Guillaume Legendre

Virginie Ringa

Arnaud Fauconnier

Prof Xavier Fritel
email: xavier.fritel@univ-poitiers.fr

Menopause, Hormone Treatment and Urinary Incontinence at Midlife Authors

Keywords: Urinary Incontinence, Stress Urinary Incontinence, Urge Incontinence, Overactive bladder, Menopause, Hormone therapy, OEstrogen, Epidemiology

Whether there is any association between urinary incontinence and menopause is the subject of debate, partly due to the fact it is difficult to tell the difference between the effects of menopause and those of ageing. For some time it was hoped that hormonal treatment for menopause would be beneficial for urinary incontinence because there are hormonal receptors in the urinary tract. The goal of this survey of current knowledge on the subject is to explore thoroughly the relationship between menopause and urinary incontinence .

Our study is based on a review of the literature dealing with the epidemiology of urinary incontinence in women aged between 45 and 60, and the effects of hormonal treatment with respect to the symptoms of involuntary loss of urine.

Analysis of the epidemiological data drawn from large cohorts shows that on the one hand, the menopause has little if any impact on the risk of urinary incontinence, and on the other hand that the effects of oestrogen medication on urinary incontinence vary according to how it is administered and the type of incontinence. The effect of oral hormone treatments for menopause is rather negative with respect to stress incontinence. Vaginal treatment appears to be beneficial for overactive bladder symptoms.

Introduction

The menopause is a universal physiological event related with the drop in ovarian hormone secretions (oestrogens and progestin) that occurs as the stock of ovarian follicules is depleted.

Menopause is diagnosed when menstruation has stopped for 12 successive months, and the average age at which it occurs is between 47 and 51 [START_REF] Melby | Culture and symptom reporting at menopause[END_REF]2]. Its clinical expression varies very considerably between socio-cultural groups and individuals [START_REF] Delanoe | Class, gender and culture in the experience of menopause. A comparative survey in Tunisia and France[END_REF]. These variations depend on many factors, such as women's social status, their nutrition, life style (smoking) and weight, not forgetting genetic factors [START_REF] Delanoe | Class, gender and culture in the experience of menopause. A comparative survey in Tunisia and France[END_REF]. It is difficult to draw the distinction between the effects of menopause and those of ageing. In addition to the menopausal "syndrome" itself, which mainly comprises vasomotor symptoms and vaginal dryness, urinary symptoms including incontinence (UI) have been attributed to menopause. The hormonal dependence of the genital tract tissues has been evoked in order to explain the appearance of these female low urinary tract symptoms (FLUTS) at the menopause. OEstrogen receptors have been found not only in the pelvic floor muscles but also in the uro-genital ligaments and detrusor muscle cells along with the connective tissues and all the fascias that maintain a stable relationship between the various organs [2].

Urinary incontinence can be defined as "the complaint of any involuntary leakage of urine" [START_REF] Haylen | An International Urogynecological Association (IUGA)/International Continence Society (ICS) joint report on the terminology for female pelvic floor dysfunction[END_REF]. The standard classification gives three main types of UI: Stress Urinary Incontinence (SUI), Urge Urinary Incontinence (UUI) and Mixed Urinary Incontinence (MUI). Stress urinary incontinence is characterised by involuntary loss of urine without any previous feeling of a need to void, which takes place on the occasion of a physical stress (cough, lifting something heavy, or any other physical activity). Urge urinary incontinence (or urgent need to void) is characterised by involuntary loss of urine preceded by an urgent and irresistible need to void resulting in uncontrollable leakage of urine. Mixed urinary incontinence is the association in variable proportions of SUI and UUI.

UI may be experienced simply as a nuisance or as a real handicap [START_REF] Bartoli | Impact on quality of life of urinary incontinence and overactive bladder: a systematic literature review[END_REF][START_REF] Saadoun | Negative impact of urinary incontinence on quality of life, a cross-sectional study among women aged 49-61 years enrolled in the GAZEL cohort[END_REF]. Women frequently present the symptoms of UI, but the estimates for its prevalence in the general population vary widely, from 8 to 30 %. This variability is due not only to the very heterogeneous nature of the populations studied but also to the varying definitions of UI used in the studies.

Many different factors are associated with an increased risk of UI [START_REF] Menezes | Predictors of female urinary incontinence at midlife and beyond[END_REF]. The main of these are ageing, pregnancy, childbirth, past history of pelvic and perineal surgery, obesity, and chronic pathologies (diabetes, cognitive problems, etc.). There is more debate about the roles of menopause and hormone replacement therapy (HRT) [START_REF] Robinson | Estrogens and the lower urinary tract[END_REF][START_REF] Berecki-Gisolf | Symptoms reported by women in midlife: menopausal transition or aging?[END_REF]. As mentioned above, it is difficult to differentiate between the role of ageing and that of the menopause, and equally to differentiate the true role of menopause from that of treatments instigated at the time of menopause, such as HRT.

