Eole Lapeyre

Aurélie Calabrèse
email: aurelie.calabrese@univ-amu.fr

Can pulsed lighting help adults with dyslexia to read better: Testing LeFloch and Ropars's hypothesis

) a text reading comprehension score. To control for individual differences in reading skills, all participants also completed a series of French standardized tests which were used to compute an individual reading impairment score. Finally, visual acuity and eye dominance were measured to control for participants' visual function. We found no effect of lighting conditions on either of the two reading measures. This was true for all reading impairment score values and irrespective of whether participants showed eye dominance or not. Furthermore, even under pulsed lighting, individuals with dyslexia did not reach the reading performance of skilled adult readers, regardless of their eye dominance. In conclusion, we failed to show a clear positive impact of pulsed lighting on the reading skills of adults with dyslexia. This set of results does not support the visual hypothesis of dyslexia proposed by Le Floch and Ropars.

Introduction

Developmental dyslexia (hereafter, dyslexia) is the most frequent neurodevelopmental disorder, affecting approximately 7% of the population [START_REF] Peterson | Seminar: Developmental Dyslexia[END_REF]. Signi cant and persistent decoding and word-reading di culties, which may also impair reading comprehension and result in poor spelling performance, are the main symptoms of dyslexia (Lyons et al., 2003). De cit hypotheses, such as the auditory hypothesis [START_REF] Tallal | Auditory temporal perception, phonics, and reading disabilities in children[END_REF], the phonological hypothesis (Denis-Noël et al., 2020), the visual hypothesis [START_REF] Franceschini | Visuo-spatial attention de cit in children with reading di culties[END_REF] or the cerebellar/motor hypothesis [START_REF] Nicolson | Developmental dyslexia: The cerebellar de cit hypothesis[END_REF] have been proposed as possible causal explanations of dyslexia, leading to a huge number of studies (for a review, see (Colé & Sprenger-Charolles, 2021)). The aim of the present study was to test the hypothesis recently proposed by Le [START_REF] Floch | asymmetry of the Maxwell spot centroids in adults without and with dyslexia[END_REF], who developed a Hebbian pulse-width modulated lighting set-up to ease dyslexia symptoms.

Several visual hypotheses of dyslexia had previously been proposed. These include: the visual-attention span de cit hypothesis (Bosse, Tainturier & Valdois, 2007), the orientation of visual attention de cit hypothesis [START_REF] Facoetti | Visual spatial attention and speech segmentation are both impaired in preschoolers at familial risk for developmental dyslexia[END_REF][START_REF] Franceschini | Visuo-spatial attention de cit in children with reading di culties[END_REF] and, more recently, the iconic memory and visual-short-term memory transfer de cit hypothesis [START_REF] Castet | Letter and symbol identi cation: No evidence for letter-speci c crowding mechanisms[END_REF][START_REF] Castet | Dyslexia Research and the Partial Report Task: A First Step toward Acknowledging Iconic and Visual Short-term Memory[END_REF]. The common idea to all these hypotheses is that the associated visual de cit would disrupt the generation of the orthographic codes of the words being read, resulting in impaired multi-letter string processing [START_REF] Martelli | Crowding, reading, and developmental dyslexia[END_REF]. As far as orthographic coding is concerned, letter orientation is an important feature of any orthographic system and makes it possible to distinguish between different letters of an alphabet, such as b and d (Colé & Sprenger-Charolles, 2021). This is not the case for images. Thus, [START_REF] Dehaene | Why do children make mirror errors in reading? Neural correlates of mirror invariance in the visual word form area[END_REF] reported that, in adults, prior presentation of a left-right inverted item facilitates its recognition when it is a picture (e.g., a horse) but not when it is a French word written backwards (e.g., mon vs. nom; my vs.name in English). In their review on mirror image processing, [START_REF] Fernandes | Mirrors are hard to break: A critical review and behavioral evidence on mirror-image processing in developmental dyslexia[END_REF] emphasize the processing speci city of left-to-right inverted letters (b, d) compared to those inverted vertically (b, p). When learning to read in a given orthographic system, children must therefore inhibit previously acquired automatisms, leading to competition in the visual word form area (Dehaene, et al., 2010).

Physicists Le [START_REF] Floch | asymmetry of the Maxwell spot centroids in adults without and with dyslexia[END_REF] proposed another visual hypothesis of dyslexia, positing that this disorder may be attributed to a speci city in the anatomy of the eye. They suggested that individuals with dyslexia may present a weak ocular dominance due to a lack of asymmetry in the central region of their retinas. This lack of asymmetry would result in an undetermined afterimage dominance, with the primary and mirror images coexisting and possibly causing confusion regarding the identi cation of mirror letters (e.g., b and d). They consequently reported that the blue cone-free areas at the center of each retina (retinal Maxwell's centroids) in skilled adult readers with established eye dominance are asymmetrical.

Using a foveascope of their design, they outlined a noise-stimulated afterimage of Maxwell's centroids, revealing dissimilar blue cone-free areas in skilled readers: a circular one in the dominant eye and an elliptical one in the non-dominant eye. In contrast, this asymmetry was absent in adults with dyslexia, for whom the authors reported circular blue cone-free areas in both eyes and no eye dominance, despite them having normal ocular status. The authors hypothesized that the lack of asymmetry they found in individuals with dyslexia might prevent them from having the expected ocular dominance.

While the typical brain receives visual information from both optic nerves, it relies preferentially on the input from the dominant eye [START_REF] Hubel | Eye, Brain, and Vision[END_REF]. It is only at the end of the developmental period (~ 8 years) that children's eye dominance stabilizes, bringing about a decrease in mirror reversal errors [START_REF] Cornell | Spontaneous mirror-writing in children[END_REF]. According to Le Floch and Ropars' hypothesis, the lack of asymmetry found in individuals with dyslexia would prevent the brain from focusing preferentially on one input, leading to perturbations in the brain's central connectivity. This supports the idea of a poor lateralization of the brain in individuals with dyslexia [START_REF] Bishop | Cerebral Asymmetry and Language Development: Cause, Correlate, or Consequence? Science[END_REF]. Again according to this hypothesis, instead of correctly perceiving only one input, individuals would receive two equally strong inputs and may therefore be unable to properly decode visual inputs such as letters (Le Floch & Ropars, 2020). Indeed, under natural lighting, individuals with dyslexia would perceive both the primary and the afterimage of the stimulus that is presented to them.

There is some evidence that temporal modulations of light frequency and luminance can in uence perception of visual afterimages [START_REF] Fiorentini | Vision with stabilized images and intermittent illumination[END_REF][START_REF] Gerling | Duration of visual afterimages on modulated backgrounds: Postreceptoral processes[END_REF][START_REF] Matteson | After-images observed with intermittent background illumination[END_REF]. Based on such mechanisms, Le Floch and Ropars developed a set-up using Hebbian pulse-width modulated lighting in order to erase the disturbing afterimage perceived by individuals with dyslexia, thus removing the mirror image (Le Floch & Ropars, 2017). After running a letter identi cation task on ve dyslexic participants, they concluded that pulse-width modulated lighting can compensate for the lack of binocular rivalry in this population and thus improve overall reading skills. However, this result may be subject to caution for several reasons. First, only a very small sample of participants with dyslexia was tested and no description of the diagnosis or functional de cit of these participants was provided. Second, the authors only tested letter recognition performance, which does not necessarily transfer to the higher-level skill of reading. Lastly, the authors gave very little detail about the characteristics of the lighting conditions that they used, making it impossible to replicate their results.

Therefore, the goal of the present work was to assess the impact of pulsed lighting on the reading skills of a larger sample of individuals with dyslexia. Importantly, throughout this work, dyslexia was considered as a multifactorial condition, ranked along a continuous scale rather than a dichotomic one (controls vs. dyslexics). Indeed, according to current understanding, dyslexia appears to result from a multifactorial disorder involving many genes, which may result in a reading de cit under unfavorable environmental conditions (e.g., opacity of the orthographic system) [START_REF] Pennington | From single to multiple de cit models of developmental disorders[END_REF][START_REF] Pennington | Individual prediction of dyslexia by single versus multiple de cit models[END_REF][START_REF] Van Bergen | The intergenerational multiple de cit model and the case of dyslexia[END_REF]. This framework allows us to better understand and account for the reported heterogeneity in reading pro les of adults with dyslexia [START_REF] Cavalli | Screening for Dyslexia in French-Speaking University Students: An Evaluation of the Detection Accuracy of the Alouette Test[END_REF][START_REF] Laasonen | Adult Dyslexia and Attention De cit Disorder in Finland--Project DyAdd: WAIS-III cognitive pro les[END_REF][START_REF] Parrila | University Students with a Signi cant Historyof Reading Di culties:What Is and Is Not Compensated?[END_REF]. Thus, in [START_REF] Cavalli | Screening for Dyslexia in French-Speaking University Students: An Evaluation of the Detection Accuracy of the Alouette Test[END_REF], 18% of a sample of university students with dyslexia achieved normative text reading uency performances, while exhibiting a performance de cit in single-word and pseudoword reading. Another example of the heterogeneity of reading pro les in the dyslexic population comes from [START_REF] Parrila | University Students with a Signi cant Historyof Reading Di culties:What Is and Is Not Compensated?[END_REF], who reported that 75% of their sample of adults with dyslexia achieved reading comprehension scores within the norms (when reading without time pressure), while 93% continued to show de cits in written word recognition and decoding skills. This heterogeneity in the manifestations of dyslexia (possibly due to compensatory mechanisms, (see [START_REF] Colé | Les compensations de l'adulte dyslexique de niveau universitaire[END_REF] for a discussion) led us to calculate an overall reading impairment score for each participant. This measure, based on four reading test scores (namely, an isolated-word reading test, a pseudoword reading test and two reading uency tests), ranged from 0 (no impairment) to 4 (pathological threshold reached on all tests). In line with the current conception of dyslexia developed by [START_REF] Pennington | From single to multiple de cit models of developmental disorders[END_REF], this score allowed us to consider reading performance on a continuum rather than on a dichotomous scale and to evaluate the impact of each lighting condition as a function of individual reading pro les.

