2 = C √ s and c 2 = Ct/2, one has ∀r ∈ [s, t], ∀x ∈ R, K 1 exp - (x -x 0 ) 2 2c 1 ≤ ρ r (x) ≤ K 2 exp - (x -x 0 ) 2 2c 2 . (A.10)
where ρ r denotes either p r or pr . The four limits in (A.7) can be obtained similarly, and we focus on the one of sup r∈[s,t]

a(F -1 r (u)) ∂uF -1 r (u) |F -1 r (u) -F -1 r (u)| p-1 .
Up to modifying K 1 > 0 and decreasing c 1 > 0, we get from (A.10) that

∀r ∈ [s, t], ∀x ≤ x 0 -1, K 1 (x 0 -x) exp - (x -x 0 ) 2 2c 1 ≤ ρ r (x) ≤ K 2 (x 0 -x) exp - (x -x 0 ) 2 2c 2 , which leads to ∀x ≤ x 0 -1, K 1 c 1 exp - (x -x 0 ) 2 2c 1 ≤ G r (x) ≤ K 2 c 2 exp - (x -x 0 ) 2 2c 2 ,
where G r denotes either F r or Fr . Thus, the inverse function satisfies

x 0 --2c 2 log( u K 2 c 2 ) ≤ F -1 r (u) ≤ x 0 --2c 1 log( u K 1 c 1 ) (A.11)
for u small enough. The two last inequalities imply that when x → -∞,

∀r ∈ [s, t], F -1 r (F r (x)) ≥ x 0 --2c 2 log( K 1 c 1 K 2 c 2 ) - (x -x 0 ) 2 2c 1 and sup r∈[s,t] |x -F -1 r (F r (x))| = x→-∞ O(x).
With the boundedness of a and (A.10), we easily deduce that lim

x→-∞ sup r∈[s,t] a(x)p r (x)|x -F -1 r (F r (x))| p-1 = 0.
Since, by (A.11), F -1 r (u) converges to -∞ uniformly in r ∈ [s, t] as u tends to 0, we conclude that lim

u→0 + sup r∈[s,t] a(F -1 r (u)) ∂ u F -1 r (u) |F -1 r (u) -F -1 r (u)| p-1 = 0.
Proof of Lemma 2.7. By Jensen's inequality,

E |E(W t -W τt | Xt )| p ≤ E [|W t -W τt | p ] ≤ C N p/2 .
Let us now check that the left-hand-side is also smaller than C t p/2 N p . To do this, we will study

E (W t -W τt )g( Xt )
where g is any smooth real valued function.

In order to continue, we need to do various estimations on the Euler scheme and its stochastic derivatives. Let η t = min{t i ; t ≤ t i } denote the discretization time just after t. We have D u Xt = 0 for u > t, and

D u Xt = 1 {t≤ηu} σ( Xτt ) + 1 {t>ηu} 1 + σ ′ ( Xτt )(W t -W τt ) + b ′ ( Xτt )(t -τ t ) D u Xτt for u ≤ t.
Then by induction, one clearly obtains that for u ≤ t,

D u Xt = σ( Xτu ) Ēu,t , Ēu,t =            1 if τ t ≤ η u 1 + b ′ ( Xτt )(t -τ t ) + σ ′ ( Xτt )(W t -W τt ) if η u = τ t Nτ t T -1 i= Nηu T 1 + b ′ ( Xt i )(t i+1 -t i ) + σ ′ ( Xt i )(W t i+1 -W t i ) if η u < τ t × 1 + b ′ ( Xτt )(t -τ t ) + σ ′ ( Xτt )(W t -W τt ) .
Note that Ē satisfies the following properties: 1. Ēu,t = Ēη(u),t and 2. Ēt i ,t j Ēt j ,t = Ēt i ,t for t i ≤ t j ≤ t. We also introduce the process E defined by

E u,t = exp t u b ′ (X s ) - 1 2 σ ′ (X s ) 2 ds + t u σ ′ (X s )dW s .
The next lemma, the proof of which is postponed at the end of the present proof states some useful properties of the processes E and Ē.

Lemma A.2 Let us assume that b, σ ∈ C 2 b . Then, we have:

sup 0≤s≤t≤T E E -p s,t + E E p s,t ≤ C, sup 0≤s≤t≤T E Ēp s,t ≤ C, (A.12) sup 0≤s,u≤t≤T E |D u Ēs,t | p + |D u E s,t | p ≤ C, (A.13) sup 0≤t≤T E E 0,t -Ē0,t p ≤ C N p 2 , (A.14)
where C is a positive constant depending only on p and T .