The relationship postulated between UI and hormone deficiencies induced by menopause is based mainly on "physiological" data, i.e. the existence of (oestrogen and progestin) hormonal receptors in the epithelial tissues of bladder, urethra and trigone and also in the vagina, uterosacral ligaments, levator ani and puborectal muscles [START_REF] Quinn | The effects of hormones on urinary incontinence in postmenopausal women[END_REF]. The other argument supporting this relation is clinical observation of an increase in the prevalence of overactive bladder (OAB) syndrome after the menopause. The lack of oestrogens could contribute in various ways to other urinary symptoms that arise: oestrogens play a role in (i) the increase in epithelial cellular trophicity in the vagina, urethra and bladder (ii) the increase in peri-urethral vascularisation (an important factor for regulation of closing pressure), (iii) the increase in maximum closing pressure and (iv) the increase in concentration and sensitivity of adrenergic receptors with modification of the -adrenergic receptors/-adrenergic receptor ratio in favour of the -adrenergic receptors [START_REF] Ewies | Topical vaginal estrogen therapy in managing postmenopausal urinary symptoms: a reality or a gimmick?[END_REF][START_REF] Kobata | Estrogen therapy influence on periurethral vessels in postmenopausal incontinent women using Dopplervelocimetry analysis[END_REF][START_REF] Bella | Hormonal influence on periurethral vessels in postmenopausal incontinent women using Doppler velocimetry analysis[END_REF]. Progestin receptors are also to be found in the entire female genital tract, although in less constant fashion than oestrogen receptors.

Progestin appears to have a detrimental effect on continence, by reducing muscle tone of bladder and urethra [START_REF] Quinn | The effects of hormones on urinary incontinence in postmenopausal women[END_REF].

The aim of our review is to analyse the data in the literature with respect to the relationship between menopause and UI. We present the main epidemiological data dealing with the relationship between prevalence, incidence and remission of UI, and the menopause. We then try to supply answers to the two main questions raised by this relationship: What consequences do hormonal changes at the menopause have on UI symptoms? What consequences do hormonal treatments have on continence?

Methods:

A review of the literature was carried out by consulting the Medline database entries between January 2000 and June 2012. Articles were selected by cross-referencing the following keywords: "urinary incontinence, stress urinary incontinence, urge incontinence, overactive bladder, menopause, oestrogen therapy". In this selection priority was given to meta-analyses, literature reviews, randomised controlled trials and cohort studies. The level of evidence (LE) scale proposed by the Oxford Centre for Evidence-Based Medicine (www.cebm.net) was used to class the articles selected.

The terminology used complies with International Continence Society (ICS) and

International UroGynecological Association (IUGA) recommendations.

Results

3.1.

Study selection Among the 488 articles initially selected during the Medline search, 29 articles were finally retained, including 3 meta-analyses, 4 literature reviews, 5 randomised controlled trials and 12 cohort studies. Figure 1 gives details on this selection of articles.

Prevalence or incidence of UI according to age

The prevalence of urinary incontinence is proportional to age. The distribution of the various types of UI changes with age.Several articles report a peak in the prevalence of SUI at midlife

(LE3). The majority of cases before the age of 50 are SUI, but after that age it represents a minority [START_REF] Hannestad | A community-based epidemiological survey of female urinary incontinence: the Norwegian EPINCONT study. Epidemiology of Incontinence in the County of Nord-Trondelag[END_REF]. After 60 it is mixed UI that predominates. The prevalence of UI at the time of menopause varies from 8 to 27 % depending on the population studied and the definition used for UI [START_REF] Hannestad | A community-based epidemiological survey of female urinary incontinence: the Norwegian EPINCONT study. Epidemiology of Incontinence in the County of Nord-Trondelag[END_REF][START_REF] Sherburn | Is incontinence associated with menopause?[END_REF][START_REF] Mcgrother | Storage symptoms of the bladder: prevalence, incidence and need for services in the UK[END_REF][START_REF] Hagglund | Changes in urinary incontinence and quality of life after four years. A population-based study of women aged 22-50 years[END_REF] (LE2).

Only a few cohort studies specifically address the menopause period (45-60 years). Mac Grother et al. report an annual incidence of UI of 8% in a population of 108 women aged between 40 and 59 in the general population [START_REF] Mcgrother | Storage symptoms of the bladder: prevalence, incidence and need for services in the UK[END_REF] (LE3). Two other cohorts of women under the age of 60 found annual incidence rates at 4 and 5 % [15, 17] (LE3). In another cohort of 2860 femmes between 40 and 60, the authors report an annual incidence rate of 6% [START_REF] Moller | Incidence and remission rates of lower urinary tract symptoms at one year in women aged 40-60: longitudinal study[END_REF] (LE3). A longitudinal study of a cohort of nurses aged 36 to 55 by Townsend et

al. found an annual incidence of 7% [19] (LE3).

There is considerable disparity between the cohort studies concerning the annual rate of remission. On the one hand Hagglund [START_REF] Hagglund | Changes in urinary incontinence and quality of life after four years. A population-based study of women aged 22-50 years[END_REF], Samuelsson [START_REF] Samuelsson | Five-year incidence and remission rates of female urinary incontinence in a Swedish population less than 65 years old[END_REF], and Townsend [START_REF] Townsend | Incidence and remission of urinary incontinence in middle-aged women[END_REF] report annual remission rates for UI of 4, 6, and 7% respectively (LE3). On another hand other authors report higher annual remission rates, at 25 to 29% [START_REF] Mcgrother | Storage symptoms of the bladder: prevalence, incidence and need for services in the UK[END_REF][START_REF] Moller | Incidence and remission rates of lower urinary tract symptoms at one year in women aged 40-60: longitudinal study[END_REF] (LE3). These results can be explained in part by the definition of remission of UI depending on the studies: complete disappearance or lessening of the severity of UI symptoms.