According to Le Floch and Ropars' (2017) hypothesis, the likelihood of observing an absence of eye dominance in adults with dyslexia should be greater than in skilled readers. Therefore, individuals with high reading impairment scores would be more likely to show an absence of eye dominance than skilled readers with no reading impairment. Furthermore, using pulsed lighting would signi cantly enhance reading performance in the dyslexic group by erasing the perception of disturbing afterimages of some of the letters present in the words being read. Hence, Le Floch and Ropars' hypothesis suggests that there should be a positive effect of pulsed lighting on reading performance, which would interact with the reading impairment score (the greater the impairment, the larger the facilitating effect) and/or the presence of ocular dominance (the facilitating effect would only be present in the absence of eye dominance).

Methods

Participants

We tested 23 participants with developmental dyslexia (9 males) and 19 chronological age-matched control readers (8 males) with no reported reading di culties. Age ranged from 19 to 27 years, with a mean (±SD) of 21±2 years in both groups. All participants were native French speakers, with normal or corrected-to-normal vision and normal audition. All were university students with equivalent schoolleaving quali cations and ranged from rst-year postgraduate to fth-year postgraduate. Participants with dyslexia were recruited from the Aix-Marseille University Disability Service and had all been diagnosed by a speech therapist earlier in life. Comorbidities with dyslexia were distributed as follows: dysorthographia (67%), speci c oral language disorder (22%), dyscalculia (19%), developmental coordination disorder (15%) and attention de cit hyperactivity disorder (11%). Written consent was obtained from all participants when they were enrolled. All received monetary compensation for their participation. The experiment followed the tenets of the Declaration of Helsinki and was approved by the local ethics committee of Aix-Marseille University.

Experimental protocol

Each participant came to the laboratory for two 90-minute sessions: a pre-test session, followed within a month by the experimental testing session. All tests were administered by a single experimenter in an empty experimental room (e.g., no windows or shelves) to avoid any distractions. During the pre-test session, a series of pre-tests was run followed by a series of tests intended to assess the individual reading impairment score. During the experimental session, the in uence of pulsed lighting on two main components of reading performance was estimated: sentence reading speed, estimated using the MNREAD chart [START_REF] Mans Eld | A new reading-acuity chart for normal and low vision. Ophthalmic and Visual Optics/Noninvasive Assessment of the Visual System Technical Digest[END_REF], and text reading comprehension, measured with a French standardized test [START_REF] Brèthes | Text Reading Fluency and Text Reading Comprehension Do Not Rely on the Same Abilities in University Students With and Without Dyslexia[END_REF]. Details of the experimental protocol are detailed below and summarized in Fig. 1.

Pre-tests

All participants completed a series of French tests, which were administered using a standard procedure (Fig. 1) [START_REF] Brèthes | Text Reading Fluency and Text Reading Comprehension Do Not Rely on the Same Abilities in University Students With and Without Dyslexia[END_REF]. The French version of the Adult Reading History Questionnaire (ARHQ) and Anamnesis (Le y & [START_REF] Le Y | Reliability and Validity of the Adult Reading History Questionnaire[END_REF] were used to screen for dyslexia. IQ levels were estimated both verbally using the French version of the Peabody Picture Vocabulary Test (PPVT, EVIP in French) [START_REF] Dunn | Échelle de vocabulaire en images Peabody: EVIP[END_REF] and non-verbally with Raven's Matrices [START_REF] Raven | Manual for Raven's Progressive Matrices and Vocabulary Scales[END_REF]. Phonological short-term memory and phonemic awareness were also tested with the EVALEC toolset (Sprenger-Charolles et al., 2005). The Alouette test (Lefavrais, 1967) was used to assess reading skills. Because, it consists of a series of real words embedded in meaningless but grammatically and syntactically correct sentences, dyslexic and poor readers cannot use contextual information to compensate for their written word recognition di culties. Here, we used a standardized version of the test developed for adults by [START_REF] Cavalli | Screening for Dyslexia in French-Speaking University Students: An Evaluation of the Detection Accuracy of the Alouette Test[END_REF].

Participants were asked to read the 265-word text aloud as rapidly and as accurately as possible within a maximum of 3 minutes. The results allowed us to derive a reading e ciency score combining speed and accuracy. Finally, three more standardized reading tests were run [START_REF] Brèthes | Text Reading Fluency and Text Reading Comprehension Do Not Rely on the Same Abilities in University Students With and Without Dyslexia[END_REF]: an isolated word reading test (to measure the e ciency of the orthographic procedure), a pseudoword reading test (to assess the e ciency of the decoding procedure) and a reading uency test. As shown in Table 1, readers with dyslexia performed signi cantly worse than controls on every skill, except for verbal IQ. Reading Impairment Score estimation

For each participant, we used the Alouette test and three additional reading tests (Fig.

Visual tests

For each participant, binocular visual acuity was measured with prescribed correction, if any, using the ETDRS letter chart [START_REF] Ferris | New visual acuity charts for clinical research[END_REF]. Corrected visual acuity was normal for all participants, with a mean value of 0.0 ± 0.1 logMAR. Eye dominance was estimated using a simple dominant eye test, derived from the distance hole-in-the-card test [START_REF] Miles | Ocular Dominance in Human Adults[END_REF]. Participants were asked to extend their arms and place their hands together at a 45-degree angle to create a triangular opening between their thumbs and fore ngers. The hand opening was centered on a distant object and the participants had both eyes open. Closing each eye one at a time, the dominant eye was de ned as the one that made it possible to keep the target object centered within the opening. Overall, 83.3% of our participants showed a strong ocular dominance, 9.5% showed a weak ocular dominance and 4.7% showed no ocular dominance. Detailed results are given in Fig. 2 and Appendix 1.

Sentence reading speed assessment MNREAD is a standardized reading test designed to estimate reading speed as a function of print size through a series of short sentences. Participants are presented with 60-character sentences (~ 10 words) and asked to read them out loud, one at a time and as fast and accurately as possible. Each new sentence is presented in a smaller type size than the previous one until it becomes so small that reading is impossible. All sentences were printed on cardboard in Times font on three lines of justi ed text. In our study, MNREAD was presented at a distance of 80 cm in its regular polarity (black print on a white background).

Text reading comprehension assessment

Our second outcome measure was text reading comprehension performance, assessed with a new French standardized task which consists in the silent reading of a newspaper article from the French newspaper, Le Monde (~ 500 words). The text is read without time constraints and is followed by a set of questions [START_REF] Brèthes | Text Reading Fluency and Text Reading Comprehension Do Not Rely on the Same Abilities in University Students With and Without Dyslexia[END_REF]; https://osf.io/zmf82/). This newly validated test makes it possible to evaluate literal comprehension as well as inferential comprehension skills, namely text-connecting inference skills (which require the participant to integrate text information in order to establish local cohesiveness) and knowledge-based inference skills (which make it possible to establish links between the text content and the reader's personal knowledge). Once participants had nished reading, they answered a series of oral multiple-choice or open questions. They were not allowed to go back to the text when answering the questions. A comprehension score was assessed, ranging from 0 (i.e, no comprehension) to 100 (perfect comprehension).

Lighting conditions

In order to assess the in uence of lighting on the two outcome measures (reading accessibility index and text reading comprehension score), each reading test was administered using different test versions under three lighting conditions: (1) a control condition with regular lighting, (2) a pulsed light condition, and (3) a combination condition with both regular and pulsed light. Control lighting was provided by a standard desk lamp, set to its standard luminance (136 lumens measured in the laboratory with a Konica Minolta LS-150 photometer). Pulsed lighting was provided using the Lili Lamp, designed by the Lili for life company and based on the patented technology from Le Floch and Ropars. Lili projects a pulsed luminous ux that can be customized in terms of both pulsation frequency and on/off time balance. Frequency and balance parameters were optimally chosen for each participant before the experiment. Settings were considered optimal when participants felt comfortable reading a mock text presented in front of them on a sheet of paper. The settings chosen for each participant are reported in Appendix 1.

The luminance of Lili (116 lumens) was noticeably lower than in our control condition. To avoid any luminance bias, we added a third condition, combining both light settings, in which the luminance (131 lumens) matched our control condition, while pulsed lighting with optimal settings was also provided. The order of the lighting conditions was randomized across participants and the lighting arrangements hid the different lamps from the participants' view.

