Blandine Boisselier

Yannick Marie

Marianne Labussière

Pietro Ciccarino

Virginie Desestret

Xiaowei Wang

Laurent Capelle

Jean-Yves Delattre

Dr Marc Sanson
email: marc.sanson@psl.ap-hop-paris.fr

John Wiley

COLD-PCR HRM: A HIGHLY SENSITIVE DETECTION METHOD FOR IDH1 MUTATION

Keywords: IDH1 mutations, COLD PCR, High Resolution Melting, glioma, diagnostic marker AII: astrocytoma, COLD PCR: co-amplification al lower denaturation temperature PCR, HRM: high resolution melting, IDH1: isocitrate dehydrogenase 1, OAII: oligoastrocytoma, IDH1 R132H mutation, COLD PCR, High Resolution Melting, glioma AII: astrocytoma, COLD PCR: co-amplification al lower denaturation temperature PCR, HRM: high resolution melting, IDH1: isocitrate dehydrogenase 1, OAII: oligoastrocytoma, IDH1 R132H mutation, COLD PCR, High Resolution Melting, glioma

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

Introduction

In recent years, recurrent mutations of the gene encoding isocitrate dehydrogenase 1 (IDH1; MIM# 147700) have been described at a high frequency in gliomas [Balss, et al., 2008;Bleeker, et al., 2009;Hartmann, et al., 2009;Ichimura, et al., 2009;Parsons, et al., 2008;Sanson, et al., 2009;Yan, et al., 2009]. Mutations in the IDH1 gene always affect the codon 132, which in more than 90% of cases result in a substitution of arginine by a histidine residue, IDH1 R132H [Ichimura, et al., 2009;Sanson, et al., 2009;Yan, et al., 2009]. Since it is almost exclusively found in gliomas, IDH1 mutation has a strong diagnostic value [Yan, et al., 2009]. IDH1 mutation is also a powerful independent prognostic factor in gliomas [Dubbink, et al., 2009;Nobusawa, et al., 2009;Sanson, et al., 2009]: patients with an IDH1 R132H mutated tumor having a longer survival when compared with those non mutated for the same grade. Given its diagnostic and prognostic potential, the determination of IDH1 status in glioma samples is gaining increasing attention in clinical practice.

Conventional PCR followed by Sanger sequencing is currently the gold standard in identifying IDH1 mutations in tumor DNA. Typically, clinical tumor samples are contaminated by normal tissue, thereby diluting the total amount of genetic that is present, and techniques are often required to identify low level alterations within an excess of wildtype DNA. This is particularly true for gliomas which are highly infiltrating tumors [Furnari, et al., 2007]. Conventional PCR enables mutation detection from a very small amount of sample but does not selectively amplify the mutant sequences. Therefore, unless the mutation exceeds a 20-25% abundance relative to wild-type alleles, conventional PCR followed by downstream methods such as Sanger sequencing will fail to detect mutations in clinical samples [Li, et al., 2009;Zuo, et al., 2009]. Enrichment methods are thus necessary to increase the mutant DNA /wild type DNA ratio. Tumor-cell enrichment methods, such as cell sorting or microdissection, are expensive and time-consuming [Zuo, et al., 2009], and therefore inadequate for clinical routine use.

Recently, co-amplification at lower denaturation temperature PCR (COLD PCR) has been described as a powerful method to identify low-level mutations in the TP53 gene [Li, et al., 2008;Milbury, et al., 2009]. This approach uses a critical temperature (Tc) during the PCR process in order to enrich mutations at any position of the amplified sequence. Indeed, a single nucleotide mismatch anywhere along a double-stranded DNA sequence generates a small change in the melting temperature for that sequence, with mutated sequences melting at a lower temperature than wild-type sequences [Li, et al., 2008]. During COLD PCR, the denaturation temperature is set to Tc, thereby mutation-containing sequences are preferentially denatured and available for primers binding and subsequent amplification. High Resolution DNA Melting (HRM) analysis is a relatively new and rapid method for detecting DNA sequence variants following an initial amplification [Erali, et al., 2008;Vossen, et al., 2009;Wittwer, 2009]. Gene scanning by HRM depends on the recognition of changes in the shape of the amplicon melting curve that result from heterozygous sequence alterations [Erali, et al., 2008]. At the end of PCR amplification, samples are heated to 95°C to ensure a complete denaturation. Next, the temperature is reduced to allow cross-hybridization between wild-type and mutant sequences, leading to the formation of mismatched heteroduplexes, which melt at a lower temperature than wild-type and mutant homoduplex [Vossen, et al., 2009]. In this study, we evaluated the sensitivity of different COLD PCR HRM assays for the detection of IDH1 R132H mutation in glioma samples.

Material and methods

DNA and tumors samples

Tumor samples were selected from the Pitié-Salpêtrière brain tumor database. Tumor DNA was extracted from frozen tissues using the QIAmp DNA according to the manufacturer instructions (Qiagen). Samples used in this work have previously been tested for the presence of IDH1 mutation (reference sequence NM_005896.2) by direct sequencing, as already described [Parsons, et al., 2008;Yan, et al., 2009].

Determination of T c

We applied the fast COLD PCR assays for detecting IDH1 mutation based on the methodology described in the literature [Li, et al., 2008], with several modifications. We first determined the new denaturation temperature Tc for the reaction. We used the same primers as for conventional PCR assay, which produced an amplicon of 172 bp. To identify the optimal critical denaturation temperature Tc, a series of COLD PCR reactions at graded temperatures below Tm were performed. We then set the Tc at 81°C, the lowest temperature that reproducibly yielded a substantial PCR. In fast COLD PCR, the reaction protocol began with 20 cycles of conventional amplification for an initial build-up of all amplicons, followed by 30 COLD PCR cycles to selectively enrich for Tm-reduced mutant sequences (Figure 1). COLD PCR assays were performed on a LightCycler480 (Roche Diagnostics Corporation) to ensure a precise control of temperature during the experiments.

Comparison of sensitivities of the different assays to detect IDH1 mutation

To determine if COLD PCR could enhance the sensitivity of IDH1 R132H mutation detection, we compared the sensitivity of three experimental protocols. Exon 4 of the IDH1 To compare the sensitivity of the different assays, we performed a serial dilution study using a gDNA sample from a grade II oligoastrocytoma patient containing an IDH1 R132H mutation as the source of the mutant allele. This mutation-containing gDNA sample was serially diluted into wild-type gDNA (ie blood constitutive gDNA from the same patient) to the following percentages: 25%, 10%, 8%, 5%, 4%, 2%, 1%, 0.5%, 0.25%, 0.1%, and 0.05%.

In addition, wild-type gDNAs (n=7) were included in each experiment. The same experiment was reproduced with several other mutated DNA samples: another CGT CAT mutation (Arg132His), and also with other, less frequent mutations, such as CGT CTT (Arg132Leu) and CGT AGT (Arg132Ser).

First round of amplification

Each PCR amplification reaction contained 50ng gDNA. PCR cycling conditions for the first round consisted of an initial denaturation step at 94°C for 5 min; 40 cycles of 94°C for 30s; 60°C for 1 min; and 72°C for 1 min30; and final extension at 72°C for 7 min. The reactions were carried out using a Mastercycler (Eppendorf). Conventional PCR reactions contained final reagent concentrations as follows: 1x PCR Master Mix (Abgene), 0.25 µM forward and reverse primers (Invitrogen; Table I) and DNA template. COLD PCR cycling conditions on LightCycler480 (Roche Diagnostics Corporation) are summarized as follows: 96°C, 10 min; 20 cycles of 95°C, 15s; 60°C, 30s, then 30 cycles of 81°C, 15s; 60°C, 30s.

Second round of amplification

PCR amplification was performed with the LightCycler480 (Roche Diagnostics Corporation). Each reaction contained diluted PCR amplicons (1/1000), 0.25 µM forward and reverse primers (Invitrogen, Table I), 3 mM MgCl 2 and LightCycler480 HRM Master (Roche Diagnostics Corporation). PCR HRM cycling conditions were as follows: 96°C, 10min; 40 cycles of 95°C, 30s; 60°C, 30s. COLD PCR HRM cycling conditions are summarized as follows: 96°C , 10min; 20 cycles of 95°C, 15s; 60°C, 30s, then 30 cycles of 81°C (Tc), 15s; 60°C, 30s. After amplification, a post amplification melting curve program was initiated by heating to 95°C for 1 min, cooling to 40°C for 1 min, and increasing the temperature to 95°C while continuously measuring fluorescence at 25 acquisitions per degree. Each PCR run contained a negative (no template) control and each amplification was duplicated.

High Resolution Melting (HRM) analysis and direct sequencing

At the end of the second round of amplification, fluorescent melting curves were analyzed usingLC480 Gene Scanning software V1.2.9 (Roche Diagnostics Corporation). All curves were analyzed following normalization, temperature shifting, automated grouping, and the inspection of difference plots. The grouping software uses a curve shape-matching algorithm in order to identify wild type from mutant samples. The 0.2 value was chosen for the grouping sensitivity in all experiments. Products of each assay were then submitted to the sequencing reaction using the BigDye Terminator v3.1 Sequencing Kit (Applied Biosystems) as previously described [Sanson, et al., 2009]. After a purification step using BigDye Xterminator Purification Kit (Applied Biosystems), both forward and reverse sequences were determined on an ABI prism 3730 DNA analyzer (Perkin Elmer).

Tumor sample selection and immunohistochemical detection of IDH1 R132H

Tumors were retrieved from the neuropathology department database according to the following criteria: histological diagnosis of WHO grade II gliomas; presence of IDH1 R132H mutation confirmed by Sanger sequencing; availability of frozen and paraffin embedded samples of tumor core and tumor edge, with edge biopsies considered as free of infiltrated tumor cells by standard HE staining.

Immunohistochemical staining for IDH1 R132H was performed on 4 µm paraffin sections of formalin-fixed tumor samples using mouse monoclonal anti-R132H-IDH1 antibody culture supernatant ( a generous gift from Pr A. von Deimling), as previously published [Capper, et al., 2010;Capper, et al., 2009]. Local immunohistochemistry protocol was validated on gliomas samples which were previously analysed by Sanger sequencing for IDH1 R132H mutation (positive and negative controls). Labelling was defined as positive (at least one cluster of positive tumor cells) or negative (no positive tumor cells detected).

Results

Determination of the sensitivity with dilution studies

To identify the most sensitive method for detecting IDH1 mutation, we evaluated two techniques after a first stage of conventional PCR: the PCR HRM assay and the COLD PCR HRM assay. In a third assay, we replaced the first conventional PCR by a run of COLD PCR (Figure 2). The sensitivity of the three assays was compared using a dilution series obtained by mixing DNA from heterozygous positive control carrying the IDH1 R132H mutation with wild-type DNA from peripheral blood.

Conventional PCR HRM assay detects mutant DNA at a concentration of 25% in a background of wild-type DNA (Figure 3A). In contrast, the COLD PCR HRM assay detects IDH1 R132H mutation at a much lower concentration: mutant DNA diluted into wild-type DNA to a 2% abundance was still clearly differentiated from normal sequences (Figure 3B). Therefore, the COLD PCR HRM assay produced an approximately 10-fold improvement in the IDH1 mutation detection, as compared to the conventional PCR HRM analysis. Replacing the first conventional PCR amplification by a run of COLD PCR further enhanced selective enrichment of the mutant DNA and improved the detection of IDH1 R132H mutation. In these conditions, we were able to detect mutant DNA at a concentration as low as 0.25% in mixture with wild-type DNA (Figure 3C). Thus, the double COLD PCR HRM assay for the detection of IDH1 R132H mutation was 100-fold more sensitive than the PCR HRM assay.