Prevalence or incidence of UI according to menopausal status

The American prospective cohort Study of Women's Health Across the Nation (SWAN) followed 1529 women, without any UI symptoms at inclusion, annually for 6 years. The women were classed according to their hormonal status at the end of the study as follows: premenopause, early peri-menopause (irregular menses), late peri-menopause ( [START_REF] Waetjen | Association between menopausal transition stages and developing urinary incontinence[END_REF]. In another study covering 2415 women from the SWAN cohort who were incontinent (at least one episode of UI per month) at inclusion, Waetjen reports that the worsening of UI symptoms observed after the menopause is not attributable to the menopause itself [START_REF] Waetjen | Factors associated with worsening and improving urinary incontinence across the menopausal transition[END_REF] but to an increase in weight. The menopause appears to be associated with an increase in weight and this would explain the increase in the prevalence of UI (LE2).

The study by Sherburn et al. of the Australian cohort of 1897 femmes, The Melbourne

Women's Midlife Health Project, addressed the impact of the menopause on UI symptoms.

Simultaneous cross-sectional and longitudinal analysis of the data did not reveal any statistical relationship between UI and menopause (LE3). In the longitudinal analysis of 438 women from the same cohort who were followed for 7 years the menopause was not associated with any increase in the incidence of UI as a whole (LE3) [START_REF] Sherburn | Is incontinence associated with menopause?[END_REF].

The results obtained by another longitudianl study, the National Survey of Health and Development (NSHD) concerning 1211 women between 48 and 54 years of age did not find any association between menopause and UUI (LE3). SUI alone was associated with the hormonal status, since in this study the peri-menopause was associated with an increased prevalence of urine leakage symptoms due to stress, as compared with the group of postmenopausal women (OR= 1, 39 [1, 4-1, 7]) (LE3). This study, in which it is not possible to distinguish clearly between the effects of age and those of menopause because there is no comparison of the incidence rates between the different groups, nevertheless appears to reveal an effect of ageing independent of that of the menopause with respect to the occurrence of UUI or SUI (LE3) [START_REF] Mishra | Menopausal transition and the risk of urinary incontinence: results from a British prospective cohort[END_REF].

A large Chinese cross-sectional study of nearly 20 000 women aged 20 to 99 shows an association between SUI and menopause (OR= 1, 26 [1, 04-1, 52]), even when age is taken into account as a risk factor (LE4) [START_REF] Zhu | The epidemiological study of women with urinary incontinence and risk factors for stress urinary incontinence in China[END_REF].

Prevalence or incidence of UI according to hormone levels

A recent cohort study looked at the effect of changes in oestrogen concentrations on urinary symptoms during the transition to menopause. A big drop in oestrogen levels is associated in significant fashion with an improvement in UI symptoms (LE2) [START_REF] Gopal | Association of change in estradiol to lower urinary tract symptoms during the menopausal transition[END_REF]. Moreover, analysis of the annual fluctuation in endogenous oestrogen levels during the menopausal period using the data from SWAN did not find any association with the onset of UI or worsening of UI symptoms (LE2) [START_REF] Waetjen | Serum estradiol levels are not associated with urinary incontinence in midlife women transitioning through menopause[END_REF]. A Swedish cohort study of 6917 women aged between 50 and 59 even found a positive association between a high oestradiol level and UI (LE2) [START_REF] Teleman | The relation between urinary incontinence and steroid hormone levels in perimenopausal women. A report from the Women's Health in the Lund Area (WHILA) study[END_REF].

3.5.

Prevalence or incidence of UI depending on menopause hormonal treatment 3.5.1. Data obtained from random studies of women treated for urinary symptoms A meta-analysis based on eleven randomised studies of the effects of oestrogen (versus placebo) on overactive bladder symptoms found a significant association between oestrogen treatments and an improvement in the urinary symptoms: less pollakiuria and nocturia, fewer episodes of UUI and urine leaks, improvement in the latence of the first urge to void and increase in bladder capacity (LE1) [START_REF] Cardozo | A systematic review of the effects of estrogens for symptoms suggestive of overactive bladder[END_REF]. However the effects vary according to the mode of administration of the oestrogens. Local application (vaginal or intravesical) is associated in significant fashion with an improvement in all the urinary symptoms (pollakiuria, nocturia, UUI, bladder capacity). Systemic administration results in only partial improvement of these symptoms, and statistically is associated only with a drop in the number of episodes of incontinence and later perception of the first urge to void.

A review of the literature covering several randomised trials (comparisons between different protocols for oestrogen treatment or versus placebo) confirms the advantage of vaginal treatments in case of post-menopausal lower urinary tract symptoms, especially in the presence of vaginal atrophy (LE2) [START_REF] Ewies | Topical vaginal estrogen therapy in managing postmenopausal urinary symptoms: a reality or a gimmick?[END_REF]. The findings of recent studies of the effect of oestrogens in association with anti-muscarini agents did not agree. A controlled study of 229 femmes presenting an overactive detrusor did not show any benefit for topical vaginal oestrogen therapy associated with tolterodine versus tolterodine alone for overactive bladder symptoms (LE2) [START_REF] Serati | Is there a synergistic effect of topical oestrogens when administered with antimuscarinics in the treatment of symptomatic detrusor overactivity?[END_REF]. Another randomised controlled trial in 80 women with overactive bladder syndrome, also comparing the effect of topical vaginal oestrogen therapy associated with tolterodine with that of tolterodine alone, found a significant improvement of objective parameters and quality of life when oestrogens were associated with tolterodine (LE2) [START_REF] Tseng | Randomized comparison of tolterodine with vaginal estrogen cream versus tolterodine alone for the treatment of postmenopausal women with overactive bladder syndrome[END_REF].