Statistical analysis

Statistical analysis was performed using R statistics (R core team 2020, version 4.0.2). Linear mixedeffects (LME) models were used to assess the effect of the lighting conditions on each of our outcome measures (i.e., reading accessibility index and comprehension score). LME models are especially appropriate here, since they make it possible to consider the random effects associated with individual participants and their repeated measures. Two LME models were tted, one for each outcome measure, which was set as the dependent variable. In both models, reading impairment score, ocular dominance strength and lighting condition were used as independent variables (i.e., xed effects). Their interactions were also included in the models. Non-verbal IQ score (Raven) was initially included in the models, but was then discarded from nal models because it showed no signi cant in uence on the effect of lighting condition. The random structure of both models included a random intercept for participants, which assumed a different baseline performance level for each individual. Both the reading accessibility index and comprehension score were transformed into natural logarithm (ln) units to satisfy the assumptions of parametric statistical tests [START_REF] Howell | Statistical methods for psychology[END_REF][START_REF] Tabachnick | Using multivariate statistics[END_REF]. Optimal model structures were assessed using the Akaike Information Criterion (AIC) and likelihood-ratio tests [START_REF] Zuur | A protocol for data exploration to avoid common statistical problems[END_REF]. The signi cance of the xed effects was estimated using t-values. Results were considered signi cant for tvalues larger than 2 or smaller than - 2, corresponding to a 5% signi cance level in a two-tailed test [START_REF] Baayen | Mixed-effects modeling with crossed random effects for subjects and items[END_REF][START_REF] Gelman | Data analysis using regression and multilevel/hierarchical models[END_REF]). In the Results section, xed-effects estimates are reported along with their t-values and 95 percent con dence intervals [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF]).

An a posteriori power analysis was conducted with the mixedpower package in R (Kumle et al., 2021). This indicated that a sample size of 40 participants would provide a statistical power above 80%.

Results

Analysis 1 -Reading Accessibility Index (ACC)

According model 1, the average ACC of expert readers (Reading Impairment = 0) with a strong ocular dominance and under control lighting was 0.97 (exp(-0.032) -Table 2). For these same readers, switching to pulsed + control lighting led to a non-signi cant increase in ACC by a factor of 1.008 (exp(0.008); t = 0.361; 95% CI = [-0.033;0.048]), with an average index value of 0.98 exp(-0.032 + 0.008).

Similarly, switching to pulsed lighting only led to a non-signi cant change (t = -0.139; Fig. 3). This was also true for expert readers with weak ocular dominance, for whom switching to pulsed + control lighting or pulsed lighting did not signi cantly modify ACC (t = - According to this second model, the average score of expert readers (Reading Impairment = 0) with strong ocular dominance under control lighting condition was 56.9 (exp(4.042) -Table 3). When the same participants were switched to the other conditions, the average scores were unchanged: 56.8 (exp(4.042 -0.001); t=-0.003) in the pulsed + control condition and 56.6 (exp (4.042 - 0.006); t=-0.029) in the pulsed condition. This was also true for expert readers with weak ocular dominance, for whom switching to pulsed + control lighting or pulsed lighting did not signi cantly modify ACC (t = -0.106 and t = -0.481 respectively). Similarly, changing lighting conditions did not signi cantly modify the comprehension scores of readers with a reading impairment of 1 or above (t = 0.74 and t = 0.811). Finally, changes in reading impairment score did not signi cantly alter the comprehension score (t = -1.956; Fig. 4).

Discussion

aim of the present study was to test Le Floch and Ropars' hypothesis according to which dyslexia is mainly triggered by visual processing de cits, namely an absence of eye dominance, which induce the aberrant perception of afterimages for individual letters (Le Floch & Ropars, 2017). According to these authors, dyslexic readers would be expected to achieve reading performances similar to those of controls under pulsed lighting conditions. To test this hypothesis, we assessed the reading performance of young adults with and without dyslexia reading under pulsed lighting, while taking into account the presence (or absence) of eye dominance as well as the functional impact of dyslexia on their reading abilities. Our main result is that we found no effect of pulsed lighting on the two reading scores we used, namely the MNREAD reading accessibility index (ACC) and a text reading comprehension score. This was true for all values of the reading impairment score, irrespective of whether participants showed eye dominance or not. In other words, we failed to show a clear positive impact of pulsed lighting on the reading performance of a group of young adults with dyslexia.

Indeed, individuals with dyslexia did not reach control readers' performance, even in the pulsed lighting condition. This suggests that pulsed lighting is not su cient to alleviate the reading de cit experienced by dyslexics and to restore reading skills. Over the years, different visual remediation strategies have also been explored to eliminate and/or reduce visual impairment, through visual aids or training. For instance, colored overlays have been used to alleviate visual stress (i.e., distortions) in dyslexic reader, but no clearcut effects have been observed [START_REF] Evans | A systematic review of controlled trials on visual stress using Intuitive Overlays or the Intuitive Colorimeter[END_REF][START_REF] Henderson | Questioning the bene ts that coloured overlays can have for reading in students with and without dyslexia[END_REF]. Lawton and Shelley-Temblay (2017) administered visual training to dyslexic children in order to restore the functioning of the magnocellular pathway (activated by eye movement), which is thought to be de cient and responsible for a lack of synchronization with the parvocellular pathway (activated during xations). Using exercises focusing on movement detection and its direction, such training is intended to improve the sensitivity and synchronization of magnocellular processing. Another type of training currently being investigated involves the use of action video games. These games, which feature fast-moving elements and targets, along with numerous spatially and temporally unpredictable events, place considerable emphasis on the peripheral visual eld as well as on the global visual processing of elements and impose a high motor load. Thanks to these features, they strongly solicit the player's attentional, visual and visual-attentional capacities, improvements which would then transfer to reading. In particular, action video games might increase the useful size of the visual eld and improve the discrimination of rapid sequences of visual stimuli, of the sort that occur in reading. Letters in words must be accurately selected from surrounding graphemes through the rapid orientation of visual attention before their correct letter-sound translation for word decoding can occur. The second important result of this study is that the majority of the participants with dyslexia (87%)

showed strong ocular dominance and this was true even of those with the highest reading impairment score of 4. This proportion is higher than in Le Floch & Ropars, (2017), who reported that only 53% of their participants with dyslexia showed ocular dominance (also measured with the hole-in-the-card test). While it may be argued that our sample did not respond differently to pulsed lighting because it mainly showed strong ocular dominance, this result actually implies that dyslexia may not be related to a lack of eye dominance, contrary to the suggestion made by Le Floch & Ropars, (2017). In the 60's, a number of authors had already suggested that the absence of ocular dominance might play a pathogenetic role in developmental dyslexia [START_REF] Benton | Dyslexia And Dominance[END_REF][START_REF] Bettman | Cerebral Dominance in Developmental Dyslexia: Role of the Ophthalmologist[END_REF]. However, con icting results led this hypothesis to fall out of favor until it was lent renewed interest through the work of [START_REF] Stein | Visual dyslexia[END_REF]. More recently, Zeri et al. (2011) postulated that these con icting results may be due to a lack of consistency between the results of different dominance tests, which assess different dimensions of eye dominance. For instance, while the hole-in-the-card test estimates sighting eye dominance (i.e., the preferential choice of one eye during monocular tasks), other tests, such as the Dunlop test, evaluate motor ocular dominance (related to binocular con ict in binocular vision) [START_REF] Evans | Dyslexia and vision[END_REF]. In their thorough investigation of the relationship between ocular dominance and reading performance, Zeri et al., (2011) con rmed that different tests yield different stability measures and concluded that test selection is critical when assessing eye dominance stability. However, their results lent no support to the idea that there is a link between stability of ocular dominance and reading performance for any of the tests they considered. The present results, obtained with a standard test of monocular vision (chosen to match Le Floch and Ropars' methodology and to simplify the comparison of the results), are in line with these ndings.

Finally, we used a continuous reading impairment score in order to grade the overall de cit experienced by dyslexic readers. This score is intended to represent a reader's level of impairment on the basis of a certain number of reading skills, namely, reading speed and accuracy, reading uency, orthographic reading procedure and reading decoding procedure. Our results show that individuals with a high reading impairment score have a poorer reading accessibility index than readers with higher scores. In other words, the more pathological markers of dyslexia a reader has, the harder it is to achieve a reading accessibility index comparable to that of expert readers. This result suggests that considering dyslexic readers' impairment level on a continuous scale using a reading impairment score of this type may be an e cient and sensitive way to grade the overall de cit experienced by such readers. In addition, the present work highlights the use of the reading accessibility index (ACC) to assess the speci c impact of dyslexia on reading speed. While, the MNREAD test was rst designed to assess reading performance in normally sighted and visually impaired individuals, our results suggest that ACC may be sensitive enough to provide a measure of reading speed impairment in individuals with dyslexia. Interestingly, the signi cant body of literature available on MNREAD for control readers of all ages would allow researchers to measure the impact of developmental dyslexia on reading performance through comparisons with age-speci c normative data (Baskaran et al., in press;[START_REF] Calabrèse | Baseline MNREAD Measures for Normally Sighted Subjects From Childhood to Old Age[END_REF].