We next define the localization given by

ψ = ϕ E -1 0,t E 0,t -Ē0,t .
Here ϕ : R →[0, 1] is a C ∞ symmetric function so that

ϕ(x) = 0, if |x| > 1 2 1, if |x| < 1 4 . One has E (W t -W τt )g( Xt ) = E (W t -W τt )g( Xt )ψ + E (W t -W τt )g( Xt )(1 -ψ) = t τt E ψg ′ ( Xt )D u Xt du + E g( Xt ) t τt D u ψdu + E (W t -W τt )g( Xt )(1 -ψ)
where the second equality follows from the duality formula (see e.g. Definition 1.3.1 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]). Since for τ t ≤ u ≤ t

E ψg ′ ( Xt )D u Xt = E ψg ′ ( Xt )σ( Xτt ) = t -1 E t 0 D s g( Xt ) ψσ( Xτt ) D s Xt ds = t -1 E g( Xt ) t 0 ψσ( Xτt )σ -1 Xτs Ē-1 s,t δW s ,
one deduces

E W t -W τt | Xt = t -1 t τt E t 0 ψσ( Xτt )σ -1 Xτs Ē-1 s,t δW s Xt du + E t τt D u ψdu Xt + E (W t -W τt ) (1 -ψ)| Xt . (A.15)
Here δW denotes the Skorohod integral. In order to obtain the conclusion of the Lemma, we need to bound the L p -norm of each term on the right-hand-side of (A.15). In particular, we will use the following estimate (which also proves the existence of the Skorohod integral on the left side below) which can be found in Proposition 1.5.4 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]: . By Jensen's inequality for p ≥ 2, we have

t 0 ψσ( Xτt )σ -1 Xτs Ē-1 s,t δW s p ≤ C(p) ψσ( Xτt )σ -1 Xτ• Ē-1
F • p 1,p ≤ t p/2-1 t 0 E[|F s | p ]ds + t p-2 t 0 t 0 E[|D u F s | p ]dsdu, (A.17)
and we will use this inequality to upper bound (A.16). When 1 ≤ p ≤ 2, we will use alternatively the following upper bound

F • p 1,p ≤ t 0 E[F 2 s ]ds p/2 + t 0 t 0 E[(D u F s ) 2 ]dsdu p/2
that comes from Hölder's inequality.

For ψ > 0, Ē0,t ≥ 1 2 E 0,t > 0. From Hypothesis 3.1, there are constants 0 < σ ≤ σ < ∞ such that 0 < σ ≤ σ ≤ σ, and one has

t 0 E ψσ( Xτt )σ -1 Xτs Ē-1 s,t p ds ≤ σ σ p t 0 E ψ p Ē-p 0,t Ēp 0,η(s) ds ≤ 2σ σ p E[E -2p 0,t ] t 0 E[| Ē0,η(s) | 2p ]ds ≤ Ct,
by using the estimates (A.12).

Next, we focus on getting an upper bound for

t 0 t 0 E D u ψσ( Xτt )σ -1 Xτs Ē-1 s,t p dsdu. (A.18)
To do so, we compute the derivative using basic derivation rules, which gives

D u ψσ( Xτt )σ -1 Xτs Ē-1 s,t = D u ψσ( Xτt )σ -1 Xτs Ē-1 s,t + ψσ ′ ( Xτt )D u Xτt σ -1 Xτs Ē-1 s,t -ψσ( Xτt )σ -2 σ ′ Xτs σ( Xτu ) Ēu,τs Ē-1 s,t 1 u≤τs -ψσ( Xτu )σ -1 Xτs Ē-2 s,t D u Ēs,t . (A.19)
One has then to get an upper bound for the L p -norm of each term. As many of the arguments are repetitive, we show the reader only some of the arguments that are involved. Let us start with the first term. We have