Finally, a recent randomised trial comparing the efficiency of topical vaginal oestrogen therapy with that of oxybutynin for the treatment of overactive bladder syndrome found that the two products were similar in terms of efficiency (LE2) [START_REF] Nelken | Randomized trial of estradiol vaginal ring versus oral oxybutynin for the treatment of overactive bladder[END_REF].

A review of the literature covering eight controlled studies and 14 prospective non-controlled studies concluded that oestrogen treatment is not efficient for SUI (LE2) [START_REF] Sultana | Estrogen and urinary incontinence in women[END_REF].

3.5.2. Data resulting from secondary analyses of randomised trials in the general population Among the post-menopausal women enrolled in the Womens' Health Initiative study (WHI), HRT significantly increased the incident risk at one year of all types of UI (SUI, UUI, MUI) in women who were continent at the time they were included (LE2). This effect of HRT is particularly distinct for SUI and less so if at all for UUI [START_REF] Hendrix | Effects of estrogen with and without progestin on urinary incontinence[END_REF]. In women taking oral oestrogens and progestin, the relative risk of de novo UI is [START_REF] Hendrix | Effects of estrogen with and without progestin on urinary incontinence[END_REF].

Steineauer et al. carried out a subsidiary analysis of the Heart Estrogen/Progestin

Replacement Study (a double-blind randomised study assessing the effect of HRT for prevention of cardiovascular risks) concerning patients with no UI at inclusion [START_REF] Steinauer | Postmenopausal hormone therapy: does it cause incontinence?[END_REF]. A significant relationship between HRT and the onset of both UUI and SUI was found during the first four months of treatment. This detrimental effect of HRT persisted throughout the 4 years of the study, with a cumulated additional risk of 12% and 16 % respectively for UUI and SUI compared with treatment by placebo (LE2).

The results of these studies were included and confirmed in the Cochrane Database [START_REF] Cody | OEstrogen therapy for urinary incontinence in post-menopausal women[END_REF] metaanalysis which covered a total of 19 313 incontinent women among whom 9417 were treated by oestrogens, in 33 studies (16 of which addressed specifically SUI). There was no analysis by type of UI. A worsening of UI symptoms was found when HRT was taken systemically (RR= 1.32 [1.17-1.42]) (LE1). However, topical vaginal oestrogen therapy contributed towards remission of UI by improving both the episodes of urge urinary incontinence (0.74 [0.64-086]) and pollakiuria (LE1).

Data obtained from cohort studies

The data of the Nurses' Health Study addressing the impact of HRT on UI in a cohort of nurses aged between 30 and 55 at the time of inclusion were analysed by Grodstein et al. [START_REF] Grodstein | Postmenopausal hormone therapy and risk of developing urinary incontinence[END_REF].

The occurrence of incident UI was associated in significant fashion with systemic hormone therapy whatever the type of HRT and mode of administration: oral oestrogen (RR= 1.54 Two randomised controlled trials concerning the oral route versus placebo to treat UI and including 83 and 67 post-menopausal women were unable to demonstrate any effect of oestrogens with respect to urodynamic parameters or urinary symptoms, whether in terms of objective measurements (pad-test, cystometrography, prolifometry) or in terms of subjective measurements (assessed by approved quality of life questionnaires) [START_REF] Jackson | The effect of oestrogen supplementation on postmenopausal urinary stress incontinence: a double-blind placebo-controlled trial[END_REF] (LE2).

Influence of hormonal treatment after surgery for SUI

Certain studies looked at the impact of oestrogen therapy when surgery is used for SUI. A prospective randomised study comparing the effects of adjuvant topical vaginal oestradiol or not for six months immediately after surgical correction of SUI using a sub-urethral tape (TVT-O) in 183 post-menopausal patient, found that pollakiuria (2 vs. 11 %, p=0.02) and urge urinary incontinence (3 vs. 12 %, p= 0.01) alone were improved by hormone therapy (LE2) [START_REF] Liapis | The use of oestradiol therapy in postmenopausal women after TVT-O anti-incontinence surgery[END_REF]. The objective parameters of the urodynamic tests remained unchanged in both groups (LE2). In a randomised controlled trial Zullo et al. found exactly the same results with urge urinary incontinence alone being improved by topical vaginal oestrogen therapy (LE2) [START_REF] Zullo | Vaginal estrogen therapy and overactive bladder symptoms in postmenopausal patients after a tension-free vaginal tape procedure: a randomized clinical trial[END_REF].

No study of the long-term advantages of oestrogen after installation of a sub-urethral tape was found.

Discussion

The results of our review concerning the link between menopause and UI do not all agree, but this overview shows that the menopause has little if any impact on the risk of UI in general, when confounding factors such as age or changes in weight are taken into account. It appears that SUI decreases after the menopause while UUI or mixed UI increase at this time. Oral HRT has a rather detrimental effect on stress urinary incontinence, by doubling the risk of de novo SUI. Concerning UUI, HRT has less of an effect, which is variable according to the population in question and the type of treatment. Topical vaginal oestrogen therapy is the only treatment that seems to have a really beneficial effect on urge urinary incontinence or overactive bladder symptoms.

Most of these results are drawn from longitudinal analyses of large cohorts enabling a distinction to be drawn between the effect of age and that of the menopause. Longitudinal studies are the only way to examine the timing, and consequently any cause and effect relationship between events. Cross-sectional studies cannot assess time-dependent variables such as age, weight and menopausal status. Nevertheless, the large cohort studies included in this review present several problems. For example, the definition of UI is not identical in all the studies because follow-up of the cohorts often started prior to the harmonisation of the terminologies proposed by ICS and IUGA. The frequency of UI as a "functional complaint" depends on the tools used. Moreover, few of the studies draw a distinction between the various types of UI (SUI, UUI, MUI), their frequency or their severity. Finally in this type of cohort, the lack of any overall assessment of incontinence risk factors (because the studies were not designed initially to study this problem specifically) can result in confounding biases.