As in Le Floch and Ropars, the present study investigated reading using the French orthographic system. It is known that the characteristics of orthographic systems may have an in uence on the impairment of reading processes in dyslexia. For instance, it has been shown that brain activity during reading or reading-related tasks differs in people classi ed as dyslexic compared to skilled readers [START_REF] Maisog | A Meta-analysis of Functional Neuroimaging Studies of Dyslexia[END_REF][START_REF] Richlan | Functional abnormalities in the dyslexic brain: A quantitative meta-analysis of neuroimaging studies[END_REF] and that the amplitude of this difference is modulated by orthographic depth [START_REF] Martin | Dyslexic brain activation abnormalities in deep and shallow orthographies: A meta-analysis of 28 functional neuroimaging studies[END_REF]. Therefore, it remains to be tested whether our results can be applied to other orthographic systems, the orthographic depth of which may differ from French. Furthermore, while our focus was on sentence and text reading skills, further investigation of the effect reported by Le Floch and Ropars should be conducted using simpler tasks. For instance, it would be possible to use isolated words and pseudowords, the identi cation of which requires precise orthographic coding and cannot be accomplished using semantic or contextual knowledge, unlike in the case of text reading. This may shed some light on the speci city of any potential bene ts of pulsed lighting in dyslexia. Finally, while this study focuses on young adults, future investigations should consider assessing the impact of pulsed lighting on a cohort of children of different ages and reading expertise.

Declarations

Author contributions AC and PC conceived and designed the analysis. EL and JBM collected the data. EL and AC performed the data analysis. EL, PC and AC wrote the manuscript.

Figures

Table 1 .

 1 Comparison of our two sample populations (readers with and without dyslexia) for each of the inclusion pre-tests. Mean and standard deviation are given for each test, along with the p-value of the Kolmogorov-Smirnov test comparing our two sample distributions. The test results showing no signi cant difference between the two groups are highlighted in grey. The tests used to estimate the reading impairment score are highlighted in green.

		Control readers	Readers with dyslexia	p-value
		(mean ± SD)	(mean ± SD)
	ARHQ	0.29 ± 0.10	0.61 ± 0.12	<0.0001
	Verbal IQ (EVIP)	38 ± 5	34 ± 5	0.25
	Non-verbal IQ (Raven)	50 ± 6	46 ± 4	0.01
	honological short-term memory	0.97 ± 0.13	0.72 ± 0.14	<0.001
	Phonemic awareness	1.02 ± 0.14	0.61 ± 0.21	<0.0001
	Alouette test	545 ± 85	376 ± 67	<0.0001
	Pseudoword reading test	134 ± 36	81 ± 30	<0.0001
	Word reading test	111 ± 21	81 ± 24	<0.001
	Text reading fluency	225 ± 41	154 ± 34	<0.001

  The reading time (in seconds) and accuracy (in number of misread words) of each presented sentence were measured and used to derive a reading speed value in words per minute. For each test, a Reading Accessibility Index value was then derived as follows: mean reading speed across the 10 largest physical print sizes on the chart, normalized by the value for a group of normally sighted

young adults

[START_REF] Calabrèse | Development of a Reading Accessibility Index Using the MNREAD Acuity Chart[END_REF][START_REF] Calabrèse | MnreadR, an R package to analyze MNREAD data[END_REF]

. This rst outcome measure is a single-valued score that represents each participant's visual access to commonly encountered printed material, where 0.0 means no access to print, 1.0 represents average normal performance and values greater than 1.0 indicate that individuals exceed the mean for normally sighted young adults.

Table 2 .

 2 This is consistent with well-established reports of impaired reading speed in individuals with dyslexia. It also suggests that, for any given individual, the more skills affected by dyslexia are labelled as impaired, the more reading speed is impaired. Still according to model 1, we found no signi cant change in ACC Fixed effects estimates from LME model 1. The intercept estimate represents the logtransformed accessibility index when all factors included in the model are at their reference level (categorical variable) or at 0 (continuous variable). Reference levels are: control lighting condition, strong ocular dominance and reading impairment score of 0. Interactions are represented by the symbol ":". Factors showing a signi cant effect on reading time are in bold font.

	0.809; 95% CI = [-0.105;0.042] and t = -1.324; 95%

when going from control lighting to pulsed + control for any individual with a reading impairment of 1 or above, (t = 1.337; 95% CI = [-0.004;0.025]), and observed only a barely signi cant increase in ACC (by a factor of 1.018 -exp(0.008)) when switching to pulsed lighting only (t = 2.273; 95% CI = [0.003;0.033]).

Table 3 .

 3 Fixed effects estimates from LME model 2. The dependent variable is the log-transformed comprehension score. The intercept estimate represents the log-transformed comprehension score when all factors included in the model are at their reference level (categorical variable) or at 0 (continuous variable). Reference levels are: control lighting condition, strong ocular dominance and reading impairment score of 0. Interactions are represented by the symbol ":". Factors showing a signi cant effect on comprehension score are in bold font.

	Estimate St. Error t-value	95% CI

  Effective visual attention would improve the perception of visual stimuli and increase the development of letter-sound connections. Using action video games with children with dyslexia[START_REF] Franceschini | Action Video Games Make Dyslexic Children Read Better[END_REF], reported a signi cant improvement in these children's reading uency skills. However,(Łuniewska et al., 2018) failed to replicate this effect. Overall, even though a number of studies have reported that individuals with dyslexia may use a different visual sampling strategy to read texts (e.g., longer xation durations, shorter saccades and fewer skipped words during rst-pass reading[START_REF] Franzen | Individuals with dyslexia use a different visual sampling strategy to read text[END_REF])), the question of which visual de cit causes this phenomenon remains unanswered.

Data availability statement

Data will be fully available upon request to the corresponding author (AC) at aurelie.calabrese@univamu.fr

Additional Information

Competing Interests Statement

The authors declare no competing interest. Effect of reading impairment score on score for three different lighting conditions. Data points show the raw data. Lines represent the ts estimated by model 2 and shaded areas their 95% con dence intervals. Lighting conditions are color-coded as follows: control condition is represented in yellow, pulsed condition in green and pulsed + control in purple.

Supplementary Files

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Lapeyreatalsupplementarymaterial.pdf


Ludovic Goudenege
email: ludovic.goudenege@math.cnrs.fr

Andrea Molent
email: andrea.molent@uniud.it

Antonino Zanette
email: antonino.zanette@uniud.it

Computing XVA for American basket derivatives by Machine Learning techniques

Keywords: XVA, Gaussian Process Regression, Basket option, Control variates

 for the one-dimensional American derivatives. The evaluation of the XVA for basket derivatives is particularly challenging as the presence of several underlings leads to a high-dimensional control problem. We tackle such an obstacle by resorting to Gaussian Process Regression, a machine learning technique that allows one to address the curse of dimensionality effectively. Moreover, the use of numerical techniques, such as control variates, turns out to be a powerful tool to improve the accuracy of the proposed methods. The paper includes the results of several numerical experiments that confirm the goodness of the proposed methodologies.

discussed methods for calculating XVA are not exempt from this limitation.

Newer techniques for evaluating derivatives make use of machine learning methods. In this regard, some authors employ neural networks. For example, Lapeyre and Lelong [START_REF] Lapeyre | Neural network regression for Bermudan option pricing[END_REF] study the Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation. Becker et al. [START_REF] Becker | Deep optimal stopping[END_REF][START_REF] Becker | Pricing and hedging American-style options with Deep Learning[END_REF][START_REF] Becker | Solving high-dimensional optimal stopping problems using Deep Learning[END_REF] develop deep learning methods for pricing and hedging American-style and, more generally, for solving optimal control problems. Other authors exploit Gaussian Process Regression (GPR), a Machine Learning technique that allows for estimations from scattered data in large dimensional spaces. In this regard, we mention the work of Ludkovski [START_REF] Ludkovski | Kriging metamodels and experimental design for Bermudan option pricing[END_REF], who evaluates Bermudan options by fitting the continuation values through GPR. More recently, Goudenège et al. [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF] propose three GPR-based algorithm, termed GPR-MC, GPR-Tree and GPR-EI, for pricing American options on a basket of assets following multi-dimensional Black-Scholes dynamics.

The literature on the computation of the XVA for high dimensional derivatives is rather sparse. As far as the computation of the credit adjustments are concerned, She and Grecu [START_REF] She | Neural network for CVA: Learning future values[END_REF] compute CVA and DVA by employing neural network as a universal approximator. Crépey and Dixon [START_REF] Crépey | Gaussian process regression for derivative portfolio modeling and application to credit valuation adjustment computations[END_REF] exploit GPR to speed up the computation of the CVA for derivatives portfolios. Gnoatto et al. [START_REF] Gnoatto | Deep xVA solver-a neural network based counterparty credit risk management framework[END_REF] exploit artificial neural network to compute the XVA for large portfolios of derivatives. Despite the importance of this topic, to our knowledge, no one has ever studied the calculation of XVA for American basket options, which are probably the most popular option involving several assets.

In this paper, we aim to fill this gap, by proposing an approach based on a suitable probabilistic formulation of the XVA, derived from the model of Burgard and Kjaer [START_REF] Burgard | PDE representations of options with bilateral counterparty risk and funding costs[END_REF], which exploits the GPR-MC and the GPR-EI algorithms for option pricing to overcome the curse of dimensionality. We point out that we have chosen to consider Burgard and Kjaer's model as it is particularly suitable as, unlike other models, the American option exercise strategy is shaped to take into account the probability of default of any of the agents. Moreover, depending on the choice of the mark-to market value, two possible kinds of models are considered: a linear and a non-linear. Furthermore, the computation accuracy is increased by exploiting suitable control variate for both the riskless and the risky price. Numerous numerical tests demonstrate the reliability and accuracy of the proposed procedures when different derivatives are considered.