D u ψ = ϕ ′ E -1 0,t E 0,t -Ē0,t D u E -1 0,t E 0,t -Ē0,t ,
and

D u E -1 0,t E 0,t -Ē0,t = E -2 0,t D u E 0,t Ē0,t -E -1 0,t D u Ē0,t .
From the estimates in (A.12) and (A.13), we obtain sup

u∈[0,t] D u ψ p ≤ ϕ ′ ∞ C(p). (A.20) Since Ē-1 s,t = Ē0,η(s) Ē-1 0,t and Ē0,t ≥ 1 2 E 0,t > 0 if ϕ ′ E -1 0,t E 0,t -Ē0,t = 0, we have E D u ψσ( Xτt )σ -1 Xτs Ē-1 s,t p ≤ 2σ σ p D u ψ p 2p E E -1 0,t Ē0,η(s) 2p 1/2 .
Similar bounds hold for the three other terms. Note that the highest requirements on the derivatives of b and σ will come from the terms involving D u Ē in (A. [START_REF] Kohatsu-Higa | Weak approximations: A Malliavin calculus approach[END_REF]). Gathering all the upper bounds, we get that ψσ( Xτt )σ -1 Xτ• Ē-1 We are now in position to conclude. Using Jensen's inequality, the results (A.15), (A.12), (A.14), (A.20) and the definition of ϕ together with Chebyshev's inequality, we have for any k > 0 that

E E W t -W τt | Xt p ≤ C t -p (t -τ t ) p t 0 ψσ( Xτt )σ -1 Xτs Ē-1 s,t δW s p p + (t -τ t ) p-1 t τt D u ψ p p du + E(|W t -W τt | 2p )4 k/2 E(|E 0,t -Ē0,t | 2k )E(E -2k 0,t ) 1/4 ≤ C t -p/2 (t -τ t ) p + (t -τ t ) p + 1 N (2p+k)/4 ≤ C 1 t p/2 N p + 1 N p 2 + k 4 .
Proof of Lemma A.2. The upper bounds (A.12) on E and Ē are obvious since b ′ and σ ′ are bounded. Now, let us remark that Ē and E satisfy Thus, (A.14) can be easily obtained by noticing that ( Xt , Ē0,t ) is the Euler scheme for the SDE (X t , E 0,t ) which has Lipschitz coefficients, and by using the strong convergence order of 1/2 (see e.g. [START_REF] Kanagawa | On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations[END_REF]).

E u,t = 1 + t u σ ′ (X s )E u,s dW s + t u b ′ (X s )E u,
The estimate (A.13) on D u E is given, for example, by Theorem 2.2.1 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]. On the other hand, we have for η(s) ≤ u ≤ t D u Ēηs,t = σ ′ ( Xτu ) Ēηs,τu + In order to obtain a L p (Ω) estimate, we then use (A.12), b, σ ∈ C 2 b and Gronwall's lemma.

B Proofs of Section 3

Proof of Proposition 3.4. We use the dual representation of the Wasserstein distance (0.6) deduced from Kantorovitch duality theorem (see for instance Theorem 5.10 p58 [START_REF] Villani | Optimal transport. Old and new[END_REF]) :

W p p (µ, ν) = sup φ∈L 1 (ν) E φ(x)µ(dx) - E φ(x)ν(dx)
where φ(x) = inf y∈E (φ(y) + |y -x| p ).

We also denote by (X s,x t ) t∈[s,T ] the solution to (0.1) starting from x ∈ R at time s ∈ [0, T ] and by ( Xt j ,x t

) t∈[t j ,T ] the Euler scheme starting from x at time t j with j ∈ {0, . . . , N }. It is enough to check that

w k def = W p (L( Xs 1 , . . . , Xs k , X s k , Xs k s k+1 , . . . , X s k , Xs k sn ), L( Xs 1 , . . . , Xs k-1 , X s k-1 , Xs k-1 s k , . . . , X s k-1 , Xs k-1 sn )) is smaller than C sup 0≤t≤T,x∈R W p (L( Xx t ), L(X x t )) since W p (L(
) -f ( Xs 1 , . . . , Xs k-1 , X s k-1 , Xs k-1 s k , . . . , X s k-1 , Xs k-1 sn ) ≤ E inf y k ∈R {f ( Xs 1 , . . . , Xs k-1 , y k , X s k ,y k s k+1 , . . . , X s k ,y k sn ) + max k≤j≤n |X s k ,y k s j -X s k , Xs k s j | p } -f ( Xs 1 , . . . , Xs k-1 , X s k-1 , Xs k-1 s k , . . . , X s k-1 , Xs k-1 sn ) ≤ E inf y k ∈R {f k ( Xs 1 , . . . , Xs k-1 , y k ) + C|y k -Xs k | p } -f k ( Xs 1 , . . . , Xs k-1 , X s k-1 , Xs k-1 s k ) ≤ CE W p p (L(X s k-1 ,x s k ), L( Xs k-1 ,x s k ))| x= Xs k-1 ≤ C sup x∈R W p p (L( Xx s k -s k-1 ), L(X x s k -s k-1 )) ≤ C sup 0≤t≤T,x∈R W p p (L( Xx t ), L(X x t )).