To both confirm the selective mutation enrichment and to exclude false positive results, amplicons produced at the end of the three assays were submitted to direct sequencing using the Sanger method. The sequencing chromatograms are presented in Figure 3D. In all three cases, direct sequencing confirmed the presence of the IDH1 R132H mutation. After II).

IDH1 R132H
immunohistochemistry was positive in 3 cases (30%, 2 grade II oligodendrogliomas, and one grade II oligoastrocytoma) showing a cytoplasmic granular staining as previously described [Capper, et al., 2010]. Figure 4A

Discussion

Gliomas are characterized by a highly invasive phenotype, with tumor cells invading the brain at a distance far from the bulk of tumor [Furnari, et al., 2007]. Therefore biopsy samples are often contaminated by normal brain tissue, and mutated DNA is diluted in a the background of wild-type DNA from surrounding brain tissue, vasculature, and infiltrating lymphocytes [Dobrowolski, et al., 2009]. Because IDH1 R132H mutation is almost restricted to gliomas [Yan, et al., 2009], it has a major diagnostic potential. Therefore a fast and reliable scanning technique to detect this mutation in clinical samples with very few tumor cells (and appearing therefore normal at neuropathological analysis) is particularly important. In addition IDH1 R132H mutation is a major prognostic factor [Sanson, et al., 2009;van den Bent, et al., 2010]. Simple HRM has been very recently reported as a fast and sensitive strategy, detecting IDH1 and IDH2 mutations in a 90% normal DNA background [Horbinski, et al., 2010]. In this study, we used fast COLD PCR to selectively amplify IDH1 R132H mutated DNA and we showed that double COLD PCR HRM assay is a highly reliable method to detect IDH1 mutation in samples with very few tumor cells (up to 0.25% mutated/non mutated DNA). In addition, this assay is fast (less than three hours) and therefore particularly suitable for routine diagnosis purposes in neuropathology.

Despite IDH1 R132H accounts for more than 90% of IDH1 mutations reported in gliomas, other mutations have been described: CGT→AGT, CGT→CTT CGT→GGT, CGT→TGT [Balss, et al., 2008;Gravendeel, et al., 2010;Sanson, et al., 2009;Yan, et al., 2009]. Fortunately, all these mutations, except the CGT→GGT change, result in a lower Tm, mutations-could be performed, though the selective amplification ratio will probably be lower than in fast COLD PCR, because amplification, and thus enrichment in mutant sequences, begins earlier during cycling in fast COLD PCR than in full COLD PCR [Li, et al., 2008;Milbury, et al., 2009].

We showed that double COLD PCR HRM (followed by sequencing in case of aberrant HRM profiles) is a powerful method to evidence the presence of tumor cells with IDH1 mutation in apparently "blank" biopsies of grade II gliomas, and is much more sensitive than immunohistochemistry for IDH1 R132H . Such procedure may be particularly useful in lesions radiologically highly suggestive of glioma, whose biopsy appears non contributive because of the very low proportion of tumor cells. In such cases, finding an IDH1 mutation will confirm the diagnosis of glioma, avoiding a further invasive procedure to establish diagnosis.

Moreover, such approach could also be suitable for IDH1 R132H mutation detection in the cerebral spinal fluid thus confirming glioma diagnosis without the need for surgery.

MUTATION

Blandine Boisselier 1,2,3 , Yannick Marie 1,2,3 , Marianne Labussière 1,2,3 , Pietro Ciccarino 1,2,3 , Virginie Desestret 1,2,3,5 , XiaoWei Wang 1,2,3 , Laurent Capelle 4 , Jean-Yves Delattre 1,2,3,6 , Marc

Introduction

In recent years, recurrent mutations of the gene encoding isocitrate dehydrogenase 1 (IDH1; MIM# 147700) have been described at a high frequency in gliomas [Balss, et al., 2008;Bleeker, et al., 2009;Hartmann, et al., 2009;Ichimura, et al., 2009;Parsons, et al., 2008;Sanson, et al., 2009;Yan, et al., 2009]. Mutations in the IDH1 gene always affect the codon 132, which in more than 90% of cases result in a substitution of arginine by a histidine residue, IDH1 R132H [Ichimura, et al., 2009;Sanson, et al., 2009;Yan, et al., 2009]. Since it is almost exclusively found in gliomas, IDH1 mutation has a strong diagnostic value [Yan, et al., 2009]. IDH1 mutation is also a powerful independent prognostic factor in gliomas [Dubbink, et al., 2009;Nobusawa, et al., 2009;Sanson, et al., 2009]: patients with an IDH1 R132H mutated tumor having a longer survival when compared with those non mutated for the same grade. Given its diagnostic and prognostic potential, the determination of IDH1 status in glioma samples is gaining increasing attention in clinical practice.

Conventional PCR followed by Sanger sequencing is currently the gold standard in identifying IDH1 mutations in tumor DNA. Typically, clinical tumor samples are contaminated by normal tissue, thereby diluting the total amount of genetic that is present, and techniques are often required to identify low level alterations within an excess of wildtype DNA. This is particularly true for gliomas which are highly infiltrating tumors [Furnari, et al., 2007]. Conventional PCR enables mutation detection from a very small amount of sample but does not selectively amplify the mutant sequences. Therefore, unless the mutation exceeds a 20-25% abundance relative to wild-type alleles, conventional PCR followed by downstream methods such as Sanger sequencing will fail to detect mutations in clinical samples [Li, et al., 2009;Zuo, et al., 2009]. Enrichment methods are thus necessary to increase the mutant DNA /wild type DNA ratio. Tumor-cell enrichment methods, such as cell sorting or microdissection, are expensive and time-consuming [Zuo, et al., 2009], and therefore inadequate for clinical routine use.

Recently, co-amplification at lower denaturation temperature PCR (COLD PCR) has been described as a powerful method to identify low-level mutations in the TP53 gene [Li, et al., 2008;Milbury, et al., 2009]. This approach uses a critical temperature (Tc) during the PCR process in order to enrich mutations at any position of the amplified sequence. Indeed, a single nucleotide mismatch anywhere along a double-stranded DNA sequence generates a small change in the melting temperature for that sequence, with mutated sequences melting at a lower temperature than wild-type sequences [Li, et al., 2008]. During COLD PCR, the denaturation temperature is set to Tc, thereby mutation-containing sequences are preferentially denatured and available for primers binding and subsequent amplification. High

Resolution DNA Melting (HRM) analysis is a relatively new and rapid method for detecting DNA sequence variants following an initial amplification [Erali, et al., 2008;Vossen, et al., 2009;Wittwer, 2009]. Gene scanning by HRM depends on the recognition of changes in the shape of the amplicon melting curve that result from heterozygous sequence alterations [Erali, et al., 2008]. At the end of PCR amplification, samples are heated to 95°C to ensure a complete denaturation. Next, the temperature is reduced to allow cross-hybridization between wild-type and mutant sequences, leading to the formation of mismatched heteroduplexes, which melt at a lower temperature than wild-type and mutant homoduplex [Vossen, et al., 2009]. In this study, we evaluated the sensitivity of different COLD PCR HRM assays for the detection of IDH1 R132H mutation in glioma samples.

Material and methods

DNA and tumors samples

Tumor samples were selected from the Pitié-Salpêtrière brain tumor database. Tumor DNA was extracted from frozen tissues using the QIAmp DNA according to the manufacturer instructions (Qiagen). Samples used in this work have previously been tested for the presence of IDH1 mutation (reference sequence NM_005896.2) by direct sequencing, as already described [Parsons, et al., 2008;Yan, et al., 2009].

Determination of T c

We applied the fast COLD PCR assays for detecting IDH1 mutation based on the methodology described in the literature [Li, et al., 2008], with several modifications. We first determined the new denaturation temperature Tc for the reaction. We used the same primers as for conventional PCR assay, which produced an amplicon of 172 bp. To identify the optimal critical denaturation temperature Tc, a series of COLD PCR reactions at graded temperatures below Tm were performed. We then set the Tc at 81°C, the lowest temperature that reproducibly yielded a substantial PCR. In fast COLD PCR, the reaction protocol began with 20 cycles of conventional amplification for an initial build-up of all amplicons, followed by 30 COLD PCR cycles to selectively enrich for Tm-reduced mutant sequences (Figure 1).

COLD PCR assays were performed on a LightCycler480 (Roche Diagnostics Corporation) to ensure a precise control of temperature during the experiments.

Comparison of sensitivities of the different assays to detect IDH1 mutation

To determine if COLD PCR could enhance the sensitivity of IDH1 R132H mutation detection, we compared the sensitivity of three experimental protocols. Exon 4 of the IDH1 To compare the sensitivity of the different assays, we performed a serial dilution study using a gDNA sample from a grade II oligoastrocytoma patient containing an IDH1 R132H mutation as the source of the mutant allele. This mutation-containing gDNA sample was serially diluted into wild-type gDNA (ie blood constitutive gDNA from the same patient) to the following percentages: 25%, 10%, 8%, 5%, 4%, 2%, 1%, 0.5%, 0.25%, 0.1%, and 0.05%.

In addition, wild-type gDNAs (n=7) were included in each experiment. The same experiment was reproduced with several other mutated DNA samples: another CGT CAT mutation (Arg132His), and also with other, less frequent mutations, such as CGT CTT (Arg132Leu)

and CGT AGT (Arg132Ser).

First round of amplification

Each PCR amplification reaction contained 50ng gDNA. PCR cycling conditions for the first round consisted of an initial denaturation step at 94°C for 5 min; 40 cycles of 94°C for 30s; 60°C for 1 min; and 72°C for 1 min30; and final extension at 72°C for 7 min. The reactions were carried out using a Mastercycler (Eppendorf). Conventional PCR reactions contained final reagent concentrations as follows: 1x PCR Master Mix (Abgene), 0.25 µM forward and reverse primers (Invitrogen; Table I) and DNA template. COLD PCR cycling conditions on LightCycler480 (Roche Diagnostics Corporation) are summarized as follows:

96°C, 10 min; 20 cycles of 95°C, 15s; 60°C, 30s, then 30 cycles of 81°C, 15s; 60°C, 30s.

Second round of amplification

PCR amplification was performed with the LightCycler480 (Roche Diagnostics Corporation). Each reaction contained diluted PCR amplicons (1/1000), 0.25 µM forward and reverse primers (Invitrogen, Table I), 3 mM MgCl 2 and LightCycler480 HRM Master (Roche Diagnostics Corporation). PCR HRM cycling conditions were as follows: 96°C, 10min; 40 cycles of 95°C, 30s; 60°C, 30s. COLD PCR HRM cycling conditions are summarized as follows: 96°C , 10min; 20 cycles of 95°C, 15s; 60°C, 30s, then 30 cycles of 81°C (Tc), 15s; 60°C, 30s. After amplification, a post amplification melting curve program was initiated by heating to 95°C for 1 min, cooling to 40°C for 1 min, and increasing the temperature to 95°C while continuously measuring fluorescence at 25 acquisitions per degree. Each PCR run contained a negative (no template) control and each amplification was duplicated.