The remainder of the paper is organized as follows. In Section 2 we introduce the model for XVA on American basket options. In Section 3 we describe the proposed procedures. In Section 4 we discuss numerical results. Finally, in Section 5, we conclude.

Total value adjustment for American basket options

In this Section, we describe a PDE-based model for the total value adjustment when American options are concerned and we discuss a probabilistic interpretation that we are going to exploit for our approaches. We stress out that the model we develop here is inspired by the framework previously introduced by Burgard and Kjaer [START_REF] Burgard | PDE representations of options with bilateral counterparty risk and funding costs[END_REF] and developed by Arregui et al. [START_REF] Arregui | PDE models and numerical methods for total value adjustment in European and American options with counterparty risk[END_REF][START_REF] Arregui | A Monte Carlo approach to American options pricing including counterparty risk[END_REF], which is very interesting among the others because it allows the exercise strategy of the American option to be influenced by the possibility of default of each agent. This phenomenon, which is certainly plausible in reality, is not present in other models. For example, the model by De Graaf et al. [START_REF] De Graaf | Efficient computation of exposure profiles for counterparty credit risk[END_REF][START_REF] De Graaf | Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method[END_REF], which is usually employed be other authors, considers the strategy for the risky option to be the same as the strategy for options without default risk but, in our opinion, this does not seem to be the right choice. This aspect is pointed out in the following Remark.

Remark 1. Consider an American call option, which is at the money at the time of issue. Now, suppose an agent buys such an option from a counterparty that provides a null recovery rate and that is going to default before the maturity of the option almost sure. It is well known that if the underlying does not pay dividends, and the counterparty does not default, it is never optimal to exercise an American call option before maturity. So, if the agent employs the standard strategy, he will achieve a payoff equal to zero almost sure, as the default of the counterparty will occur before maturity and the option will lose all its value (the recovery rate is zero). On the other hand, if he exercises the option immediately, he will obtain a positive payoff (the option is in-the-money), so this strategy is better than the classical one. This simple example shows that the optimal strategy for exercising an American option must take the default risk into account.

Let S = (S t ) t∈[0,T ] = S 1 t , . . . , S d t t∈[0,T ]
denote a d-dimensional stochastic process following the multi-dimensional Black-Scholes model. Under the risk neutral probability Q, the dynamics of each underlying is given by

dS i t = (r -η i ) S i t dt + σ i S i t dW i t , i = 1, . . . , d, (2.1)
with S 0 = s 1 0 , . . . , s d 0 ∈ R d + the spot price, r the (constant) interest rate, η = (η 1 , . . . , η d ) the vector of dividend rates, σ = (σ 1 , . . . , σ d ) the vector of volatilities, W a d-dimensional correlated Brownian motion and ρ ij the instantaneous correlation coefficient between W i t and W j t . Let us consider an American option issued at time 0 with maturity T and let H : R d + → R denote the payoff function. Let us term B the issuer and C the buyer of the option. For the moment, we suppose that none of the two agents can default. We approximate the value of the risk-less American option by a Bermudan option which can be exercise at the times t n = n • ∆t for n = 0, . . . , N with ∆t = T /N and N ∈ N. By employing standard arguments, one can prove that

V (t n , S t ) = max (C (t, S t ) , H (S t )) ,
where C (t, S t ) stands for the continuation value. In particular, C (t, S t ) restricted to the time interval ]t n , t n+1 [ is equal to C n (t, S t ), the solution of the following PDE, defined in ]t n , t n+1 [for n = 0, . . . , N -1 :

∂C n ∂t + A (C n ) -rC n = 0, with C n = C n (t n+1 , x) for x = (x 1 , . . . , x d ) and A (C n ) = d i=1 (r -η i ) x i ∂C n ∂x i + d i=1 σ 2 i x 2 i 2 ∂ 2 C n ∂x 2 i + d-1 i=1 d j=i+1 ρ i,j σ i σ j x i x j ∂ 2 C n ∂x i ∂x j .
The terminal condition is

C n (t n+1 , x) = H (x) if n + 1 = N, V (t n+1 , x) otherwise.
Now, let us suppose that both agents B and C can default. We take the point of B, and we denote the risky option price by V t, S t , J B t , J C t , with J B and J C two independent jump processes that change value, from 0 to 1, at the time the corresponding agent defaults.

Let M t = M (t, S t ) represent the close-out mark-to-market value, that is, the monetary value of the contract used as the basis for settlement. Let us define M + = max (M, 0) and M -= min (M, 0). Following Burgard and Kjaer [START_REF] Burgard | PDE representations of options with bilateral counterparty risk and funding costs[END_REF], in case of default of one counterparty, the risky values are defined as follows: [START_REF] Antonelli | Approximate value adjustments for European claims[END_REF] the recovery rate of C respect to the default of B;

• if the issuer B defaults first, V (t, S t , 1, 0) = M + t + R B M - t , with R B ∈ [0,
• if the buyer C defaults first,

V (t, S t , 0, 1) = R C M + t + M - t , with R C ∈ [0, 1]
the recovery rate of B respect to the default of C.

Let λ B and λ C be the constant default intensities of B and C, respectively, and s F the funding cost of B. According to Burgard and Kjaer [START_REF] Burgard | PDE representations of options with bilateral counterparty risk and funding costs[END_REF], if the derivative can be used as a collateral, then s F = 0, and if it cannot, then s F = (1 -R B ) λ B . Following Arregui et al. [START_REF] Arregui | PDE models and numerical methods for total value adjustment in European and American options with counterparty risk[END_REF][START_REF] Arregui | A Monte Carlo approach to American options pricing including counterparty risk[END_REF]

∂ Ĉn ∂t + A Ĉn -r Ĉn = (λ B + λ C ) Ĉn + s F M + -λ B R B M -+ M + -λ C R C M + + M -, (2.2)
with the terminal condition

Ĉn (t n+1 , x) = H (x) if n + 1 = N, V (t n+1 , x, 0, 0) otherwise.
We proceed backward in time. Suppose V n (t n+1 , x) is known and we aim to compute V n (t n , x). By the Feynman-Kac formula applied to equation (2.2), (see e.g. Platen and Heath [START_REF] Platen | A benchmark approach to quantitative finance[END_REF]), we have

Ĉn (t n , x) = E Q tn+1 tn e -r0(u-tn) g (u, S u ) du + e -r0∆t Ĉn t n+1 , S tn+1 | S tn = x , (2.3)
with

r 0 = r + λ B + λ C , g (u, S u ) = -s F M + u -λ B R B M - u + M + u -λ C R C M + u + M - u = M + u (λ B + λ C R C -s F ) + M - u (λ C + λ B R B ) = M + u c p + M - u c m , c p = λ B + λ C R C -s F and c m = λ C + λ B R B .
In particular, as λ B , λ C , R C and R B are positive quantities and s F is equal to 0 or (1 -R B ) λ B , thus c p and c m are positive values.

We approximate the integral in (2.3) by a two points trapezoidal quadrature rule:

Ĉn (t n, , x) ≈ E Q e -r0∆t g t n+1, , S tn+1 + g (t n, , S tn ) 2 ∆t + e -r0∆t Ĉn t n+1 , S tn+1 | S tn = x = e -r0∆t E Q ∆t 2 g t n+1, , S tn+1 + Ĉn t n+1 , S tn+1 | S tn = x + ∆t 2 g (t n, , S tn ) ,
and thus

V (t n, , x, 0, 0) ≈ max e -r0∆t E Q ∆t 2 g t n+1, , S tn+1 + V t n+1 , S tn+1 , 0, 0 | S tn = S + ∆t 2 g (t n, , x) , H (x)
(2.4) Now, we distinguish two cases: M u = V (u, S u ), that is the value of the risk-free derivative, and M u = V (u, S u , 0, 0) , that is the value of the defaultable derivative.

Case M = V

We suppose that the values of V have already been computed in a suitable domain. If M u = V (u, S u ), we can compute V t n, , S tn, , 0, 0 explicitly, by replacing M with the pre-computed values of V and by approximating the expectation in (2.4) by a suitable numeric technique.

Case

M = V If M u = V (u , , S u ), then V (t n, , x, 0, 0) ≈ max E (x) + ∆t 2 V (t n, , x, 0, 0) + c p + V (t n, , x, 0, 0) -c m , H (x) , (2.5)
with

E (x) = e -r0∆t E Q ∆t 2 V t n+1 , S tn+1 , 0, 0 + c p + V t n+1 , S tn+1 , 0, 0 -c m + V t n+1 , S tn+1 , 0, 0 | S tn = x .
(2.6) We define Ṽ (t n, , x) as the solution of the implicit equation problem

Ṽ (t n, , x) = max E (x) + ∆t 2 Ṽ (t n, , x) + c p + Ṽ (t n, , x) -c m , H (x) , (2.7)
and we employ it as an approximation of V (t n, , x, 0, 0). Equation (2.7) is implicit -V appears both on left and the right side of the equation -and non linear. The following proposition discuss how to solve it.