Part of the results of this review is based on secondary analyses of randomised trials in which the main objective was not to study UI but instead the cardiovascular morbidity of HRT. So the level of evidence contributed by these secondary analyses is lower.

Certain pathophysiological data would tend to indicate a worsening of UI due to lack of oestrogen. However, in contradiction with this postulate, the epidemiological data obtained with the cohorts shows that the menopause has little if any effect on urinary symptoms. The results of several large randomised trials in the general population reveal an aggravation of UI in case of systemic HRT. This detrimental effect of HRT needs to be seen in the light of what is observed during pregnancy, when an increase in the prevalence of UI is observed [START_REF] Fritel | Female urinary incontinence, from pregnancy to menopause, a review of epidemiologic and pathophysiologic findings[END_REF]. This greater prevalence of UI during pregnancy could be explained by the increase in oestrogen concentrations.

Topical vaginal oestrogen therapy alone appears to have a beneficial effect on overactive bladder symptoms. This effect could be explained by the improvement in terms of vaginal dryness and a drop in recurrent urinary infections [START_REF] Robinson | Estrogens and the lower urinary tract[END_REF][START_REF] Ewies | Topical vaginal estrogen therapy in managing postmenopausal urinary symptoms: a reality or a gimmick?[END_REF]. The learned societies do not recommend the use of oestrogens (whether or not associated with progesterone) for the treatment of UI. However, in case of overactive bladder symptoms in a post-menopausal woman, topical vaginal treatment may be proposed [START_REF] Fritel | Diagnosis and management of adult female stress urinary incontinence: guidelines for clinical practice from the French College of Gynaecologists and Obstetricians[END_REF][START_REF] Thuroff | EAU guidelines on urinary incontinence[END_REF].

Conclusion

Female urinary incontinence is a complex and dynamic phenomenon, related with age and a many other factors that can change with time. In order to gain greater insight, longitudinal studies are necessary, with several years or even decades of follow-up in order to clarify its evolution and risk factors.

The onset and/or worsening of UI at the menopause that is expected on the basis of physiological observations is not confirmed by the results of most of the epidemiological studies that mostly cover a large number of women followed up over many years. Moreover, correction of the lack of oestrogen by HRT gives rise to paradoxical results that depend on the type of UI and mode of administration. Systemic oestrogen therapy results in an increase in UI

[ 1 . 34 [ 1 . 3 . 5 . 4 .

 1341354 44-1.65]), transdermal oestrogen (1.68 [1.41-2.00]), oral oestrogen and progestin (1.24-1.44]) and transdermal oestrogen and progestin therapies (1.46 [1.16-1.84]) (LE3). The overall risk of UI remained low however, with an annual incidence of 1.6%, and the effects of HRT disappeared when treatment ceased.A recent cohort study of post-menopausal women found an association between UI symptoms and duration of oestrogen treatment. Prolonged oral oestrogen therapy (5 years or more) is associated with a worsening of leakage symptoms (OR= 3.99[1.21-13.1]) and also an increase in the frequency of handicapping UI (3.97 [1.02-15.4])[START_REF] Northington | Self-reported estrogen use and newly incident urinary incontinence among postmenopausal community-dwelling women[END_REF] (LE3). Data obtained from studies concerning intermediate markers Certain older studies report that the use of oral oestrogens appears to increase the urethral closing pressure and could thus improve SUI symptoms in 65 to 70% of women[START_REF] Rud | Urethral pressure profile in continent women from childhood to old age[END_REF] (LE5).


E Bacry

S Delattre

M Hoffmann

J F Muzy

Modeling microstructure noise with mutually exciting point processes

Keywords: Microstructure noise. Point processes. Hawkes processes. Bartlett spectrum. Signature plot. Epps effect. Mathematical Subject Classification: 60G55, 62M10, 62M15

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales.

More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

Introduction

With the availability of a huge number of quality high frequency data, there is a fast growing literature devoted to the modelling of intra-daily asset prices behaviour. Since Bachelier and the seminal work of Black and Scholes, most popular models at a coarse time scale -for daily data, say -are Brownian diffusions, see for instance the classical textbooks of Musiela [START_REF] Musiela | Martingale Methods in Financial Modelling[END_REF] or Bouchaud and Potters [START_REF] Bouchaud | Theory of Financial Risk and Derivative Pricing[END_REF] and the references therein. In particular, diffusion models aim at describing more or less faithfully the volatility dynamics, characterized by stylized facts such as volatility clustering or leverage effect [START_REF] Bouchaud | Theory of Financial Risk and Derivative Pricing[END_REF]. A key issue that naturally emerges when one studies high frequency data is the problem of improving volatility estimation or covariance estimation between two asset returns over a given time period, thanks to the massive amount of data available at such scales nowadays. The discrete nature of time trade arrivals and of price variations (prices are point processes living on a tick grid), the presence of so-called microstructure noise (described as strong mean reversion effects at small scales) makes this question highly non trivial. At a very high frequency, prices variations are also characterized by well documented stylized facts like the signature plot behaviour and the Epps Effect [START_REF] Epps | Comovements in stock prices in the very short run[END_REF].