High Resolution Melting (HRM) analysis and direct sequencing

At the end of the second round of amplification, fluorescent melting curves were analyzed usingLC480 Gene Scanning software V1.2.9 (Roche Diagnostics Corporation). All curves were analyzed following normalization, temperature shifting, automated grouping, and the inspection of difference plots. The grouping software uses a curve shape-matching algorithm in order to identify wild type from mutant samples. The 0.2 value was chosen for the grouping sensitivity in all experiments.

Results

Determination of the sensitivity with dilution studies

To identify the most sensitive method for detecting IDH1 mutation, we evaluated two techniques after a first stage of conventional PCR: the PCR HRM assay and the COLD PCR HRM assay. In a third assay, we replaced the first conventional PCR by a run of COLD PCR (Figure 2). The sensitivity of the three assays was compared using a dilution series obtained by mixing DNA from heterozygous positive control carrying the IDH1 R132H mutation with wild-type DNA from peripheral blood.

Conventional PCR HRM assay detects mutant DNA at a concentration of 25% in a background of wild-type DNA (Figure 3A). In contrast, the COLD PCR HRM assay detects IDH1 R132H mutation at a much lower concentration: mutant DNA diluted into wild-type DNA to a 2% abundance was still clearly differentiated from normal sequences (Figure 3B).

Therefore, the COLD PCR HRM assay produced an approximately 10-fold improvement in the IDH1 mutation detection, as compared to the conventional PCR HRM analysis. Replacing the first conventional PCR amplification by a run of COLD PCR further enhanced selective enrichment of the mutant DNA and improved the detection of IDH1 R132H mutation. In these conditions, we were able to detect mutant DNA at a concentration as low as 0.25% in mixture with wild-type DNA (Figure 3C). Thus, the double COLD PCR HRM assay for the detection of IDH1 R132H mutation was 100-fold more sensitive than the PCR HRM assay.

To both confirm the selective mutation enrichment and to exclude false positive results, amplicons produced at the end of the three assays were submitted to direct sequencing using the Sanger method. The sequencing chromatograms are presented in Figure 3D. In all three cases, direct sequencing confirmed the presence of the IDH1 R132H mutation. After II).

IDH1 R132H
immunohistochemistry was positive in 3 cases (30%, 2 grade II oligodendrogliomas, and one grade II oligoastrocytoma) showing a cytoplasmic granular staining as previously described [Capper, et al., 2010].

Discussion

Gliomas are characterized by a highly invasive phenotype, with tumor cells invading the brain at a distance far from the bulk of tumor [Furnari, et al., 2007]. Therefore biopsy samples are often contaminated by normal brain tissue, and mutated DNA is diluted in a the background of wild-type DNA from surrounding brain tissue, vasculature, and infiltrating lymphocytes [Dobrowolski, et al., 2009]. Because IDH1 R132H mutation is almost restricted to gliomas [Yan, et al., 2009], it has a major diagnostic potential. Therefore a fast and reliable scanning technique to detect this mutation in clinical samples with very few tumor cells (and appearing therefore normal at neuropathological analysis) is particularly important. In addition IDH1 R132H mutation is a major prognostic factor [Sanson, et al., 2009;van den Bent, et al., 2010]. Simple HRM has been very recently reported as a fast and sensitive strategy, detecting IDH1 and IDH2 mutations in a 90% normal DNA background [Horbinski, et al., 2010]. In this study, we used fast COLD PCR to selectively amplify IDH1 R132H mutated DNA and we showed that double COLD PCR HRM assay is a highly reliable method to detect IDH1 mutation in samples with very few tumor cells (up to 0.25% mutated/non mutated DNA). In addition, this assay is fast (less than three hours) and therefore particularly suitable for routine diagnosis purposes in neuropathology.

Despite IDH1 R132H accounts for more than 90% of IDH1 mutations reported in gliomas, other mutations have been described: CGT→AGT, CGT→CTT CGT→GGT, CGT→TGT [Balss, et al., 2008;Gravendeel, et al., 2010;Sanson, et al., 2009;Yan, et al., 2009]. Fortunately, all these mutations, except the CGT→GGT change, result in a lower Tm, rendering them theoretically detectable by our simple double COLD PCR assay. Indeed we were able to detect some of these less frequent mutations (supplementary figure mutations-could be performed, though the selective amplification ratio will probably be lower than in fast COLD PCR, because amplification, and thus enrichment in mutant sequences, begins earlier during cycling in fast COLD PCR than in full COLD PCR [Li, et al., 2008;Milbury, et al., 2009].

We showed that double COLD PCR HRM (followed by sequencing in case of aberrant HRM profiles) is a powerful method to evidence the presence of tumor cells with IDH1 mutation in apparently "blank" biopsies of grade II gliomas, and is much more sensitive than immunohistochemistry for IDH1 R132H . Such procedure may be particularly useful in lesions radiologically highly suggestive of glioma, whose biopsy appears non contributive because of the very low proportion of tumor cells. In such cases, finding an IDH1 mutation will confirm the diagnosis of glioma, avoiding a further invasive procedure to establish diagnosis.

Moreover, such approach could also be suitable for IDH1 R132H mutation detection in the cerebral spinal fluid thus confirming glioma diagnosis without the need for surgery.

  amplified either by conventional PCR or by fast COLD PCR. A second round of amplification was then performed either by PCR HRM, or by fast COLD PCR HRM (Figure 2).

  contained final reagent concentrations as follows: LightCycler480 HRM Master (Roche Diagnostics Corporation), 0.25 µM forward and reverse primers (Invitrogen), 3 mM MgCl 2 and DNA template.

  HRM with 0.25% of mutant DNA/wild DNA, IDH1 R132H mutation was evident as clearly as for 25% with conventional PCR HRM and 2% with COLD PCR HRM.The same results were obtained with another glioma with CGT CAT mutation (Arg132His), and also with less frequent IDH1 132 mutations CGT CTT (Arg132Leu) and CGT AGT (Arg132Ser) (supplementary figure).Comparison of PCR HRM, COLD PCR HRM and double COLD PCR HRM and IDH1 R132H immunohistochemistry in biopsies of tumor edges.To definitely validate the clinical interest of our technique, we selected 10 pairs of tumor core and tumor edge (one astrocytoma, 5 oligodendrogliomas, and 4 oligoastrocytomas), and performed IDH1 R132H mutation detection by both immunohistochemistry and molecular techniques (Table

  rendering them theoretically detectable by our simple double COLD PCR assay. Indeed we were able to detect some of these less frequent mutations (supplementary figure). For the detection of CGT→GGT, a full COLD PCR assay -allowing the enrichment of all possible

Figure 1 :

 1 Figures legends

Figure 2 :

 2 Figure 2: Assays performed to detect IDH1 R132H mutation in tumor samples.

Figure 3 :

 3 Figure 3: Comparison of conventional PCR HRM (A), COLD HRM (B) and double COLD HRM (C) for the detection of IDH1 R132H mutation. Mutant DNA was

Figure 4 :

 4 Figure 4: Detection of IDH1 R132H mutation in grade II gliomas edges. A. IDH1 R132H

  Pierre et Marie Curie-Paris 6, Centre de Recherche de l'Institut du Cerveau et de la Moëlle épinière (CRICM) UMR-S975, Paris 75013, France 2. INSERM U 975, Paris 75013, France 3. CNRS, UMR 7225, Paris 75013, France 4. AP-HP, Groupe Hospitalier Pitié-Salpêtrière, Department of Neurosurgery, Paris 75013, France 5. AP-HP, Groupe Hospitalier Pitié-Salpêtrière, Laboratoire de Neuropathologie R. Escourolle, Paris 75013, France. 6. AP-HP, Groupe Hospitalier Pitié-Salpêtrière, Service de Neurologie 2, Paris 75013, France

  amplified either by conventional PCR or by fast COLD PCR. A second round of amplification was then performed either by PCR HRM, or by fast COLD PCR HRM (Figure 2).

  contained final reagent concentrations as follows: LightCycler480 HRM Master (Roche Diagnostics Corporation), 0.25 µM forward and reverse primers (Invitrogen), 3 mM MgCl 2 and DNA template.

  HRM with 0.25% of mutant DNA/wild DNA, IDH1 R132H mutation was evident as clearly as for 25% with conventional PCR HRM and 2% with COLD PCR HRM.The same results were obtained with another glioma with CGT CAT mutation (Arg132His), and also with less frequent IDH1 132 mutations CGT CTT (Arg132Leu) and CGT AGT (Arg132Ser) (supplementary figure).Comparison of PCR HRM, COLD PCR HRM and double COLD PCR HRM and IDH1 R132H immunohistochemistry in biopsies of tumor edgesTo definitely validate the clinical interest of our technique, we selected 10 pairs of tumor core and tumor edge (one astrocytoma, 5 oligodendrogliomas, and 4 oligoastrocytomas), and performed IDH1 R132H mutation detection by

  Figure 4A presents the results of immunohistochemistry staining for patient 8 in the tumor core and in the biopsy edge (lower panel). PCR HRM was positive in 8 cases out of 10 (80.0%; Figure 4B upper panel). In contrast, both COLD PCR HRM (data not shown) and double COLD PCR HRM were positive in all cases (100%; Figure 4B, lower panel, and table II).

Figure 1 :

 1 Figures legends

Figure 2 :

 2 Figure 2: Assays performed to detect IDH1 R132H mutation in tumor samples.

Figure 3 :

 3 Figure 3: Comparison of conventional PCR HRM (A), COLD HRM (B) and double COLD HRM (C) for the detection of IDH1 R132H mutation. Mutant DNA was

Figure 4 :

 4 Figure 4: Detection of IDH1 R132H mutation in grade II gliomas edges. A. IDH1 R132H

  Figure 1 167x64mm (300 x 300 DPI) Page 38 of 43

Table I :

 I Primers used for first and second rounds of amplification

			Human Mutation	Page 36 of 43
				Amplicon
		Primer
				length
	First	Forward	CGGTCTTCAGAGAAGCCATT
	amplification	Reverse	CACATACAAGTTGGAAATTTCTGG	172 bp
	Second amplification	Forward Reverse F o r	CGGTCTTCAGAGAAGCCATT GCAAAATCACATTATTGCCAAC	129 bp
		P
		e e r
			R e v i e w
		John Wiley & Sons, Inc.

Table II : Comparison of IDH1 R132 immunohistochemistry with PCR HRM and double COLD PCR HRM for the detection of IDH1 R132H mutation in tumor edges.

 II For all techniques, results are expressed as positive (+) or negative(-). COLD PCR HRM (data not shown) gave the same results as double COLD PCR HRM.

John Wiley & Sons, Inc.Human Mutation

Acknowledgements

Work supported by the Association pour la Recherche sur les Tumeurs Cérébrales (ARTC) and the Association pour la Recherche contre le Cancer (ARC). ML is supported by a grant from the Association pour la Recherche contre le Cancer (ARC). We are indebted to Pr A von Deimling for providing the mouse monoclonal anti-R132H-IDH1 antibody and to Alice Brunker for editing the manuscript.

Acknowledgements

Work supported by the Association pour la Recherche sur les Tumeurs Cérébrales (ARTC) and the Association pour la Recherche contre le Cancer (ARC). ML is supported by a grant from the Association pour la Recherche contre le Cancer (ARC). We are indebted to Pr A von Deimling for providing the mouse monoclonal anti-R132H-IDH1 antibody and to Alice Brunker for editing the manuscript.