Proposition 2.1. Let Ṽ (t n, , x) be the unique solution of the implicit equation (2.7). Then, if H (x) ≤ 0:

• if E (x) ≤ H (x) 1 -∆t 2 c m ≤ 0 then Ṽ (t n, , x) = H (x); • if H (x) 1 -∆t 2 c m < E (x) ≤ 0 then Ṽ (t n, , x) = E(x) 1-∆t 2 cm ; • if E (x) > 0 then Ṽ (t n, , x) = E(x) 1-∆t 2 cp . If H (x) > 0: • if E (x) ≤ H (x) 1 -∆t 2 c p then Ṽ (t n, , x) = H (x); • if E (x) > H (x) 1 -∆t 2 c p then Ṽ (t n, , x) = E(x) 1-∆t 2 cp . The proof of Proposition 2.1 is discussed in the Appendix A.

Gaussian Process Regression for computing XVA

According to the previous Section, the calculation of XVA requires the computation of an expected value, both in the case M = V and in the case M = V , see (2.4). This calculation involves a stochastic underlying which is a multidimensional process, potentially high dimensional. We propose to use two techniques, already successfully applied by Goudenège et al. [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF] for multidimensional option pricing problems: GPR-MC and GPR-EI.

Below, we recall the main aspects of these two methods, and we refer the interested reader to [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF] for more information.

GPR-MC

The GPR Monte Carlo approach employs Monte Carlo simulations to compute the continuation value of a Bermudan option and GPR to learn the option value at each time step.

The algorithm starts by simulating a set of trajectories of the underlyings. Let X n represent the set of P points whose coordinates represent certain possible values for the underlyings at time t n , for n = 0, . . . , N, that is

X n = {x n,p = (x n,p 1 , . . . , x n,p d ) , p = 1, . . . , P } ⊂ R d . (3.1)
The points of the sets X n are computed by employing the Halton's low-discrepancy sequence in R d and standard algorithms for simulating the undelying values in the multidimensional Black-Scholes model. Now, suppose we want to compute the continuation value of an Bermudan option but only for S tn = x n,p ∈ X n . This goal can be achieved by means of a one step Monte Carlo simulation. In particular, for each x n,p ∈ X n , we simulate a set of M points

Xn p = {x n,p,m = (x n,p,m 1 , . . . , xn,p,m d ) , m = 1, . . . , M } ⊂ R d ,
which are possible values for S tn+1 according to the law of S tn+1 |S tn = x n,p . In particular, for i = 1, . . . , d, n = 1, . . . , N , p = 1, . . . , P , m = 1, . . . , M , we define

xn,p,m i = x n,p i e (r-ηi-1 2 σ 2 i )∆t+ √ ∆tσiΣiG n,p,m , (3.2)
where G n,p,m ∼ N (0, I d ) is a standard Gaussian random vector and Σ i is the i-th row of the matrix Σ, which is defined as a square root of the correlation matrix Γ of the multidimensional Brownian increments. Thus, the risk-less option value can be approximated for each x n,p ∈ X n by the following scheme:

   V MC n (x n,p ) = max e -r∆t M M m=1 V MC n+1 (x n,p,m ) , H (x n,p ) if n < N, V MC (t n , x n,p ) = H (x n,p ) if n = N. (3.3)
Furthermore, the risky value V MC n (x n,p ) for M = V , t = t n and S tn = x n,p is approximated by the following scheme:

   V MC n (x n,p ) = max ĈMC n (x n,p ) , H (x n,p ) if n < N, V MC n (x n,p ) = H (x n,p ) if n = N, (3.4)
with

ĈMC n (x n,p ) = e -r0∆t M M m=1 ∆t 2 V MC n+1 (x n,p,m ) + c p + V MC n+1 (x n,p,m ) -c m + V MC n+1 (x n,p,m ) + + ∆t 2 V MC n (x n,p ) + c p + V MC n (x n,p ) -c m .
Finally, the risky value V MC n (x n,p ), for M = V , t = t n and S tn = x n,p , is computed according to Proposition 2.1, with E (x n,p ) approximated by Similarly, such a computation at a time step t n with n < N requires the knowledge of the value functions V MC n+1 and V MC n+1 at the next time step t n+1 at all the points of the set p=1,...,P Xn+1

E MC (x n,p ) = e -r0∆t M M m=1 ∆t 2 V MC n+1 (x n,p,m ) + c p + V MC n+1 (x n,p,m ) -c m + V MC n+1 (x n,p,m ) . ( 3
p , but, following the procedure just described, those functions are known only at the points of the set X n+1 : a multidimensional extrapolation tool is required to extending the value functions from X n to a suitable neighbourhood of such a set. For this purpose, we exploit Gaussian Process Regression, a class of non-parametric kernel-based probabilistic models that represents the input data as the random observations of a Gaussian stochastic process and it employs a Bayesian approach to perform estimation of the process at new input data. This Machine Learning techniques is well suited to our problem, as it is capable of handling randomly scattered input data and, generally, only a few input observations are needed to obtain accurate predictions. For a brief introduction to GPR, we refer the interested reader to De Spiegeleer et al. [START_REF] Spiegeleer | Machine Learning for quantitative finance: fast derivative pricing, hedging and fitting[END_REF] or to Goudenège et al. [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF], while for a more in-depth discussion, we suggest Rasmussen and Williams [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF].

Let

V GP R-MC

GPR-EI

The GPR-Exact Integration method is similar to the GPR-MC method but the continuation value is estimated through an exact computation of the expectation, based on the Gaussian distribution. By contrast with the GPR-MC method, the predictors employed in the GPR step are related to the logarithms of the underlyings. Secondly, the continuation value at these points is computed through a closed formula which comes from an exact integration.

Here, for the sake of brevity, we limit ourselves to pointing out the main elements of this algorithm, and we refer the interested readers to [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF]. The computation of the continuation value, for both risky or riskless options, is a particular case of the computation of an expectation as

E Q [Ψ (S t+τ ) |S t = x] ,
with Ψ a certain function, t, t + τ ∈ [0, T ] and τ > 0.

Let us define the input set Z = {z p , p = 1, . . . , P } consisting of P points in R d quasi-randomly distributed according to the law of the vector

σ 1 W 1 τ , . . . , σ d W d τ ⊤ .
In particular, we define

z p i = √ τ σ i Σ i h p , (3.6)
where Σ i is i-th row of the matrix Σ and h p is the q-th point of the Halton's low-discrepancy sequence in R d . Let u : Z → R be the function defined by

u (z) := Ψ x exp r -η - 1 2 σ 2 τ + z . (3.7)
The first step is to approximate the function u by training the GPR method with a Squared Exponential kernel on the set Z, so that the GPR approximation of the function u is given by

u GP R (z) = P p=1 k SE (z q , z) ω p , (3.8)
where ω 1 , . . . , ω P are weights. The continuation value can be computed by integrating the function u GP R against a d-dimensional probability density. The use of the Squared Exponential kernel allows one to easily perform such a calculation by means of a closed formula, that is:

E Q [Ψ (S t+τ ) |S t = x] ≈ P p=1 ω q σ 2 f σ d l e -1 2 (z p ) ⊤ (τ•Π+σ 2 l I d ) -1 (z p ) det (τ • Π + σ 2 l I d ) , (3.9)
where σ f , σ l , and ω 1 , . . . , ω Q are certain constants determined by the GPR approximation of the function z → u (z) considering Z as the predictor set, and Π = (Π i,j ) is the d × d covariance matrix of the vector

σ 1 W 1 T , . . . , σ d W d T ⊤
, that is Π i,j = ρ i,j σ i σ j .

Control Variates

As suggested by Goudenège et al. [START_REF] Goudenège | Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension[END_REF], control variates technique is a usefull tool to improve the accuracy of pricing methods based on GRP. Specifically, we use the European risk-less price V EU as the control variate for the American risk-less price, and the American risk-less price for the American risky price. In particular, we compute the European risk-less price by Monte Carlo simulations with antithetic variates. We explain the use of control variates technique for the computation of the risk-less American option price V , and we leave the appropriate adjustments for the risky price V to the reader. Let V EU represent the risk-less price of the European option. For a fixed time t and an underlying stocks value x, the American-European price gap is defined as the difference between the American and the European price, that is:

v (t, x) = V (t, x, 0, 0) -V EU (t, x, 0, 0) . (3.10)
The price gap is equal to zero at maturity and, at a general time t, it can be computed as

v (t, x) = sup τ ∈Tt,T E Q e -r(τ -t) K (τ, S τ ) |S t = x , (3.11)
where T t,T stands for the set of all stopping times taking values in [t, T ] and K is the exercise value gap, defined by K (t, x) = H (x) -V EU (t, x, 0, 0) . (3.12)

Therefore, the function v (t, x) can be estimated by exploiting a dynamic programming principle based on Bermudan approximation. In particular, one can use GPR-MC and GPR-EI, by replacing H with K. Finally, after computing the initial price gap v (0, S 0 ), by inverting relation (3.10), one can obtain the American price as V (0, S 0 , 0, 0) = v (0, S 0 ) + V EU (0, S 0 , 0, 0) . (3.13)

Remark 2. The computation of the European prices for the control variates technique and the expectation (2.4) are the most time demanding steps. However, these steps can easily be parallelised, thus reducing the total computational time.