High frequency volatility: microstructure noise

If X(t) stands for the price of some asset at time t (defined indifferently as the last traded price or the mid-price between best bid and best offer in the order book), the signature plot can be defined from the quadratic variation of X(t) over a time periode [0, T ] at a scale τ > 0 -the so-called realized volatility -as

C(τ ) = 1 T T /τ n=0 X (n + 1)τ -X(nτ ) 2 . (1)
The microstructure noise effect manifests through an increase of the observed daily variance when one goes from large to small scales i.e. in the limit τ → 0 (see e.g. Fig. 5(b)). This behaviour is different from what one would expect if the data were sampled from a Brownian diffusion, in which case the function plotted in Fig. 5(b) should be flat. From the perspective of statistical estimation, this leads to a simple paradox. On one side, the smaller τ , the larger is the dataset that can be use to estimate the volatility. However, how should one be using high-frequency data in order to obtain better estimates of the volatility, since the realized volatility (1) is not stable as τ decreases.

In the literature the most popular approaches attempts to model microstructure noise with the concept of latent price. One starts with a Brownian diffusion X(t) defined as an efficient price, which is latent, in the sense that it cannot be observed directly. Instead the practitioner has only access to a noisy version X(t) of X(t) that accounts for microstructure noise. The most successful version -as far as mathematical development is concerned -is the additive microstucture noise model, introduced in 2001 by Gloter and Jacod [START_REF] Gloter | Diffusion with measurement errors. i. local asymptotic normality[END_REF][START_REF] Gloter | Diffusion with measurement errors. ii. optimal estimator[END_REF] and in the context of financial data by Ait-Sahalia et al. [START_REF] Ait-Sahalia | How often to sample a continuous-time process in the presence of market microstructure noise[END_REF][START_REF] Zhang | A tale of two time scales: Determining integrated volatility with noisy high-frequency data[END_REF][START_REF] Zhang | Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach[END_REF]. Given a sampling scale τ , one rather observes

X(nτ ) = X(nτ ) + ξ n,τ , (2)
where the microstructure noise term ξ n,τ satisfies E ξ n,τ = 0 for obvious identifiability conditions. (Hereafter, E[•] denotes the expectation operator.)

The goal is then to separate the noise from the true signal X(t), from which a classical volatility estimator can be performed. This has raised a vast research program over the last decade, mostly covered by econometricians and statisticians, see [START_REF] Ait-Sahalia | Ultra high frequency volatility estimation with dependent microstructure noise[END_REF][START_REF] Seo | Realized volatility and colored market microstructure noise[END_REF][START_REF] Diebold | On the correlation structure of microstructure noise in theory and practice[END_REF][START_REF] Robert | A new approach for the dynamics of ultra high frequency data: the model with uncertainty zones[END_REF][START_REF] Rosenbaum | A new microstructure noise index[END_REF][START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF][START_REF] Bandi | Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations[END_REF][START_REF] Bandi | Separating microstructure noise from volatility[END_REF][START_REF] Barndorff-Nielsen | Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise[END_REF][START_REF] Jacod | Microstructure noise in the continuous case: The pre-averaging approach[END_REF][START_REF] Munk | Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise[END_REF][START_REF] Munk | Lower bounds for volatility estimation in microstructure noise models[END_REF][START_REF] Reiß | Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise[END_REF][START_REF] Hoffmann | Nonparamtreic estimation of the volatility under microstructure noise[END_REF] and the references therein. Whereas representation [START_REF] Ait-Sahalia | How often to sample a continuous-time process in the presence of market microstructure noise[END_REF] produces an elegant pilot model to describe microstructure noise effects at the scale of a few minutes, it cannot faithfully reproduce the data as they are observed on a microscopic scale of a few seconds: for instance, the discreteness of price changes is left out and the mathematical artefact of forcing C(τ ) to explode when τ → 0 becomes unavoidable.

High frequency correlations : the Epps effect

Another important feature concerns the way different asset price movements are correlated. Stong correlations between asset returns at daily or larger time scales usually exist, and modern portfolio theory precisely relies on such dependences. One expects that coarse scale correlations originate from intraday strongly correlated movements, and this naturally raises the question of how to get better estimates of such correlations by using high frequency data. Again, an analogous paradox as for volatility estimation under microstructure noise arises. In Fig. 6, we display the last traded ask prices of futures Bund 10Y and Bobl 5Y over a period of a few hours, prices increments are clearly correlated in their intra-day variations. If X 1 (t) and X 2 (t) are the prices of two assets, a correlation coefficient estimator over a time period [0, T ] can be naturally defined from high frequency price increments as

ρ(τ ) = C 12 (τ ) C 1 (τ ) C 2 (τ )
, where

C 12 (τ ) = 1 T T /τ n=0 X 1 (n + 1)τ -X 1 nτ X 2 (n + 1)τ -X 2 nτ
and C 1 (τ ) and C 2 (τ ) denote the realized volatility of X 1 (t) and X 2 (t) respectively at scale τ . In Fig. 7(c), we plot ρ(τ ) as a function of τ for the couple Bund/Bobl. It is striking to observe that the correlation coefficient is an increasing function of the time resolution and that correlation almost vanish at a very high frequency. This phenomenon, first reported by Epps [START_REF] Epps | Comovements in stock prices in the very short run[END_REF], is the so-called Epps effect. Few approaches however address the Epps effect in the literature.