C Some properties of diffusion bridges

Let us suppose that the SDE dX t = b(X t )dt+σ(X t )dW t , X 0 = x has a transition density p t (x, y) which is positive and of class C 1,2 with respect to (t, x) ∈ R * + × R. We check later in this section that this holds under Hypothesis 3.1. Then, the law of the diffusion bridge with time horizon T is given by (see for instance Fitzsimmons, Pitman and Yor [START_REF] Fitzsimmons | Markovian bridges: construction, Palm interpretation, and splicing[END_REF])

E[F (X u , 0 ≤ u ≤ t)|X T = y] = E F (X u , 0 ≤ u ≤ t) p T -t (X t , y) p T (x, y) , 0 ≤ t < T ,
where F : C([0, t], R) → R is a bounded measurable function. Indeed for g : R → R measurable and bounded, using that X T has the density p T (x, y) then the Markov property at time t, one checks that

E E F (X u , 0 ≤ u ≤ t) p T -t (X t , y) p T (x, y) y=X T g(X T ) = E F (X u , 0 ≤ u ≤ t) R g(y)p T -t (X t , y)dy = E [F (X u , 0 ≤ u ≤ t)E[g(X T )|X t ]] = E [F (X u , 0 ≤ u ≤ t)g(X T )] .
We thus focus on the change of probability measure

dP y dP Ft = p T -t (X t , y) p T (x, y) =: M t , so that E[F (X u , 0 ≤ u ≤ t)|X T = y] = E y [F (X u , 0 ≤ u ≤ t)
] where E y denotes the expectation with respect to P y . We define ℓ t (x, y) = log p t (x, y). The process (M t ) t∈[0,T ) is a martingale, and by Itô's formula, we get dM t = M t ∂ x ℓ T -t (X t , y)σ(X t )dW t , which gives

M t = exp t 0 ∂ x ℓ T -s (X s , y)σ(X s )dW s - 1 2 t 0 ∂ x ℓ T -s (X s , y) 2 σ(X s ) 2 ds .
Girsanov Theorem then gives that for all y ∈ R, (W y t = W t -t 0 ∂ x ℓ T -s (X s , y)σ(X s )ds) t∈[0,T ) is a Brownian motion under P y , so that (W X T t ) t∈[0,T ) is a Brownian motion independent of X T . Moreover, we have

dX t = [b(X t ) + ∂ x ℓ T -t (X t , y)σ(X t ) 2 ]dt + σ(X t )dW y t , (C.1)
which gives precisely the diffusion bridge dynamics.

Conversely, we would like now to reconstruct the diffusion from the initial and the final value by using diffusion bridges. We have the following result.

Proposition C.1 We consider an SDE dX t = b(X t )dt + σ(X t )dW t , X 0 = x with a transition density p t (x, y) positive and of class C 1,2 on (t, x) ∈ R * + × R. Let (B t , t ≥ 0) be a standard Brownian motion and Z T be a random variable with density p T (x, y) drawn independently from B. We assume that pathwise uniqueness holds for the SDE

dZ x,y t = [b(Z x,y t ) + ∂ x ℓ T -t (Z x,y t , y)σ(Z x,y t ) 2 ]dt + σ(Z x,y t )dB t , Z x,y 0 = x, t ∈ [0, T ), (C.2)
for any x, y ∈ R, and set Z t = Z x,Z T t for t ∈ [0, T ). Then, (Z t ) t∈[0,T ] and (X t ) t∈[0,T ] have the same law.

A consequence of this result is that (Z t , t ∈ [0, T ]) has continuous paths, which gives that lim t→T -Z x,y t = y a.s., dy-a.e. Proof . Let t ∈ [0, T ) and F : C([0, t], R) → R and g : R → R be bounded and measurable functions. Since pathwise uniqueness for the SDE (C.2) implies weak uniqueness, we get

E [F (Z x,y u , 0 ≤ u ≤ t)] = E y [F (X u , 0 ≤ u ≤ t)] = E F (X u , 0 ≤ u ≤ t)
p T -t (X t , y) p T (x, y) .