Products of each assay were then submitted to the sequencing reaction using the BigDye Terminator v3.1 Sequencing Kit (Applied Biosystems) as previously described [Sanson, et al., 2009]. After a purification step using BigDye Xterminator Purification Kit (Applied Biosystems), both forward and reverse sequences were determined on an ABI prism 3730 DNA analyzer (Perkin Elmer).

Tumor sample selection and immunohistochemical detection of IDH1 R132H

Tumors were retrieved from the neuropathology department database according to the following criteria: histological diagnosis of WHO grade II gliomas; presence of IDH1 R132H mutation confirmed by Sanger sequencing; availability of frozen and paraffin embedded samples of tumor core and tumor edge, with edge biopsies considered as free of infiltrated tumor cells by standard HE staining.

Immunohistochemical staining for IDH1 R132H was performed on 4 µm paraffin sections of formalin-fixed tumor samples using mouse monoclonal anti-R132H-IDH1 antibody culture supernatant ( a generous gift from Pr A. von Deimling), as previously published [Capper, et al., 2010;Capper, et al., 2009]. Local immunohistochemistry protocol was validated on gliomas samples which were previously analysed by Sanger sequencing for IDH1 R132H mutation (positive and negative controls). Labelling was defined as positive (at least one cluster of positive tumor cells) or negative (no positive tumor cells detected).


Jean-Louis Marchand

Conditioning diffusions with respect to partial observations

Keywords: Conditioned diffusion, Partial observations, Simulation

In this paper, we prove a result of equivalence in law between a diffusion conditioned with respect to partial observations and an auxiliary process. By partial observations we mean coordinates (or linear transformation) of the process at a finite collection of deterministic times. Apart from the theoritical interest, this result allows to simulate the conditioned diffusion through Monte Carlo's method, using the fact that the auxiliary process is easy to simulate.

Introduction

We are interessed in multidimensional diffusions solutions of stochastic differential equations (SDE's) generated by a Brownian motion. For a n-dimensional diffusion solution on [0, T ] of the following

dx t = b t (x t )dt + σ t (x t )dw t , x 0 = u (1)
where w is a n-dimensional Brownian motion, it is known (see e.g. [START_REF] Lyons | On conditional diffusion processes[END_REF]) that its conditional law L (x|x T = v) is given by the law of a bridge process (as extension of Brownian bridge) y solution of dy t = b t (y t )dt + σ t (y t )d wt + σ t (y t )σ t (y t ) * ∇ z log p t,T (z, v) z=yt dt, y 0 = u where w is a Brownian motion and p s,t (z, .) is the density of x t knowing x s = z. But in most cases this density is not explicitely known so that we are not able to simulate it easily. For practical purposes, e.g. parameter estimation of diffusion processes, simulation of paths corresponding to the conditional law is needed.

In their paper [START_REF] Bernard | Simulation of conditioned diffusions and applications to parameter estimations[END_REF], B.Delyon and Y.Hu studied the following equation on [0, T ]

dy t = b t (y t )dt - y t -v T -t dt + σ t (y t )d wt , y 0 = u. ( 2
)
where w is a n-dimensional Brownian motions. Under adequate assumptions, the process y is unique on [0, T ], lim t→T y t = v, a.s. and for all positive function f in C([0, T ], R n ) we have

E[f (x)|x T = v] = CE[f (y)R(y)]
where R is a functional of whole path y on [0, T ]. The quantity R(y) is computable knowing parameters b, σ, T and v. The constant C is unknown, but in practice the conditional law is estimated through

E[f (x)|x T = v] ≃ i f (y i )R(y i ) i R(y i )
where each y i is an independant sample of [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. In this case, we call the process y a bridge even if y does not have the right targeted law. If b = 0 and σ = I n (identity n-dimensional matrix), the process x is a n-dimensional Brownian motion and process y is a n-dimensional Brownian bridge so that C = R = 1. This theorem applies in the case of more than one observation. The Markov property indeed implies that the conditional law is the tensor product of each bridge.

The aim of this paper is to extend this result to solve this problem with only partial observations. The previous remark does not apply; indeed we have to treat simultaneously all conditionings . To give an idea, let w = ( w 1 w 2 ) be a 2-dimensional Brownian motion. The law of w conditioned on w 1 S = u and w 2 T = v with S < T is given by that one of y solution of dy t = d wt -

y 1 t -u S-t 1 t<S y 2 t -v T -t 1 t<T
dt, y 0 = u each coordinate is a Brownian bridge. Let us define our observations. At each deterministic positive observation time of the sequence 0 < T 1 < • • • < T k < • • • < T N = T , we get a partial information given by a linear transformation of x T k , L k x T k , where L k is a deterministic matrix in M m k ,n (R) whose m k rows form an orthonormal family. So that our aim is to be able to describe the conditional law L (x|(L k x T k = v k ) 1≤k≤N ) where v k is an arbitrary deterministic m k -dimensional vector.

We define process y to be the solution of dy t = b t (y t )dt + σ t (y t )d wt -

N k=1 P k t (y t ) yt-u k T k -t 1 (T k -ε k ,T k ) (t)dt y 0 = u (3)
where for all time t, all vector z and for all 1 ≤ k ≤ N , the matrix P k t (z) is an oblique projection and u k is any vector satisfying L k u k = v k . The correction term operates only on the interval (T k -ε k , T k ) where T k -ε k < T k for technical reasons. We will show that with a good choice for those projections (see Equation ( 6)) we have the following equivalence in law

L (x|(L k x T k = v k ) 1≤k≤N ) ∼ L (y)
with an explicit density (Theorem 1 below).

In this paper a first part is devoted to the study of general bridges which will provide us the good candidate whose law is absolutely continuous with respect to targeted one. The second one provides the main result. Some properties and proofs are postponed in the appendix to ease the reading.

Notations For the sake of readibility, we choose not to specify arguments when not necessary. For example (1) becomes dx t = b t dt + σ t dw t For all z, the matrix a t (z) is defined by

a t (z) = σ t (z)σ t (z) *
we suppose that there exists a positive number ρ such that for all (t, z)

ρ -1 I n < a t (z) < ρI n
in the sense of symmetric matrices, where I n is the n-dimensional identity matrix. The function a -1 is defined by

a -1 : [0, T ] × R n → M n (R n ) (t, x) → a t (x) -1
We define the infimum of all the ε k ε 0 = min k {ε k }

Bridges and bridges approximations

Bridges

We recall that a bridge is defined as a solution of (3)

dy t = b t (y t )dt + σ t (y t )d wt - N k=1 P k t (y t ) yt-u k T k -t 1 (T k -ε k ,T k ) (t)dt y 0 = u 0
We assume that the deterministic parameters b and σ are C 1,2 b functions (bounded with bounded derivatives). We assume that (t, z)

→ P k t (z) is a C 1,2 b
function and that for any z

L k P k t (z) = L k and ker(L k ) = ker(P k t (z)) (4)
First of all, a lemma to describe the behaviour of process y Lemma 1. The SDE (3) admits a unique solution on [0, T ] in the absolute convergence's sense meaning that

T k T k -ε k P k t (y t )(y t -u k ) T k -t dt < +∞ For all k we have L k y T k = L k u k almost surely (a.s.). Moreover for T k -ε k < t < T k , L k (y t -u k ) ≤ C k (ω)(T k -t) log log[(T k -t) -1 + e] a.s.
, where C k is a positive random variable.

Proof. Let us remark that for times in [T

k-1 , T k ] (with T 0 = 0) the SDE (3) becomes dy t = b t dt -P k t y t -u k T k -t 1 (T k -ε k ,T k ) (t)dt + σ(y t )d wt
So that we may reduce the proof to the study of (3) with only one observation time, but we here have to consider random initial conditions. If unicity holds it will lead to the result by concatenation. The proof in the case N = 1 is given in the appendix with Lemma 6.

Bridges approximations

We now introduce approximations that will be useful in the proof of the main result in next section. Let 0 < ε < ε 0 , we set

dy ε t = b ε t (y ε t )dt + σ t (y ε t )d wt - k P k t (y ε t ) y ε t -u k T k -t 1 (T k -ε k ,T k -ε) (t)dt, y ε 0 = u 0 (5)
The only difference with the Bridge Equation ( 3) is that each correction term is stopped from a distance ε from the observation time.

Lemma 2. There exists a constant 0 < κ < 1 such that

sup t∈[0,T ] E[ y ε t -y t 2 ] ≤ Cε κ for all 0 < ε < (ε 0 ∧ 1)
where C is a positive constant. The numbers C and κ depend on T , N , the (ε k ) k , the (A k ) k , and the bounds for b and σ.

Proof. Given in the appendix, the proof uses classical techniques and auxiliary processes each defined on

[T k-1 , T k ].

Result in the case of partial observation

Case where b is bounded

We aim to obtain a Delyon&Hu-type theorem that gives absolute continuity of process x solution of (1) conditioned on observations (L k x T k = v k ) 1≤k≤N with respect to a bridge process y solution of (3). We now consider a peculiar projection P , for all k and z

P k t (z) = a t (z)L * k (L k a t (z)L * k ) -1 L k (6)
We set

A k t (z) = (L k a t (z)L * k ) -1 and also β t (z) k = σ t (z) * L * k A k t (z) and η k (z) = det(A k t (z)) Let us remark that β k t (z) * β k t (z) = A k t (z) and L k σ t (z)β k t (z) = I m k (7)
where I m k is the identity m k -dimensional matrix. Here are both systems we now consider

dx t = b t (x t )dt + σ t (x t )dw t , x 0 = u dy t = b t (y t )dt + σ t (y t )d wt - N k=1 σ t (y t )β k t (y t ) L k y t -v k T k -t 1 (T k -ε k ,T k ) dt, y 0 = u ( 8
)
The result is the following Theorem 1. Suppose b, σ and a -1 to be C 1,2 b -functions. Then for any bounded continuous function f

E[f (x)|(L k x T k = v k ) 1≤k≤N ] = CE f (y) N k=1 η k (y T k ) exp - β k T k -ε k (L k y T k -ε k -v k ) 2 2ε k + T k T k -ε k - (L k y s -v k ) * L k b s (y s )ds T k -s - (L k y s -v k ) * d A k t (y t ) (L k y s -v k ) 2(T k -s) - 1≤i,j≤m k d A k (y . ) i,j , (L k y . -v k ) i (L k y . -v k ) j s 2(T k -s) ( 9
)
where C is a positive constant.

Proof. This one consists in using approximations y ε solutions of (5) of process y solution of [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]. Thanks to Girsanov's theorem, we are able to obtain an equality for all bounded continuous function f

E[f (x)G ε (x)] = E[f (y ε )H ε (y ε )]
where G ε /H ε is the density given by Girsanov's theorem. We want to prove that with a good choice for G ε and H ε , the lefthand member of the last inequality converges to the conditional expectation, and the righthand one converges to what appears in the Theorem 1.