Besides the problems related to the estimation of coarse scale asset properties, we see that modelling high frequency price dynamics is a source a many challenging questions. Other approaches consist in defining "fine" scale models and addressing directly the price dynamics at the tick level, see for instance Engle and collaborators, [START_REF] Engle | Autoregressive conditional duration: A new model for irregularly spaced transaction data[END_REF][START_REF] Engle | Trades and quotes: a bivariate point process[END_REF] who introduce the ACM-ACD model or Bauwens and Hautsch [START_REF] Bauwens | Modelling financial high frequency data using point processes[END_REF], Bowsher [START_REF] Bowsher | Modelling scurity market eents in continuous time: intensity based, multivariate processes[END_REF] who construct intensity based point processes for modelling high frequency data. In this context, Hewlett [START_REF] Hewlett | Clustering of order arrivals, price impact and trade path optimisation[END_REF] introduces a model for measuring trade arrivals that is formally very close to our construction. However, Hewlett's approach is mostly order book oriented: it focuses on the imbalance properties of an asset and leaves out the directional behaviour of the prices. In the same way, the aforementioned literature that uses point processes is essentially specialized to high frequency data description, or focuses on trades arrivals dynamics. In particular, the questions related to intermediate or asymptotic behaviour of the statistical price properties as the resolution scale varies are not considered. We plan to address in the present paper this next logical step.

Objective and content of the paper

We introduce a "fine-to-coarse" model that starts from the description of the changes of prices in continuous time and that allows one, from the microscopic properties of the model to recover a large scale diffusion behaviour. More precisely, be defining a (multivariate) tick-by-tick model by means of marked point processes with appropriate stochastic intensities, we are able to control its features at all scale (i.e. its aggregation properties). Our model relies on multivariate Hawkes processes [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF][START_REF] Daley | An introduction to the theory of point processes[END_REF] associated with positive and negative jumps of the asset prices. Notably, by coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales.

The paper is organized as follows: in Section 2 we define a univariate version of the model and compute the expected signature plot. We discuss how to simulate the model in practice and how to estimate its parameters from real data. The vector (bivariate) version of the model is defined in Section 3. We show that a closed form expression for the price variations correlation functions can be explicitly obtained for all time scales and time lags. From such quantity one can deduce all second order properties of the process like individual signature plots, the Epps effect or lead-lag effects. We discuss the large scale diffusive limit of our model as far as its correlations properties are concerned and make the link with further mathematical asymptotic results (that are presented in a separate forthcoming paper). Comparisons to empirical data are provided in Section 5 in both 1D and 2D frameworks. Section 6 is devoted to a conclusion and prospects for future research.

2 The model in the univariate case

Construction of the model

We start with two point processes N 1 (t) and N 2 (t) for t ∈ [0, T ] that represent respectively the sum of positive and negative jumps of some asset price X(t) over some time horizon [0, T ]:

X(t) = N 1 (t) -N 2 (t).
If N 1 (t) and N 2 (t) are two independent Poisson processes with intensity µ, it is easy to show that the model diffuses at large scale, i.e., when T → ∞, by introducing the scaling factor T 1/2 , we obtain the following limit in distribution: lim

T →+∞ 1 √ T X(tT ) (d) = 2µB(t), t ∈ [0, 1],
where √ 2µ is the diffusive or macroscopic volatility, that accounts for the activity of negative and positive jumps, hence the factor 2 in the limit. According to Eq. ( 1), the corresponding mean signature plot is flat: for all τ > 0, we have E[ C(τ )] = 2µ. In order to account for the previously reported noise microstructure features, intuitively and as confirmed by empirical observations, one has to introduce some mean reversion in the small scales, while ensuring that this mean reversion effect vanishes on large scale. This can be naturally done within the context of (multivariate) Hawkes process [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF][START_REF] Daley | An introduction to the theory of point processes[END_REF] as follows.

For technical reason, we extend in a first step the time horizon [0, T ] over the whole real line R = (-∞, +∞). Let λ i (t) for t ∈ R be the stochastic intensities of two counting processes N i (t), i = 1, 2, such that at time t:

λ i (t) = lim ∆→0 ∆ -1 E [N i (t + ∆) -N i (t) | F t ] (3)
where F t stands for the filtration generated by the history of the processes

N 1 (t), N 2 (t
λ i (t) = µ i + t -∞ ϕ ii (t -u)dN i (u) + t -∞ ϕ ij (t -u)dN j (u). (4)
The so-obtained process can be shown to be well defined and to admit a version with stationary increments under the stability condition all the eigenvalues of the matrix

{||ϕ ij || 1 }are < 1, (5)
where ϕ 1 = R ϕ(t)dt, see [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF][START_REF] Daley | An introduction to the theory of point processes[END_REF]. Mean reversion can be translated by the fact that the more X(t) goes up, the greater the intensity λ 2 (t) and conversely, the more X(t) goes down, the greater the intensity λ 1 (t). This leads to the following simplified version of previous model (where only mean-reverting terms were kept):

λ 1 (t) = µ + t -∞ ϕ(t -s)dN 2 (s) (6)
λ 2 (t) = µ + t -∞ ϕ(t -s)dN 1 (s) (7)
where µ is an exogenous intensity and ϕ(t) a positive kernel which is causal (i.e., Supp(ϕ) ⊂ R + ). Eqs. ( 6) and ( 7) define two mutually exciting point processes that are stationary and stable under the condition ||ϕ|| 1 < 1.