Thus, we have

E [F (Z u , 0 ≤ u ≤ t)g(Z T )] = E F (X u , 0 ≤ u ≤ t) R p T -t (X t , y)g(y)dy = E [F (X u , 0 ≤ u ≤ t)g(X T )] .
Hence the finite-dimensional marginals of the two processes are equal. Since (X t ) t∈[0,T ] has continuous paths and (Z t ) t∈[0,T ] has càdlàg paths (continuous on [0, T ) with a possible jump at T ), this concludes the proof.

From now on, we assume that Hypothesis 3.1 holds. We introduce the Lamperti transformation of the stochastic process (X t , t ≥ 0). We define ϕ(x) = Proof . It is well-known that we can express the transition density pt (x, ŷ) by using Girsanov theorem as an expectation on a Brownian bridge between x and ŷ. Namely, since α and its derivatives are bounded, we can apply a result stated in Gihman and Skorohod [START_REF] Gihman | Stochastic differential equations. Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF] .

This is a continuous function on R + × R 2 , and we easily conclude by using the dominated convergence theorem and α ∈ C 3 b .

By straightforward calculations, we have Proof . For x, ŷ ∈ R, we consider the following SDE

  s ds, Ēηu,t = 1 + t ηu σ ′ ( Xτs ) Ēηu,τs dW s + t ηu b ′ ( Xτs ) Ēηu,τs ds.

σ

  ′′ ( Xτr )σ( Xτu ) Ēηu,τr Ēηs,τr + σ ′ ( Xτr )D u Ēηs,τr dW r + t ηu b ′′ ( Xτr )σ( Xτu ) Ēηu,τr Ēηs,τr + b ′ ( Xτr )D u Ēηs,τr dr.

2 •

 2 ) and α(y) = b σ -σ ′ ϕ -1 (y), Xt def = ϕ(X t ) so that we haved Xt = α( Xt )dt + dW t , t ∈ [0, T ]. (C.3)By Hypothesis 3.1, ϕ is a C 5 bijection, α ∈ C 3 b and both ϕ and ϕ -1 are Lipschitz continuous. We denote by pt (x, ŷ) the transition density of X and lt (x, ŷ) = log(p t (x, ŷ)).Lemma C.2 The density pt (x, ŷ) is C 1,2 with respect to (t, x) ∈ R * + × R. Besides, we have ∂ x lt (x, ŷ) = ŷx t α(x) + g t (x,ŷ), where g t (x, ŷ) is a continuous function on R + × R 2 such that ∂ xg t (x, ŷ) and ∂ ŷ g t (x, ŷ) exist and ∀T > 0, sup t∈[0,T ], x,ŷ∈R |∂ x g t (x, ŷ)| + |∂ y g t (x, ŷ)| < ∞.

  p t (x, y) = 1 σ(y)pt (ϕ(x), ϕ(y)), and p t (x, y) is thus positive and C 1,2 with respect to (t, x). The diffusion bridge (C.1) is thus well defined. Since ∂ x ℓ t (x, y) = 1 σ(x) ∂ x lt (ϕ(x), ϕ(y)), we get by Itô formula from (C.1)d Xt = [α( Xt ) + ∂ x lT -t ( Xt , ϕ(y))]dt + dW y t , dW y t = dW t -∂ x lT -t ( Xt , ϕ(y))dt. Therefore, as one could expect, the Lamperti transform on the diffusion bridge coincides with the diffusion bridge on the Lamperti transform.Proposition C.3 Let Hypothesis 3.1 hold. There exists a deterministic constant C such that∀T ∈ (0, T ], x, x ′ , y, y ′ ∈ R, sup t∈[0,T ) |Z x,y t -Z x ′ ,y ′ t | ≤ C(|xx ′ | ∨ |yy ′ |),and in particular, pathwise uniqueness holds for (C.2).

  d(|∆ t | ∨ |ŷ -ŷ′ |) ≤ C(|∆ t | ∨ |ŷ -ŷ′ |), for some positive constant C. Gronwall's lemma gives then |∆ t | ≤ e CT (|x -x′ | ∨ |ŷ -ŷ′ |). This gives in particular pathwise uniqueness for (C.4).Now, let us assume that (Z x,y t ) t∈[0,T ) solves (C.2). Then, ϕ(Z x,y t ) solves (C.4) with x = ϕ(x) and ŷ = ϕ(y), and we necessarily have Z x,y t = ϕ -1 ( Ẑϕ(x),ϕ(y) t) by pathwise uniqueness. Both ϕ and ϕ -1 are Lipschitz, and we denote by K a common Lipschitz constant. Then, we get|Z x,y t -Z x ′ ,y ′ t | = |ϕ -1 ( Ẑϕ(x),ϕ(y) t )ϕ -1 ( Ẑϕ(x ′ ),ϕ(y ′ ) t )| ≤ K 2 e CT (|xx ′ | ∨ |yy ′ |),which gives the desired result.