We set for all z ∈ R n

h ε t (z) = N k=1 β k t (z) v k -L k z T k -t 1 (T k -ε k ,T k -ε) (t)
Then for all bounded continuous function f

E[f (y ε )] = E[f (x) exp{- T 0 h ε t (x t ) * dw t + 1 2 h ε t (x t ) 2 dt}]
We are looking for a different expression of the argument of the exponential function. We use Itô's formula for T k -ε k < t < T k -ε and use [START_REF] Lyons | On conditional diffusion processes[END_REF] to get

d β k t (x t )(L k x t -v k ) 2 T k -t = 2(L k x t -v k ) * A k t (x t )L k dx t T k -t + β k t (x t )(L k x t -v k ) 2 (T k -t) 2 dt + m k T k -t dt + (L k x t -v k ) * d A k t (x t ) (L k x t -v k ) T k -t + 1≤i,j≤m k d A k i,j (x . ), (L k x . -v k ) i (L k x . -v k ) j t T k -t
The k th term of (h ε t ) * dw t coming from that one in dx t is now isolated

- 2(L k x t -v k ) * A k t σ t dw t T k -t - β k t (L k x t -v k ) 2 (T k -t) 2 dt = -d β k t (L k x t -v k ) 2 T k -t + m k T k -t dt + 2(L k x t -v k ) * A k t b t dt T k -t + (L k x t -v k ) * dA k t (L k x t -v k ) T k -t + 1≤i,j≤m k d A k i,j , (L k x -v k ) i (L k x -v k ) j t T k -t (10)
Since we have

h ε t 2 dt = k β k t (L k x t -v k ) 2 (T k -t) 2 1 (T k -ε k ,T k -ε) (t)dt and (h ε t ) * dw t = - k 1 (T k -ε k ,T k -ε) (t) (L k x t -v k ) * A k t σ t dw t T k -t we obtain -2(h ε t ) * dw t -h ε t 2
dt adding the terms given by (10). Finally, it leads us to a new expression for the density given by Girsanov's theorem

E[f (y ε )] = E exp N k=1 - β k T k -ε (L k x T k -ε -v k ) 2 2ε + β k T k -ε k (L k x T k -ε k -v k ) 2 2ε k + T k -ε T k -ε k (L k x t -v k ) * A k t b t dt T k -t + m k 2(T k -t) dt + (L k x t -v k ) * dA k t (L k x t -v k ) 2(T k -t) + 1≤i,j≤m k d A k i,j , (L k x -v k ) i (L k x -v k ) j t 2(T k -t)
In an equivalent way, even if it means changing f

E[f (y ε )ϕ ε ] = E[f (x)ψ ε ] (11)
with

ϕ ε := ϕ ε (y ε ) = k=1 N ε - m k 2 k η ε k (y ε T k -ε ) exp N k=1 - β k T k -ε k (L k y ε T k -ε k -v k ) 2 2ε k + T k -ε T k -ε k - (L k y ε t -v k ) * A k t b t dt T k -t - (L k y ε t -v k ) * dA k t (L k y ε t -v k ) 2(T k -t) - 1≤i,j≤m k d A k i,j , (L k y ε -v k ) i (L k y ε -v k ) j t 2(T k -t) (12)
and

ψ ε = C ε N k=1 η ε k (x T k -ε ) exp{- β k T k -ε (x T k -ε )(L k x T k -ε -v k ) 2 2ε } (13)
where for all

z ∈ R n η ε k (z) = det A k T k -ε (z) and C ε = k ε -m k 2
Now using it in the case where f = 1, we get formally

E[f (y ε )ϕ ε ] E[ϕ ε ] = E[f (x)ψ ε ] E[ψ ε ]
the fact that this quantity is finite is given by Proposition 1. The fact that the righthand term converges to the conditional expectation is given by Lemma 9 in the appendix. The proof relies essentially on the use of Aronson's estimates that provides gaussian bounds for transition probabilities.

The main difficulty of the proof consists in showing almost sure convergence and then uniform one for the ϕ ε . An obvious candidate for the limit is

ϕ = N k=1 ε - m k 2 k η k (y T k ) exp - β k T k -ε k (y T k -ε k )(L k y T k -ε k -v k ) 2 2ε k + T k T k -ε k - (L k y t -v k ) * A k t L k b t (y t )ds T k -t - (L k y t -v k ) * d A k t (y t ) (L k y t -v k ) 2(T k -t) - 1≤i,j≤m k d A k i,j (y . ), (L k y . -v k ) i (L k y . -v k ) j t 2(T k -t) (14)
Thanks to Lemma 10 given in the appendix, ϕ is well defined. As said before, we want to prove the following Lemma 3. There exists a decreasing sequence (ε i ) i∈N tending to 0 such that

lim i→∞ E [ ϕ εi -ϕ ] = 0
Proof. The proof is decomposed into two main parts. First one aims at showing the almost sure convergence of ϕ εi . In second part we prove that

E[ϕ εi ] tends to E[ϕ].
Finally to conclude, we will use Scheffé's lemma.

For almost sure convergence, we first use triangular inequality

|ϕ ε (y ε ) -ϕ(y)| ≤ |ϕ ε (y ε ) -ϕ ε (y)| + |ϕ ε (y) -ϕ(y)|
The second one converges to 0, this is given by Lemma 10. We now treat the term

|ϕ ε (y ε ) -ϕ ε (y)|. ϕ ε (y ε ) ϕ ε (y) = N k=1 η ε k (y ε T k -ε ) η ε k (y T k -ε ) exp - β k T k -ε k (y ε T k -ε k )(L k y ε T k -ε k -v k ) 2 -β k T k -ε k (y T k -ε k )(L k y T k -ε k -v k ) 2 2ε k + T k -ε T k -ε k - (L k y ε t -v k ) * A k t (y ε t )L k b t (y ε t ) -(L k y t -v k ) * A k t (y t )L k b t (y t ) T k -t dt - (L k y ε t -v k ) * d A k t (y ε t ) (L k y ε t -v k ) -(L k y t -v k ) * d A k t (y t ) (L k y t -v k ) 2(T k -t) - i,j d A k i,j (y ε . ), (L k y ε . -v k ) i (L k y ε . -v k ) j t -d A k i,j (y . ), (L k y . -v k ) i (L k y . -v k ) j t 2(T k -t)
We can write it respecting the order above

ϕ ε (y ε ) ϕ ε (y) N otation = N k=1 Ξ ε k exp{Υ ε k + Ψ ε k + Θ ε k + Φ ε k }
According to Lemma 2 there exists a decreasing sequence (ε i ) i∈N tending to 0 satisfying for all k that y εi T k -εi converges almost surely to y T k . From this we obtain the fact that Ξ εi k converges almost surely to 1 and Υ εi k to 0 using regularity of σ. Then for all k

|Ψ ε k | ≤ T k -ε T k -ε k L k (y ε t -y t ) * A k t (y ε t )L k b t (y ε t ) T k -t + (L k y t -v k ) * A k t (y ε t )L k b t (y ε t ) -A k t (y t )L k b t (y t ) T k -t dt
Since b and σ are bounded we use Lemma 1 to get

|Ψ ε k | ≤ C T k -ε T k -ε k y ε t -y t 2 T k -t dt 1 2 T k -ε T k -ε k (1 + log log (T k -t) -1 + e T k -t dt 1 2
where C and C ′ are positive random variables. Thanks to Lemma 2, up to an extracted subsequence

lim i→∞ T k -εi T k -ε k y εi t -y t 2 T k -t dt = 0
that leads us to convergence for all k of |Ψ εi k | to 0. Now we use Identity (34)

Θ ε k = Ly t -v 2 T -t p t (y t )dt + Ly t -v 2 T -t q t (y t )dw t + Ly t -v 2 (T -t) 2 r t (y t )dt - Ly ε t -v 2 T -t p t (y ε t )dt - Ly ε t -v 2 T -t q t (y ε t )dw t - Ly ε t -v 2 (T -t) 2 r t (y ε t )dt
where p, q, and r are all C 1,2 b functions. Hence using Lemmas 1 and 2 as above we obtain that lim εi→0 |Θ εi k | = 0 up to a subsequence. It remains to treat the term Φ ε k . Still using Identity (34), Lemmas 1 and 2 we show that lim εi→0 |Φ εi k | = 0 even if it means extracting once more a subsequence. We have obtained almost sure convergence of ϕ ε to ϕ. Then we show the convergence of the expectations. For this we set a preliminary result Proposition 1. There exist two positive constants c 1 and c 2 such that for all 0 < ε < ε 0

c 1 ≤ C ε E[ψ ε ] ≤ c 2
Proof. We give an explicit expression

C ε E[ψ ε ] = q ε (ζ 1 , . . . , ζ N ) N k=1 η ε k (ζ k ) -m k 2 exp{- β k T k -ε (ζ k )(L k ζ k -v k ) 2 2ε }dζ k
where q ε is the density of (x T1-ε , . . . , x TN -ε ). Under theorem's assumptions x is a strong Markov process, with positive transition density. For s, t ∈ [0, T ], we denote p s,t (u, z) the density of x s,u t solution of (1) initialized to be u at time s. Then thanks to Aronson's estimates there exist positive constant µ, λ, M and Λ such that the density p satisfies for s < t

µ(t -s) -n 2 e -λ z-u 2 t-s ≤ p s,t (u, z) ≤ M (t -s) -n 2 e -Λ z-u 2 t-s
Now using p we are able to write

q ε (ζ 1 , . . . , ζ N ) = p 0,T1-ε (u, ζ 1 ) . . . p TN-1-ε,TN -ε (ζ N -1 , ζ N )
Then we apply Aronson's estimates and the fact that for all i, j the coordinate A k i,j is bounded by two positive constants. We obtain bounds for

C ε E[ψ ε ] of the type λ -1 C ε exp{ N j=1 -λ L k ζ k -v k 2 2ε - λ ζ 1 -u 2 T 1 -ε - N k=2 λ ζ k -ζ k-1 2 T k -T k-1 } N k=1 dζ k
where λ is a positive constant large for the lower bound and small for the upper one. The integral can be interpreted as a gaussian expectation where

     ζ 1 ζ 2 -ζ 1 . . . ζ N -ζ N -1     
is a centered gaussian vector with covariance matrix

R ε = 1 2λ       (T 1 -ε)I n 0 . . . 0 0 (T 2 -T 1 )I n . . . . . . . . . . . . . . . 0 0 . . . 0 (T N -T N -1 )I n      
where I n is the n-dimensional identity matrix. As a remark, in the sense of symmetric matrices, there exist two positive constants c 1 and c 2 such that

c 1 I N n < R ε < c 2 I N n
where I N n is the N n-dimensional identity matrix. Thus the gaussian vector

   ζ 1 . . . ζ N    admits for covariance matrix Γ ε = G -1 R ε G - * where G =          I n 0 . . . .
0 -I n I n         
We still keep bounds for the covariance matrix

c 1 I N n < Γ ε < c 2 I N n
Now we can get bounds for

C ε E[ψ ε ] with expectations of type λ m k 2 C ε E[exp{- N k=1 λ L k X k -v k 2 2ε }]
where X k is a n-dimensional gaussian variable. Then we use Lemma 11 given in the appendix to obtain the fact that

R m1 × • • • × R mN → R (v 1 , . . . , v N ) → C ε N k=1 λ m k 2 E[exp{-λ L k X k -v k 2 2ε }]
is a gaussian density of a variable (L