Numerical experiments

In this Section we propose the results of some numerical experiments. The algorithms have been implemented in MATLAB and computations have been preformed on a server which employs a 2.40 GHz Intel Xeon processor (Gold 6148, Skylake) and 64 GB of RAM. In the remainder of this Section, we discuss 3 American derivatives: a Geometric Put, a Call on the maximum and a Swaption with floor. Finally, we stress that obtaining accurate values (in terms of relative error) for the XVA is not an easy task. The XVA is in fact obtained as the difference between two prices that are usually very close to each other. A small estimation error on prices can have a significant weight in relative terms on their difference.

Geometric Put

We start by considering a Geometric Put option, whose payoff is

H(S T ) =   K - d i=1 S i T 1 d   + .
This is a very particularly interesting case since the value of this d-dimensional option is equal to the value an appropriate one dimensional American Put option in the Black-Scholes model, as pointed out in [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF][START_REF] Goudenège | Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension[END_REF]. So, by using one-dimensional standard techniques, such as the CRR tree or a finite difference algorithm, one can obtain very accurate prices for both risk and risk-less American option. In particular, we compute the American benchmark by using both the CRR model with 4000 time steps and a PDE approach with 4000 time steps and 4000 space steps. The obtained values with these two algorithms are equal to three decimal places, so that they can be assumed reliable. The Bermudan benchmark is computed as the American one, but the option has only 41 possible exercise dates, that is t 0 = 0, t 1 = 1 /40, . . . , t 40 = 1. The GPR-MC method employs 40 time steps, 2000 points and 10 4 Monte Carlo simulations, while the GPR-EI method employs 40 time steps and 2000 points.

Table 2 shows the numerical results, which appear to be very accurate and reliable. As far as the price calculation is considered, the relative errors compared to the American benchmark never exceed (in absolute value) 0.14%, which is a very small value. The results are even more interesting when compared to the Bermudian benchmark: in this case, the relative error is always below 0.07%. We can therefore say that, in general, the Bermudian approximation and the algorithmic approximations have a similar contribution to the total error with respect to the American price. The relative error with respect to the XVA are generally larger because the XVA is obtained as the difference of almost equal quantities, so the absolute error must be related to a smaller quantity. However, for the cases considered, the absolute error on the XVA never exceeds Option prices XVA Risk-free With default risk

d M = V M = V M = V M = V American benchmark 2
Table 3: Numerical results for a Geometric American put option. Values in brackets are the computational times (in seconds). P is the number of points employed in the GPR algorithms.

2.50% and, in general, tends to increase as the problem size increases. Again, the Bermudian approximation contributes about half of the total error. To conclude, we observe that the results for M = V and M = V are very similar, both in terms of prices and XVA.

To investigate the convergence rate of the two methods, we compute the XVA by changing the number P of points employed for the sparse quasi-random grid. As one may observe from the results reported in Table 3, the GPR algorithms provide convergence to Bermudian prices with great accuracy. Moreover, due to the use of the control variate technique, very few points are needed to obtain very accurate results. Obviously, the larger the dimension, the more points are required to approach the exact value. This fact is particularly important as the computational time increases more than linearly as the number of points increases (the higher cost is due to the training of the GPR model, which is cubic). Finally, we note that the GPR-EI method is generally faster and more accurate than GPR-MC, but the latter returns more accurate results in very high dimensions, especially for d = 80.

Call on the maximum

The American option Call on the maximum is a difficult to evaluate derivative and so it has been considered by many authors, such as Schoenmakers [START_REF] Schoenmakers | Optimal dual martingales, their analysis, and application to new algorithms for Bermudan products[END_REF], Lelong [START_REF] Lelong | Dual pricing of american options by Wiener chaos expansion[END_REF], Becker et al. [START_REF] Becker | Deep optimal stopping[END_REF], Goudenège et al. [START_REF] Goudenège | Machine learning for pricing American options in highdimensional Markovian and non-Markovian models[END_REF][START_REF] Goudenège | Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension[END_REF], and Ech-Chafiq et al. [START_REF] Ech-Chafiq | Pricing Bermudan options using regression trees/random forests[END_REF]. Specifically, the payoff of such an option is given by

H(S T ) = max i=1,...,d S i T -K + .
We start the numerical analysis by considering the same model parameters as for the Geometric put, which are reported in Table 1. We stress out that, since the considered derivative is a call option and the underlying pays no dividends (η i = 0, see Table 1), early exercise is never optimal for the riskless option. Moreover, since the payoff of the derivative is always possible, we can use the closed formulas proposed by Burgard and Kjaer [START_REF] Burgard | PDE representations of options with bilateral counterparty risk and funding costs[END_REF] to compute the XVA for the European derivative. Specifically, if M = V , then

XV A EU = V EU (t, S 0 ) • 1 -e -(λB +λC )T -c p 1 -e -(λB +λC )T λ B + λ C , and if M = V , then XV A EU = V EU (t, S 0 ) • 1 -e (cp-λB -λC )T .
It is worth noting that despite the prices of an European and an American riskless options are equals, this does not also apply to their XVAs. In fact, an American option may be exercised early so to reduce the losses due to counterparty default, thus the XVA on the American option is expected to be smaller than the European one. So, we present the XVA on the European option as an upper-bound (UB). Results are shown in Table 4. We can see that both proposed methods provide very accurate values for the cases considered. When a large number of points is used (at least 500), the relative deviation between the returned values, that is the difference divided by the larger value, is less than 2%. The values obtained for XVA are all below the upper-bound, although very close to it.

Finally, for the sake of comparison, let us calculate the XVA for a Call on the maximum considering a positive dividend rate, equal for all underlyings and equal to η i = 0.02. In this specific case, there are neither benchmarks nor upper-bounds. Table 5 presents the results. In the case under consideration (positive dividend), the valuation seems to be more challenging than in the previous case (with zero dividend). In fact, at least 500 points are needed to obtain a relative deviation of less than 5%.

Swaption with floor on two portfolios

The derivatives considered in the numerical examples above are all options and therefore their payoff function and their values are always positive. The model considered in this work also admits negative values for the payout, so it is interesting to consider a case with this attribute. Let us now consider an American twoportfolio Swaption with a negative floor, i.e. a derivative in which two portfolios are swapped between counterparties, whose value can be either positive or negative. Specifically, the first portfolio consists of the first d/2 underlyings and the second portfolio consists of the remaining underlyings. For simplicity, we will assume d to be an even number. In both cases, the underlyings all have the same weight, so the value of each portfolio is equal to the average of the prices of the individual risky assets. The payout of such a derivative is given by

H(S T ) = max   2 d   d/2 i=1 S i T - d i=d/2+1 S i T   , K   .
In particular, the floor K is a negative number, thus the payout of the option can be negative. Table 6 presents the numerical results. We observe that, in the case considered, the estimated values for the XVA are much smaller than in the previous cases. The two methods return very similar values for d ≤ 40, whereas for d = 80 GPR-EI estimates of the XVA are greater than those returned by GPR-MC (approximately +20%). The lack of a benchmark makes it unclear which of the two methods is the more accurate in this case.

Conclusion

In this paper, we have discussed the problem of calculating the XVA of a derivative that depends on multiple underlyings. This issue plays an essential role in counterparty risk management, also in light of the regulations currently in force. Nevertheless, it is an element that is often overlooked due to the curse of dimensionality associated with the problem of valuing high-dimensional options. Our proposal to address this challenge is to reformulate the problem in probabilistic terms and make use of the GPR-MC and GPR-EI techniques with control variate, which have already been successfully applied in similar contexts. Numerical results show that it is possible to obtain very accurate estimates of the XVA and, in some cases, very few points are

. 5 )

 5 If we proceed backward, the functions V MC N and V MC N are known since they are equal to the payoff of the option H, so one can compute both V MC N -1 and V MC N -1 at Xn p by exploiting equations (3.3), (3.4) or (3.5).

n

  and V GP R-MC n be the GPR approximations of the functions V MC n and V MC n , obtained from the observations x n,p , V MC n (x n,p ) , p = 1, . . . , P and x n,p , V MC n (x n,p ) , p = 1, . . . , P respectively. The GPR-MC algorithm requires the replacement of V MC n+1 and V MC n+1 in the right side on (3.3), (3.4) or (3.5) with V GP R-MC n+1 and V GP R-MC n+1 respectively.

  with Ĉ the continuation value of the risky option. Similarly to what happens for the risk-free option, Ĉ (t, S t ) restricted to the time interval ]t n , t n+1 [ is equal to Ĉn (t, S t ), the solution of the following PDE, defined in ]t n , t n+1 [ for n = 0, . . . , N -1 :

	, the value V (t, S t , 0, 0) of the Bermudan
	risky option, satisfies
	V (t, S t , 0, 0) = max Ĉ (t, S t ) , H (S t ) ,

Table 1

 1

	lists all the

Table 1 :

 1 Parameters employed for the numerical experiments in the multi-dimensional Black-Scholes model. In particular,s F = (1 -R B ) λ B .parameters of the stochastic model, with the exception of the dimension d, which takes on different values from d = 2 up to d = 80. Based on the results discussed in this Section, one can observe that the two proposed methods are very accurate in the various cases considered. The quality of the results degrades slightly as the size of the problem increases, but the quality of the results is still acceptable, successfully limiting the effects of the curse of dimensionality. Overall, the results proposed by the two methods are always in agreement and very close to the benchmark (when available).

Table 2 :

 2 Numerical results for a Geometric American put option. Values in brackets are the relative errors with respect to the American benchmark. d stands for the dimension.