  Proposition 3.4 Let R n be endowed with the norm |(x 1 , . . . , x n )| = max 1≤l≤n |x l |. For any p ≥ 1, there is a constant C not depending on n such that W p (L(X s 1 , . . . , X sn ), L( Xs 1 , . . . , Xsn )) ≤ Cn sup

	0≤t≤T,x∈R

  Xs 1 , . . . , Xsn ), L(X s 1 , . . . , X sn )) ≤ , s ≤ s k ) and using(1.3), next conditioning to σ(W s , s ≤ s k-1 ) and using the dual formulation of the Wasserstein distance, one gets E f ( Xs 1 , . . . , Xs k , X s k , Xs k s k+1 , . . . , X s k , Xs k sn

	n k=1 w k . For f : R n → R a bounded measurable function and
	f (x 1 , . . . , x n ) =	inf (y 1 ,...,yn)∈R n	{f (y 1 , . . . , y n ) + max

1≤j≤n |y jx j | p }, we set f k (x 1 , . . . , x k ) = E(f (x 1 , . . . , x k , X s k ,x k s k+1 , . . . , X s k ,x k sn

)). First choosing (y 1 , . . . , y k-1 , y k+1 , . . . , y n ) = ( Xs 1 , . . . , Xs k-1 , X s k ,y k s k+1 , . . . , X s k ,y k sn ), then conditioning to σ(W s

  (Theorem 1, Chapter 3, § 13) to get that pt (x, ŷ) is positive andClearly, lt (x, ŷ) is C 1,2 in (t, x) ∈ R *+ × R (we can use carefree the dominated convergence theorem for the third term since α ∈ C 3 b ), and we have ′′ + 2αα ′ )(x + W s + s t (ŷx -W t ))ds

	lt (x, ŷ) = -	(x -ŷ) 2 2t	+	ŷ x α(z)dz + log E e -1 2	t 0 (α ′ +α 2 )(x+Ws+ s t (ŷ-x-Wt))ds -	1 2	log(2πt).
	g t (x, ŷ) = -	1 2	E e -1 2	t 0 (α ′ +α 2 )(x+Ws+ s t (ŷ-x-Wt))ds t 0 t (α E e -1 t-s 2 t 0 (α ′ +α 2 )(x+Ws+ s t (ŷ-x-Wt))ds

  T -t ( Ẑ x,ŷ t , ŷ) dt, Ẑ x,ŷ 0 = x, t ∈ [0, T ) (C.4)that corresponds to the diffusion bridge on the Lamperti transform X. We set ∆ t = Ẑ x,ŷ t -Ẑ x′ ,ŷ ′ t for t ∈ [0, T ) and x′ , ŷ′ ∈ R. We haved∆ t = ŷ -ŷ′ -∆ t Tt + g T -t ( Ẑ x,ŷ t , ŷ)g T -t ( Ẑ x′ ,ŷ ′

	d Ẑ x,ŷ t	= dB t +	ŷ -Ẑ x,ŷ t T -t	+ g

t , ŷ′ ) dt, and thus d(|∆ t | ∨ |ŷ -ŷ′ |) = sign(∆ t )1 |∆t|≥|ŷ-ŷ ′ | d∆ t . On the one hand, we observe that 1 |∆t|≥|ŷ-ŷ ′ | [sign(∆ t )(ŷ -ŷ′ ) -|∆ t |] ≤ 0. On the other hand, g t is uniformly Lipschitz w.r.t (x, ŷ) on t ∈ [0, T ] by Lemma C.2, which leads to:

This research benefited from the support of the "Chaire Risques Financiers", Fondation du Risque and the French National Research Agency (ANR) under the program ANR-08-BLAN-0218 BigMC. This research was supported by grants of the Japanese goverment.