1 X 1 + ε λ Y 1 , . . . , L N X N + ε λ Y N )
where the Y k are m k -dimensional centered normalized gaussian vectors. Moreover the two families (X k ) k and (Y k ) k are independant. Finally using bounds obtained above for Γ ε we get the fact that for all 0 < ε < ε 0

c 1 < C ε E[ψ ε ] < c 2
As a first consequence, thanks to identity (11), E[ϕ ε ] is finite so that ϕ ε E[ϕ ε ] is a density. We may also use Fatou's lemma to get

E[ϕ] ≤ lim inf ε→0 E[ϕ ε ] ≤ c 2 It takes more work to control lim sup ε→0 E[ϕ ε ].
Let J > 0 be a large number, we introduce for all process (z t ) t∈[0,T ] and for all 1

≤ k ≤ N the stopping time τ ε k τ ε k = inf{t k < t ≤ T k -ε : 1 √ T k -t exp{- L k z t -v k 2 2(T k -t) D} ≤ J -1 } = inf{t k < t ≤ T k -ε : L k z t -v k 2 2(T k -t) ≥ D -1 log J √ T k -t }
where D is a positive constant such that DI d ≤ A k . We know that such a constant exists according to assumptions on the function a. The t k are chosen to be real numbers contained in (T k-1 , T k -ε). As a convention we set τ ε k = T k if the condition is empty. Let τ ε be the first of the τ ε k such that the condition is non-empty

τ ε = inf k {τ ε k : τ ε k < T k } we set as convention τ ε = T if if for all k, τ ε k = T k . Even if it means changing f in Equation (11) E[f (y ε )1 τ ε <T ϕ ε ] E[ϕ ε ] = E[f (x)1 τ ε <T C ε ψ ε ] E[C ε ψ ε ]
We recall that

C ε ψ ε = k ε -m k 2 exp{- β k T k -ε (L k x T k -ε -v k 2 2ε }
We now consider to be on set

{τ ε = τ ε k } t k τ ε T k T k-1 T k -ε
We decompose C ε ψ ε into three parts as a product of three factors

F 1 = j<k ε -m j 2 exp{- β j Tj-ε (L j x Tj -ε -v j ) 2 2ε } F 2 = ε -m k 2 exp{- β k T k -ε (L k x T k -ε -v k ) 2 2ε } F 3 = j>k ε -m j 2 exp{- β j Tj-ε (L j x Tj -ε -v j ) 2 2ε } We are interessed in E[C ε ψ ε 1 τ ε =τ ε k ] = E[F 1 F 2 F 3 1 τ ε =τ ε k ]
We now use Markov's property to get independance between Past and Future knowing Present (cf [START_REF] Dellacherie | Chapitres I à IV, Édition entièrement refondue[END_REF] see last chapter about conditional expectations)

E[C ε ψ ε 1 τ ε =τ ε k ] = E[F 1 E[F 2 F 3 1 τ ε =τ ε k |τ ε k , x τ ε k ]] = E[F 1 E[F 2 1 τ ε =τ ε k |τ ε k , x τ ε k ]E[F 3 |x τ ε k ]] (15) = E[F 1 E[F 2 1 τ ε =τ ε k |τ ε k , x τ ε k ]E[F 3 |x T k -ε ]] (16)
In order to study the factor

E[F 2 1 τ ε =τ ε k |F τ ε k ] we introduce θ t = 1 √ T k -t exp{- β k t (L k x t -v k ) 2 2(T k -t) } For t ∈ [T k-1 , T k -ε], we set z t = L k x t -v k , p t = β k t (L k x t -v k ) . We recall that β k t = σ * t L * k A k t with A k t = (β k t ) * β k t = (L k a t L * k ) -1
. With respect to these notations, we have

p 2 t = z * t A k t z t
It is also easy to see that

dz t = L k b t dt + L k σ t dw t and then d z t = L k a t L * k dt = (A k t ) -1 dt. We use Itô's formula d(p 2 t ) = d(z * t A k t z t ) = 2z t A k t dz t + z * t dA k t z t + i,j d A k i,j , z i z j t + m k dt Then d p 2 t T k -t = 2z t A k t dz t T k -t + p 2 t dt (T k -t) 2 + z * t dA k t z t T k -t + m k dt T k -t + i,j d ∆ i,j , z i z j t T k -t
First using definitions of z, β k and A k we get

z * t A k t dz t = z * t A k t L k dx t = z * s A k t L k σ t σ -1 t dx t = z * t (β k t ) * σ -1 t b t dt + z * t (β k t ) * dw t
This leads us to the existence of two bounded adapted processes r (1) and r (2) defined on

[T k-1 , T k -ε] such that z * t A k t dz t = p t r (1)
t dt + p t r (2)
t dw t In a same way we remark that there exist two bounded adapted processes r (3) and r (4)

such that d(L k a t L * k ) = r (3)
t dt + r (4)
t dw t

we get even if it means changing r (3) and r (4)

z * t dA k t z t = z * t d(L k a t L * k ) -1 z t = p 2 t r (3)
t dt + p 2 t r (4)
t dw t Finally, we obtain existence of two bounded adapted processes r and r ′ such that

d p 2 t T -t = 2p t dw t T -t + p 2 t dt (T -t) 2 + r t p 2 t T -t dw t + dt T -t + r ′ t p 2 t + p t T -t dt
From this we deduce that quadratic variation

d p 2 t T k -t = 4p 2 t + r 2 t p 4 t + 4r t p 3 t (T k -t) 2 dt
Now we apply Itô's formula to the function θ always for t

∈ [T k-1 , T k -ε] dθ t = θ t dt 2(T k -t) - 1 2 θ t d p 2 t T k -t + 1 8 θ t d p 2 t T k -t
We deduce from the three last equations after simplification of four terms that there exists a martingale M and a bounded adapted process r ′′ both defined on

[T k-1 , T k -ε] such that dθ t = dM t + θ t r ′′ t p 2 t + p t T k -t + p 4 t + p 3 t (T k -t) 2 dt
For any η > 0, functions x → e -η z 2 2 |z| m for m = 1, 2, 3, 4 are all bounded, then there exists a constant c η such that

( T k -t θ t ) η p 2 t + p t T k -t + p 4 t + p 3 t (T k -t) 2 ≤ c η √ T k -t
This gives us the existence of a bounded adapted process π defined on [T k-1 , T k -ε] that allows us to write

dθ t = dM t + θ 1-η t (T k -t) -h α t dt with h = 1+η 2 . We now integrate it for t ∈ (τ ε k , T k -ε] θ t = θ τ ε k + M t -M τ ε k + t τ ε k π s θ 1-η s (T k -s) -h ds
This leads to the following

E[θ t 1 τ ε k <t ] ≤ J -1 + π t t k E[θ s 1 τ ε k <s ] 1-η (T k -s) -h ds where π = sup s |π s |. So E[θ1 τ ε k <t ] is bounded by the solution u of du s = ᾱu 1-η s (T k -s) -h ds, u t k = J -1
and this equation has an explicit solution

u t = η ᾱ 1 -h [(T k -t k ) 1-h -(T k -t) 1-h ] + J -η 1 η ≤ c k (T k -t k ) 1-h + J -η 1 η
where c k is a positive constant. Then for all t ∈

[T k-1 -T k -ε] E[θ t 1 t>τ ε k |τ ε k , x τ ε k ] ≤ {c k (T k -t k ) 1-h + J -η } η In particular when t = T k -ε E[F 2 1 t>τ ε k |τ ε k , x τ ε k ] = E[θ T k -ε 1 t>τ ε k |τ ε k , x τ ε k ] ≤ {c k (T k -t k ) 1-h + J -η } η
We now come back to equation (15), we get a first bound

E[C ε ψ ε 1 τ ε =τ ε k ] ≤ {c k (T k -t k ) 1-h + J -η } η E[F 1 E[F 3 |x T k -ε ]] In order to treat the factor E[F 3 |x T k -ε ] we use Aronson's estimate to get E[F 3 |F T k -ε ] ≤ G j>k 1 √ ε exp{- β j Tj-ε (L j ζ j -v j ) 2 2ε } 1 T j -T j-1 exp{-Λ j |ζ j -ζ j-1 | 2 T j -T j-1 }dζ j
where G is a positive constant. We just have to use Lemma 11 given in the appendix to obtain an positive constant upper bound. The same Lemma 11 brings us a positive constant upper bound for

E[F 1 ]
. Finally the inequation we get from equation ( 15) is the following

E[C ε ψ ε 1 τ ε =τ ε k ] ≤ G{c k (T k -t k ) 1-h + J -η } η
where G is a positive constant. From this we deduce

E[C ε ψ ε 1 T >τ ε ] = k E[C ε ψ ε 1 τ ε =τ ε k ] ≤ G max k (c k (T k -t k ) 1-h + J -η ) η
According to this last result and using the lower bound of C ε E[ψ ε ] given by Proposition 1 we finally have

E[C ε ψ ε 1 T =τ ε ] E[C ε ψ ε ] ≥ 1 -G max k (c k (T k -t k ) 1-h + J -η ) η
where G is positive constant. So using inequality (11) we obtain

E[ϕ ε 1 T =τ ε ] E[ϕ ε ] ≥ 1 -G max k (c k (T k -t k ) 1-h + J -η ) η (17)
Moreover the family (ϕ ε 1 T =τ ε ) ε is uniformly integrable. Indeed by definition of τ ε we can get upper bounds depending on J for the different factors in Expression (12) of ϕ ε or (14) of ϕ, for all 0 ≤ ε < 1

ϕ ε = N k=1 ε - m k 2 k η ε k (y ε T k -ε ) exp{- T k -ε T k -ε k (L k y ε t -v k ) * A k t L k b t dt T k -t - (L k y ε t -v k ) * dA k t (L k y ε t -v k ) 2(T k -t) - 1≤i,j≤m k d A k i,j , (L k y ε -v k ) i (L k y ε -v k ) j t 2(T k -t) - β k T k -ε k (y ε T k -ε k )(L k y ε T k -ε k -v k ) 2 2ε k }
in fact "ϕ 0 = ϕ". We recall that b and σ are bounded so is η. Then on

{τ ε = T } (L k y ε t -v k ) * A k t L k b t T k -t ≤ C √ T k -t log J (T k -t) m k 2
which is an integrable quantity in T k , and C is a positive constant depending on the choice of b and σ.

A same method gives an upper bound for the terms where quadratic variation appears. For the terms in dA k t , we decompose with respect to integrals with respect to dt and dw t

(L k y ε t -v k ) * dA k t (L k y ε t -v k ) 2(T k -t) = L * k y ε t -v k 2 T k -t r k t (y ε t )dw t + L k y ε t -v k 2 T k -t q k t (y ε t )dt
where r k and q k are bounded adapted functions. Then for fixed J, there exists a constant K such that

ϕ ε 1 T =τ ε ≤ C k exp{ T k T k -ε k L k y t -v k 2 T k -t r k t dw t - 1 2 L k y t -v k 4 (T k -t) 2 r k t 2 dt}
where C is a positive constant. Let us recall the following lemma (cf [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] p.198) Lemma 4 (Novikov). Let (M t ) t∈R be a continuous local martingale, we set for all t

Z t = exp{M t - 1 2 M t } If E[exp{ 1 2 M t }] < +∞ then we have E[Z t ] = 1
Let us remark that for all p > 0

exp{p k T k -ε T k -ε k L k y ε t -v k 4 2|T k -t| 2 r k t 2 dt}] ≤ exp{p k T k -ε T k -ε k log J (T k -t) m k 2 2 dt} ≤ C p
where C is a positive constant. Thus, we apply Novikov's lemma to get uniform integrability. Then we take the lim inf ε→0 and use Lebesgue's theorem to obtain

E[ϕ1 T =τ ε ] lim sup ε→0 E[ϕ ε ] ≥ 1 -N max k (c k (T k -ε -t k ) 1-h + J -η ) 1 η (18)
Now 1 T =τ ε converges almost surely to 1 as J tends to infinity. We are able to say after making the t k tend to

T k that lim sup ε→0 E[ϕ ε ] ≤ E[ϕ] (19)
We finish the proof by Scheffé's Lemma (cf [START_REF] Dellacherie | Chapitres I à IV, Édition entièrement refondue[END_REF] p.36)

Lemma 5 (Scheffé). Let (f n ) n∈N be a sequence of positive functions converging to f , moreover we suppose that

lim n→∞ E[f n ] = E[f ] < ∞ then the sequence (f n ) converges to f in L 1 .
Finally we conclude thanks to Lemmas 9 and 3.