		Benchmarks			GPR-MC					GPR-EI
								P
	d	American	Bermudian	125	250	500 1000	2000	125	250	500 1000 2000
	XVA, case M = V
	2	0.242	0.244	0.247 (117)	0.245 (202)	0.244 (480)	0.245 (1419)	0.244 (4732)	0.242 (103)	0.244 (126)	0.243 (151)	0.245 (320)	0.244 (841)
	10	0.177	0.178	0.183 (160)	0.180 (287)	0.180 (688)	0.179 (1784)	0.179 (5621)	0.180 (132)	0.179 (199)	0.179 (358)	0.179 (698)	0.179 (1803)
	20	0.166	0.167	0.171 (243)	0.171 (447)	0.172 (1052)	0.170 (2345)	0.169 (6592)	0.169 (221)	0.169 (344)	0.169 (692)	0.168 (1257)	0.168 (2679)
	40	0.160	0.161	0.152 (363)	0.159 (614)	0.164 (1413)	0.165 (3360)	0.164 (8537)	0.159 (360)	0.163 (515)	0.164 (1076)	0.164 (2171)	0.164 (4284)
	80	0.157	0.158	0.118 (477)	0.138 (1047)	0.155 (2287)	0.158 (4709)	0.159 (12258)	0.070 (441)	0.119 (1010)	0.145 (1885)	0.151 (3961)	0.154 (7242)
	XVA, case M =	V
	2	0.244	0.246	0.249 (125)	0.247 (194)	0.249 (518)	0.248 (1472)	0.246 (4869)	0.244 (97)	0.246 (236)	0.245 (301)	0.247 (607)	0.246 (1570)
	10	0.178	0.180	0.188 (190)	0.183 (301)	0.181 (702)	0.182 (1811)	0.181 (5528)	0.182 (132)	0.181 (367)	0.181 (680)	0.181 (1286)	0.181 (3252)
	20	0.167	0.169	0.175 (249)	0.175 (464)	0.174 (1011)	0.172 (2295)	0.171 (6680)	0.170 (212)	0.170 (600)	0.170 (1276)	0.170 (2275)	0.170 (4830)
	40	0.161	0.163	0.152 (353)	0.163 (657)	0.165 (1345)	0.167 (3365)	0.165 (8782)	0.160 (355)	0.164 (954)	0.166 (1982)	0.165 (3997)	0.165 (7853)
	80	0.158	0.159	0.119 (503)	0.141 (1160)	0.156 (2242)	0.159 (5020)	0.160 (12265)	0.071 (832)	0.120 (1804)	0.147 (3366)	0.153 (6826)	0.156

Table 4 :

 4 Numerical results for a Call on the maximum option. The confidence interval for the upper-bound UB is computed at a 99% confidence level.

	d			GPR-MC			GPR-EI	UB
				P
		125	250	500 1000 2000	125	250	500 1000 2000
	XVA, case M = V
	2 10 20 40 80 XVA, case M = 1.004 1.001 0.998 0.999 0.998 2.191 2.210 2.226 2.231 2.229 2.670 2.720 2.753 2.762 2.771 2.743 3.149 3.223 3.268 3.294 2.674 3.001 3.602 3.702 3.764 V	0.999 0.999 0.999 0.998 0.998 2.237 2.231 2.236 2.234 2.230 2.789 2.792 2.788 2.774 2.774 3.317 3.262 3.247 3.296 3.299 2.830 3.241 3.642 3.736 3.741	1.009 ± 0.001 2.252 ± 0.001 2.803 ± 0.001 3.337 ± 0.001 3.852 ± 0.001
	2 10 20 40 80	1.012 1.019 1.011 1.010 1.011 2.217 2.238 2.250 2.254 2.255 2.711 2.748 2.787 2.790 2.803 2.766 3.180 3.259 3.305 3.333 2.705 3.046 3.646 3.742 3.812	1.010 1.011 1.011 1.010 1.011 2.263 2.257 2.261 2.259 2.257 2.820 2.825 2.823 2.806 2.809 3.358 3.298 3.284 3.336 3.339 2.891 3.259 3.699 3.732 3.783	1.021 ± 0.001 2.279 ± 0.001 2.837 ± 0.001 3.377 ± 0.001 3.898 ± 0.001

Table 5 :

 5 Numerical results for a Call on the maximum option with a positive dividend rate.

	d			GPR-MC			GPR-EI
				P
		125	250	500 1000 2000	125	250	500 1000 2000
	XVA, case M = V
	2	0.817 0.805 0.800 0.806 0.804	0.805 0.801 0.802 0.802 0.803
	10	1.988 2.003 2.008 2.009 2.006	1.962 1.966 2.000 2.004 1.999
	20	2.479 2.529 2.551 2.555 2.556	2.473 2.485 2.481 2.456 2.457
	40	2.599 2.948 3.009 3.063 3.080	3.026 2.973 2.964 3.005 2.997
	80	2.577 2.850 3.407 3.487 3.557	2.811 3.033 3.461 3.422 3.488
	XVA, case M =	V
	2	0.820 0.814 0.809 0.808 0.815	0.814 0.811 0.812 0.811 0.812
	10	2.018 2.031 2.027 2.030 2.031	1.981 1.992 2.025 2.026 2.022
	20	2.505 2.555 2.581 2.581 2.588	2.503 2.514 2.508 2.488 2.484
	40	2.625 2.978 3.051 3.100 3.118	3.059 3.009 2.998 3.040 3.033
	80	2.613 2.878 3.449 3.531 3.597	2.848 3.068 3.461 3.464 3.560

Table 6 :

 6 Numerical results for a Swaption with floor on two portfolios. All the results must be multiplied by 10 -2 . sufficient to achieve very accurate results. For the considered cases, the proposed methods demonstrate to be effective for large dimensions, thus providing new methods for estimating XVA by overcoming the curse of dimensionality. so there is one and only one solution to F tn,x (z) = 0. Now, we have proved that there is one and only one solution, let us compute it. We rewrite equation (2.7) asṼ (t n, , x) = max E (x) + ∆t 2 Ṽ (t n, , x) + c p + Ṽ (t n, , x) -c m , H (x) .Case 1a: Ṽ (t n, , x) = H (x) ≤ 0. In this case, we have Case 2a: 0 < Ṽ (t n, , x) = H (x). In this case, we have

	d			GPR-MC					GPR-EI
						P
		125	250	500	1000	2000	125	250	500	1000	2000
	XVA, case M = V
	2	41.911 42.031 41.953 41.884 42.054	42.086 41.958 42.063 41.982 42.000
	10	14.604 14.938 14.693 14.762 14.750	14.820 14.307 14.551 14.691 14.739
	20	7.959	8.168	8.567	8.631	8.827	7.612	7.560	8.241	8.230	8.441
	40	4.426	4.546	4.277	4.397	4.427	3.710	4.010	4.275	4.215	4.354
	80	3.010	1.448	1.855	2.079	2.089	2.173	2.287	2.613	2.524	2.474
	XVA, case M =	V
	2	42.560 42.404 42.372 42.563 42.463	42.594 42.377 42.477 42.377 42.549
	10	14.894 15.040 14.789 14.849 14.897	14.941 14.369 14.694 14.775 14.854
	20	8.022	8.233	8.631	8.653	8.937	7.587	7.726	8.355	8.383	8.592
	40	4.446	4.589	4.282	4.414	4.559	3.681	4.081	4.265	4.249	4.412
	80	3.179	2.493	1.899	2.104	2.132	2.146	2.301	2.682	2.611	2.593

Case 1b: H (x) < Ṽ (t n, , x) ≤ 0. In this case, we have

max E (x) + ∆t 2 Ṽ (t n, , x) c m , H (x) = E (x) + ∆t 2 Ṽ (t n, , x) c m = Ṽ (t n, , x) , so Ṽ (t n, , x) = E (x) 1 -∆t 2 c m ,

which implies E ≤ 0 and

H (x) 1 -∆t 2 c m < Ṽ (t n, , x) 1 -∆t 2 c m = E (x) ≤ 0.

Case 1c: H (x) < 0 < Ṽ (t n, , x). In this case, we have

max E (x) + ∆t 2 Ṽ (t n, , x) c p , H (x) = E (x) + ∆t 2 Ṽ (t n, , x) c p = Ṽ (t n, , x) , m < 0 < E (x) . p ≤ H (x) , thus E (x) ≤ H (x) 1 -∆t 2 c p .

A Proof of Proposition 2.1 Equation (2.7) is a non-linear equation, so, first, we discuss existence and uniqueness of the solution. Let us assume that we have fixed the value of t n and x, so we can consider them as model parameters. We define the function f : R → R as

so that equation (2.7) can be rewritten as

So, in to Ṽ (t n, , x) one has to solve the equation zf tn,x (z) = 0, that is computing the zeros of the function F tn,x (z) = xf tn,x (z). We observe that F tn,x is a continuous function and it is piecewise derivable. In particular, if

then the derivative of F tn,x is given by

Therefore F tn,x is a continuous piecewise linear function. Moreover, if we assume 1-∆t 2 c p > 0 and 1-∆t 2 c m > 0 (which is true for ∆t small enough) F tn,x is strictly increasing, so it can not have more than one zero. Furthermore, we observe lim z→-∞ F tn,x (z) = -∞, lim z→+∞ F tn,x (z) = +∞,