Case where b is unbounded

Suppose now that b is locally Lipschitz with respect to x and is locally bounded. Moreover the SDE (1) admits a strong solution. We use a Girsanov theorem to reduce the problem to the case of a bounded drift.

We recall the Girsanov theorem for unbounded drifts introduced in [START_REF] Bernard | Simulation of conditioned diffusions and applications to parameter estimations[END_REF] Theorem 2. Let b, h and σ be measurable functions from R + × R n to R n , R d and R n×d locally Lipschitz with respect to x; consider the following SDE's

dx t = b t (x t )dt + σ t (x t )dw t , dy t = (b t (y t ) + σ t (y t )h t (y t ))dt + σ t (y t )d wt , x 0 = y 0 on the finite interval [0, T ].
We assume the existence of strong solution for each equation. We assume in addition that h is bounded on compact sets. Then the Girsanov formula holds: for any non negative Borel function f defined on C([0, T ], R n ), one has

E[f (y, wh )] = E[f (x, w) exp{ T 0 h * t (x t )dw t - 1 2 T 0 h t (x t ) 2 dt}] E[f (x, w)] = E[f (y, wh ) exp{- T 0 h * t (y t )d wt - 1 2 T 0 h t (y t ) 2 dt}]
where wh = wt + t 0 h s (y s )ds.

Thanks to both Theorems 1 and 2 we obtain Theorem 3. Suppose σ and a -1 to be C 1,2 b -functions. Assume that b is a locally Lipschitz with respect to x and locally bounded function. Let y be the solution of

dy t = bt (y t )dt + σ t (y t )d wt - N k=1 σ t (y t )β k t (y t ) L k y t -v k T k -t 1 (T k -ε k ,T k ) (t)dt
where b satisfies the assumptions of Theorem 1.

Then for any bounded continuous function f

E[f (x)|(L k x T k = v k ) 1≤k≤N ] = CE f (y) N k=1 η k (y T k ) exp - β k T k -ε k (L k y T k -ε k -v k ) 2 2ε k + T k T k -ε k - (L k y s -v k ) * L k bs (y s )ds T k -s - (L k y s -v k ) * d A k t (y t ) (L k y s -v k ) 2(T k -s) - 1≤i,j≤m k d A k (y . ) i,j , (L k y . -v k ) i (L k y . -v k ) j s 2(T k -s) + T 0 b * t (y t )a t (y t ) -1 dy t - 1 2 σ t (y t ) -1 bt (y t ) 2 dt
where C is a positive constant and b = b -b.

Proof. Let x be the solution of

dx t = bt (x t )dt + σ t (x t )dw t , x0 = u
Then from Theorem 2, for any bounded continuous function f and g

E[f (x)g(L 1 x T1 , . . . , L N x TN )] = E[f (x t )g(L 1 xT1 , . . . , L N xTN )e T 0 b * t (xt)at(xt) -1 dxt-1 2 σt(xt) -1 bt(yt) 2 dt ] = E[f (x t )e T 0 b * t (xt)at(xt) -1 dxt-1 2 σt(xt) -1 bt(yt)
For all 1 ≤ i ≤ n the process {( t 0 (T -s) -1 σ s (y s )d ws ) i , t ≥ 0} is a continuous local martingale whose quadratic variation τ t = t 0 n j=1 (T -s) -2 σ s (y s ) i,j ds satisfies lim t→T τ t = +∞ and τ t ≤ c T -t where c is a positive constant. Hence we just have to apply the Dambis-Dubins-Schwarz theorem that gives us the existence of a Brownian motion B i such that

t 0 (T -s) -1 σ s (y s )d ws i = B i (τ t )
The law of iterated logarithm allows us to conclude.

dy t = b t (y t )dt -P t (y t ) y t -u 1 T -t dt, y 0 = u Then for all s < t < T , E[ L(y t -u 1 ) 2 ] T -t ≤ c(1 + √ T -t E[ L(u -u 1 ] 2 ]) (20)
and

E[ y s -y t 2 ] ≤ C(t -s)(1 + √ T -s E[ L(u -u 1 ) 2 ]) (21)
where c and C are positive constants depending on T , ε 1 , bounds for b and σ.

Proof. Thanks to Identity (4), on

(T -ε 1 , T ) dL(y t -u 1 ) = L[b t dt + σ t d wt ] -L y t -u 1 T -t dt Thus d L(y t -u 1 ) 2 = 2(y t -u 1 ) * L * L[b t dt + σ t d wt ] -2 L(y t -u 1 ) 2 T -t dt + Tr(La t L * )dt
where the function T r gives the sum of all diagonal terms. Finally

d L(y t -u 1 ) 2 T -t = 2 (y t -u 1 ) * T -t L * L[b t dt + σ t d wt ] + Tr(La t L * ) T -t dt - L(y t -u 1 ) 2 (T -t) 2 dt Setting E t = E L(yt-u1) 2 T -t
, since b and σ are bounded, we get

E ′ t ≤ C 1 √ E t + 1 T -t - E t T -t (22)
where C 1 is a positive constant depending on b ∞ and σ ∞ .

E ′ t ≤ (T -t) -1 C 1 E t 2C 1 + C 1 2 + 1 -E t = (T -t) -1 (C - E t 2 ) ( 23
)
where

C = C 1 + C 2 1 2 . Thus E t -2C √ T -t ′ = E ′ t √ T -t + E t -2C 2(T -t) 3 2 ≤ 0 thanks to (23). Hence E t -2C √ T -t ≤ E T -ε1 -2C √ ε 1
that can be written

E t ≤ 2C + T -t ε 1 (E T -ε1 -2C) Similarly for t < T -ε 1 , Inequality 22 becomes E t ≤ C ′ (E 0 + 1) exp{ C ′ t ε 1 }
where C ′ is a positive constant only depending on T and bounds of b and σ. That gives us (20). By definition for s, t ∈ (T -ε 1 , T ) we have

y s -y t = t s b τ dτ + σ τ d wτ -P τ y τ -u 1 T -τ dτ
Since b and σ are bounded functions, using Minkovski's inequality

E[ y s -y t 2 ] 1 2 ≤ E[ t s b τ dτ 2 ] 1 2 + E[ t s σ τ d wτ 2 ] 1 2 + E[ t s P τ y τ -u 1 T -τ dτ 2 ] 1 2 (24)
Thanks to Doob's inequality (see e.g. [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] p.170) we get

E[ t s b τ dτ 2 ] + E[ t s σ τ d wτ 2 ] ≤ C 2 (t -s)
where

C 2 = C 2 1
is the square of the constant introduced above. In order to treat the last term in (24), we beforehand give a property Proposition 2. Let f be a real-valued process defined on a segment [a, b], then

E[( b a f s ds) 2 ] 1 2 ≤ b a E[f 2 s ] 1 2 ds Proof. Indeed E[( b a f s ds) 2 ] ≤ E[( b a |f s |ds) 2 ] = 2E[ b a s a |f u |du|f s |ds] = 2 b a s a E[|f u f s |]du ds so that E[( b a f s ds) 2 ] ≤ 2 b a s a E[(f u ) 2 ] 1 2 E[(f s ) 2 ] 1 2 du ds = ( b a E[f 2 s ] 1 2 ds) 2
Thanks to Proposition 2, assumptions (4) on matrix P and result (20)

E t s P τ y τ -u 1 T -τ dτ 2 ≤ t s E P τ (y τ -u 1 ) 2 (T -τ ) 2 1 2 ds 2 ≤ c(1 + √ T -s E[ L(u -u 1 ) 2 ])( t s dτ √ T -τ ) 2 = 4c(T -s)(1 + √ T -s E[ L(u -u 1 ) 2 ]) Finally using (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 ), we have on (T -ε 1 , T ) E[ y s -y t 2 ] ≤ (C 2 ∧ 4c)(T -s)(2 + √ T -s E[ L(u -u 1 ) 2 ]) Using Doob's inequality for s, t ∈ [0, T -ε 1 ] with s < t E[ y s -y t 2 ] ≤ C 2 (t -s)
this gives the second result (21).

Lemma 8. Let y and z be two bridges, solutions of (3) with N = 1 and different initializations. Then, there exist two constants C > 0 and 0 < α < 1 such that for all t ∈ [0, T ]

E[ y t -z t 2 ] ≤ CE[ y 0 -z 0 2 ] α (25)
(σ t (y t ) -σ t (z t )) 2 i,j dt (27)
We denote

E t = E[ y t -z t 2 ]
. We decompose the interval [0, T ] into [0, T -h] and (T -h, T ] with a parameter h that will be chosen later. On [0, T -h] with respect to both precedent Inequations (26) and ( 27) we get by using regularity of b and σ

E ′ t ≤ C 1 (E t + E t T -t ) ≤ C 2 E t T -t
where C 1 and C 2 are positive constants depending on T , b and σ. We use Gronwall's lemma to obtain

E t ≤ E 0 T h C2
For the other part (T -h, T ], we use ( 26), ( 27) and (20) to get

E ′ t ≤ C 3 (E t + E t T -t ) ≤ C 4 E t + 1 √ T -t
where C 3 and C 4 are positive constants depending on T , b and σ. Then

log(E t + 1) -log(E T -h + 1) ≤ C 4 ( √ h - √ T -t) ≤ C 4 √ h
Finally, on [0, T ]

E t ≤ E 0 T h C5 (1 + e C5 √ h ) + e C5 √ h -1 ≤ E 0 T h C5 (1 + K) + C 5 K √ h
where K is a positive constant depending on T and C 5 = C 2 ∨C 4 . We then choose h = 2E0T C 5 (K+1)

K 1 C 5 + 1 2
which minimizes the last member above. Hence

E t ≤ CE 1 2C 5 +1 0
where C is a positive constant depending on b, σ, P and T .

Proof of Lemma 2 . We now consider an interval of type [T k-1 , T k ). We introduce a process y k solution on this interval for the Bridge Equation (3) initialized at time T k-1 by the value y ε T k-1 . A picture to visualize what is going on is given by Figure 1 in page 17. We use this new process y k to write

E[ y ε t -y t 2 ] ≤ 2E[ y ε t -y k t 2 ] + 2E[ y k t -y t 2 ] ( 28
)
We will study both terms separately.

For the first one, on [T k-1 , T k -ε) the term y ε t -y k t is 0 a.s. and on [T k -ε, T k ), we can reduce the study to that of x t -y t 2 with a same initialization

y ε T k -ε at time T k -ε. E E[ y ε t -y k t 2 |y ε T k -ε ] ≤ 2E E[ y k T k -ε -y k t 2 |y ε T k -ε ] + 2E E[ y ε t -y ε T k -ε 2 |y ε T k -ε ]
1 First, the three processes follow the dynamics of the initial diffusion (1) with a different initialization for y.

2 Now, the three processes follow the dynamics of the bridge [START_REF] Dellacherie | Chapitres I à IV, Édition entièrement refondue[END_REF], that means that the correction term operates and forces these processes to get closer to the observation 3 At the end, only the processes y and y k go on following the dynamics of the bridge (3) and both tends to the obervation, while y ε follows the initial dynamics (1)

We then use Lemma [START_REF] Lyons | On conditional diffusion processes[END_REF] given in the appendix and classical technique (see e.g. [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] p.170) to obtain upper bounds

E E[ y ε t -y k t 2 |y ε T k -ε ] ≤ cε(1 + √ ε E[ L k (y ε T k-1 -u k ) 2 ]) (29)
where c is a positive constant. Now in order to treat the remaining term we use Lemma 8

E E[ y k t -y t 2 |y T k-1 , y ε T k-1 ] ≤ E[ y ε T k-1 -y T k-1 2 ] α Finally, on [T k-1 , T k ) E[ y ε t -y t 2 ] ≤ c ′ ε(1 + √ ε E[ L k (y ε T k-1 -u k ) 2 ]) + E[ y ε T k-1 -y T k-1 2 ] α (30)
where 0 < α < 1 and c ′ is a positive constant only depending on T , bounds b and σ. We show by induction that there exists some constant C such that for all 1

≤ k ≤ N E[ y ε T k -y T k 2 ] ≤ C k ε α k-1 (31)
The base case is given by Equation (29). Indeed on [0, T 1 ] processes y 1 and y are indistinguishable since they have a same initialization at time 0. Suppose now for some k that Inequality (31) holds. We now use Equation (30) to get

E[ y ε T k+1 -y T k+1 2 ] ≤ c ′ ε(1 + √ ε E[ L k+1 (y ε T k -u k+1 ) 2 ]) + E[ y ε T k -y T k 2 ] α Let us recall that L k y T k = L k u k hence E[ L k+1 (y ε T k -u k+1 ) 2 ] ≤ E[ L k+1 (y ε T k -y T k ) 2 ] + L k+1 (u k -u k+1 ) 2 ≤ c k (1 + E[ y ε T k -y T k 2 ])
where c k is a positive constant depending on L k+1 , u k and u k+1 . That gives us thanks to the induction hypothesis

E[ y ε T k+1 -y T k+1 2 ] ≤ c ′ k ε(1 + √ ε E[ y ε T k -y T k 2 ]) + E[ y ε T k -y T k 2 ] α ≤ C[ε(1 + √ εC k ε α k-1 ) + C k ε α k ]
where C is a positive constant. This concludes the proof.

Lemma 9. Let (t k,q ) 1≤k≤N 1≤q≤M k be a sequence such that t k,q ∈ (T k-1 , T k ) and for all k, (t k,q ) q is an increasing sequence. Then for all bounded continuous function g

lim ε→0 E[g(x t1,1 , . . . , x tN,M N )ψ ε ] E[ψ ε ] = E[g(x t1,1 , . . . , x tN,M N )|(L k x T k = v k ) 1≤k≤N ] Proof. Let us recall C ε ψ ε = N k=1 ε -m k 2 η ε k (x T k -ε ) exp{- β k T k -ε (x T k -ε )(L k x T k -ε -v k ) 2 2ε } where for all z ∈ R n η ε k (z) = det(A k T k -ε (z))
Let introduce Aronson's estimates (see e.g. [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF], [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] or [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]) that gives bounds for the transition density. If p s,t (u, .) (with s < t) is the density of x t knowing that x s = u, we have for all z µ

(t -s) -n 2 e --λ z-u 2 t-s < p s,t (u, z) < M (t -s) -n 2 e --Λ z-u 2 t-s
The transition densities allow to expand the density q ε of (x t1,1 , . . . , x tN,M N , x T1-ε , . . . , x TN -ε )

q ε (z 1,1 , . . . , z tN,M N , ζ 1 , . . . , ζ N ) = p 0,t1,1 (u, z 1,1 ) . . . p t1,M 1 ,T1-ε (z 0,M0 , ζ 1 ) . . . p tN,M N ,TN -ε (z N,MN , ζ N ) Then we set for ε ≥ 0 Φ ε g (ζ 1 , . . . , ζ N ) = E[g(x t1,1 , . . . , x tN,M N )|(x T k -ε = ζ k ) k ] = g(z 1,1 , . . . , z tN,M N )q ε (z 1,1 , . . . , z tN,M N , ζ 1 , . . . , ζ N ) j dz j
This application is continuous according to Aronson's estimates. From this expression it comes

I ε g I ε 1 := E[g(x t1,1 , . . . , x tN,M N )C ε ψ ε ] E[C ε ψ ε ] = C ε Φ ε g k η ε k exp{- β k T k -ε (L k ζ k -v k ) 2 2ε }dζ k C ε Φ ε 1 k η ε k exp{- β k T k -ε (L k ζ k -v k ) 2 } 2ε }dζ k
We recall that the rows of each matrix L k form an orthonormal family. We now complete arbitrarily each family into an orthonormal basis of R n . We denote P k an arbitrary matrix whose first rows are given by L k . Then we make a basis change with respect to those matrices P k for each ζ k . Thus

I ε g = C ε Φ ε g (P -1 1 ζ 1 , . . . , P -1 N ζ N ) k η ε k (P -1 k ζ k ) exp{- β k T k -ε (ζ 1:m k k -v k ) 2ε
}dζ k denoting ζ i:j k the vector composed by the coordinates from i th to j th one of ζ k . We now a second change

ζ 1:m k k = √ εξ 1:m k k + v k ζ m k +1:n k = ξ m k +1:n k So that I ε g = k ε -m k 2 Φ ε g (P -1 1 ζ 1 , . . . , P -1 N ζ N ) k η ε k exp{- β k T k -ε (ζ 1:m k k -v k ) 2 2ε }dζ k = Φ ε g P -1
β k T k -ε ξ 1:m k k 2 2 }dξ k
We now use Aronson's estimates and Lemma 11 to get an integrable uniform upper bound for q ε when 0 < ε < ε 0 . Thanks to Lebesgue's theorem we obtain the convergence for the last term We conclude thanks to the Bayes formula.

Lemma 10. Let y be solution of Equation [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]. Then almost surely for all 1 ≤ k ≤ N the following integral are absolutely convergent Lemma 11. Let (Z j ) 1≤j≤K be a family of random m j -dimensional variables and let (g j : R mj → R) 1≤j≤K be a family of densities. Then the function K j=1 R mj → R (v j ) j → E j g j (Z j -v j ) is the density of the family (V j = W j + Z j ) 1≤j≤K where each of the W j whose law is given by g j is independent with respect to the (Z j ) 1≤j≤K and (W k ) k =j .

T k T k -ε k (L k y t -v k ) * A k t (
Proof. Let f be a bounded continuous function

f (v j ) j E   j g j (Z j -v j )   j dv j = E   f (v j ) j j g j (Z j -v j )   j dv j
Then we make the change of variables w j = Z j -v j for all j f (v j where W j admits g j as density.

Lemma 7 .

 7 Let us consider Equation (3) with random initial condition u on [0, T ] with N = 1 which means only one observation time in T

Figure 1 :

 1 Figure 1: Illustration of the three different dynamics considered

  We then integrate with respect to the ξ 1:m k k

  f (w j + Z j ) j j g j (w j )   j dw j = E f (W j + Z j ) j

  Proof. Using their definition, we have on [T-ε 1 , T ] d(y t -z t ) = [b t (y t ) -b t (z t )]dt + [σ t (y t ) -σ t (z t )]d wt -P t (y t )(y t -u 1 ) -P t (z t )(z t -u 1 ) T -t dt Thus d y t -z t 2 = 2(y t -z t ) * [b t (y t ) -b t (z t )]dt + [σ t (y t ) -σ t (z t )]d wt -P t (y t )(y t -u 1 ) -P t (z t )(z t -u 1 ) -z t ) * [b t (y t ) -b t (z t )]dt + [σ t (y t ) -σ t (z t )]d wt +

		T -t	dt +	i,j	(σ t (y t ) -σ t (z t )) 2 i,j dt (26)
	In a same way, on [0, T -ε 1 ] we obtain
	d y t -z t	2 = 2(y t i,j

  y t )b t (y t )dt T k -t + (L k y t -v k ) * dA k t (y t )(L k y t -v k ) 2(T k -t)We reduce the study without loss of generality to that ofdy t = b t (y t )dt + σ t (y t )d wt -σ t (y t )β t (y t ) Ly t -v T -t 1 (T -ε1,T ) (t)dtWe then treat integrability for each term.For the first term, since b and β are bounded, we use Lemma 1 to get where C is a positive random variable. Now for all positive α, we have log log x ≤ x α . Then for α small enough, we obtain integrability of righthandside.For the second term in (33), we recall that for all z we haveA t (z) = β t (z) * β t (z) = (La t (z)L * ) -1 , hence dA t = p t dt + q t d wt + r t Ly t -v T -t dtwhere p, q and r are bounded adapted processes. So that even if it means changing p q and r(Ly t -v) * dA t (Ly t -v) T -t = Ly t -v 2 T -t p t dt + Ly t -v 2 T -t q t dw t + Ly t -v 2 (T -t) 2 r t dt(34)Using Lemma 1, we obtain that the quantitiesLyt-v 2 T -t , Lyt-v 2 (T -t)2 and Lyt-v 4 (T -t) 2 are integrable in a left neighboorhood of T . For the last term in (33), we use Itô's formula and the fact that Lσ t (z)β t (z) = I d , so that on (T -ε k , T k ) d(Ly t -v) = L[b t dt + σ t d wt ] -Ly

	Ly t -v T -t	≤ C	log log (T -t) -1 + e T -t
			+
			1≤i,j≤m k

d A k i,j (y . ), (L k y . -v k ) i (L k y . -v k ) j t 2(T k -t) (33) Proof. t -v T -t dt Hence d A i,j , (Ly . -v) i (Ly . -v) j t ≤ Ly t -v p t dt

where p is the same bounded adapted process given above. Finally i,j

d A i,j , (Ly . -v) i (Ly . -v) j t T -t ≤ Ly t -v p t T -t dt

even if it means changing p, and this last term is integrable.

dt |(L k xt k = v k ) 1≤k≤N ]g(v 1 , . . . , v N ) k dv kIt remains to apply Theorem 1.

Appendix

Lemma 6. Let us consider Equation [START_REF] Dellacherie | Chapitres I à IV, Édition entièrement refondue[END_REF] with random initial condition u on [0, T ] with N = 1 which means only one observation time in T .

, where C is a positive random variable.

Proof. We recall that parameters b and σ are locally Lipschitz functions. So that the equation admits a unique solution on both intervals [0, T -ε 1 ] and (T -ε 1 , T ) and so on [0, T ). Moreover thanks to Itô's formula, on