A simple and natural choice for ϕ is a right-sided exponential function:

ϕ(t) = αe -βt 1 R + (t) (8)
where α, β > 0 are such that

||ϕ|| 1 = α β < 1. (9)

Signature plot

We are interested only in X(t) on [0, T ] ⊂ R + and for simplicity and without loss of generality, we set1 X(0) = 0. Using Eq. ( 1), the mean signature plot can be written as

C(τ ) = E C(τ ) = 1 τ E |X((n + 1)τ ) -X(nτ )| 2 = 1 τ E[X(τ ) 2 ].
If ϕ(t) is defined as in Eq. ( 8), then one can actually obtain a closed form for the mean signature plot using the approach initiated in [START_REF] Bartlett | The spectral analysis of point processes[END_REF][START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF]. In Appendix 1, we prove the following result:

Proposition 2.1. Under the stability condition (9), we have

C(τ ) = Λ κ 2 + (1 -κ 2 ) 1 -e -γτ γτ (10)
where

Λ = 2µ 1 -||ϕ|| 1 , κ = 1 1 + ||ϕ|| 1
, and γ = α + β.

One sees in particular a cross-over from the microstructural variance

V 0 = E[ C(0)] = Λ = 2E(λ i ),
to the diffusive variance

V ∞ = E[ C(∞)] = Λκ 2 ,
In Fig. 1(b), we display a plot example of the function C(τ ).

Numerical simulations and parameter estimation

In this section, we focus on the numerical simulations and the parameters estimation of the simplified univariate model defined by Eqs ( 6) and ( 7) in the case the function ϕ is the right sided exponential function [START_REF] Bouchaud | Theory of Financial Risk and Derivative Pricing[END_REF]. Thus, there are 3 parameters, namely : θ = (µ, α, β). Simulation of this process on an interval [0, T ] can be performed using the thinning algorithm described in [START_REF] Ogata | On lewis simulation method for point processes[END_REF]. It basically consists in simulating on [0, T ] a standard Poisson process with an intensity M large enough such that it satisfies the following condition:

λ 1 (t) < M and λ 2 (t) < M, for all t ∈ [0, T ]
(which of course can be checked only a posteriori). A thinning procedure is then applied to each jump of the obtained process from the first one to the last one allowing to either reject the point (with probability M -λ 1 (t)-λ 2 (t)) or mark it as a jump of N 1 (t) with probability λ 1 (t) or of N 2 (t) with probability λ 2 (t), where t is the time of the considered jump.

A realization of the process X over T = 42 hours is represented in Fig. 1(a) with θ = (µ = 0.016, α = 0.023, β = 0.11). Let us note that µ,α and β are all expressed in the the same unit, namely seconds -1 . These particular values were chosen to match the estimated parameters on real data (see Section 5.2).

Let {t

(1) i } 0≤i<N 1 (•) (resp. {t (2)
i } 0≤i<N 2 (•) ) the upward (resp. downward) jumps of the realization of X. The estimation of the parameters can be processed in very different ways depending on what is the focus of the model. On the one hand, if one is mainly interested in the ability of the model to reproduce the mean signature plot, the parameters can be estimated using a best fit of the realized signature plot. The realized signature plot over [0, T ] is defined as

C(τ ) = 1 T T /τ n=0 X (n + 1)τ -X(nτ ) 2 , (11)
and regression estimator θ reg is then naturally given by

θ reg = Argmin θ C(τ ) -C(τ ) 2 , (12)
where C(τ ) is defined by [START_REF] Daley | An introduction to the theory of point processes[END_REF].

On the other hand, if the goal of the model is not simply to reproduce the signature plot behaviour but to mimick the arrival times themselves, it is more natural to consider the Maximum Likelihood Estimator (MLE) instead. This is possible since there is a closed formula for the likelihood. Let N (t) be a multivariate point process with conditionnal intensity λ(t) = λ θ (t) depending on a parameter θ. If, for every θ, the law of the process N (t) is absolutely continuous w.r.t. to the law of a standard stationary Poisson process, then the statistical model generated by the continuous observation of N (t) over [0, T ] has a log-likelihood function given by

L(θ) = T 0 log λ θ (t)dN (t) + T 0 1 -λ θ (t) dt, (13)
see for instance [START_REF] Jacod | Limit theorems for stochastic processes[END_REF][START_REF] Daley | An introduction to the theory of point processes[END_REF] or [START_REF] Ogata | The asymptotic behaviour of maximum likelihood estimators for stationary point processes[END_REF] in the context of Hawkes processes and the references therein. Consequently the likelihood functions θ → L 1 (θ) and θ → L 2 (θ) generated by the observation of N 1 (t) and N 2 (t) over [0, T ] are given by

L 1 (θ) = 0≤t (1) i <N 1 (T ) log µ + 0≤t (2) j <N 2 (T ) αe -β(t (1) i -t (2) j ) -(µ -1)T - 0≤t (2) j <N 2 (T ) α β 1 -e -β(T -t (2) j ) .
and similarly

L 2 (θ) = 0≤t (2) i <N 2 (T ) log µ + 0≤t (1) j <N 2 (T ) αe -β(t (2) i -t (1) j ) -(µ -1)T - 0≤t (1)
j <N 1 (T ) α β 1 -e -β(T -t (1) j ) .
and the MLE of θ based on the observation of X(t) over [0, T ] is thus given by

θ MLE = Argmin θ L 1 (θ) + L 2 (θ) . (14)
Of course, for both estimators θ reg and θ M LE the minimization (Eqs ( 11) and ( 13)) must be performed under the constraints µ > 0, α > 0, β > 0, and the stability condition :

α β < 1. ( 15
)
Let us note that whereas the regression estimator θ reg can be performed on uniformly sampled data, the ML estimator θ M LE needs to have access to the point process itself. In that sense, θ reg can be considered as a (multi-scale) low-frequency estimator. For that reason, when applied to real data, we expect it to be much more stable than θ M LE (see Section 5). Moreover, the regression estimator has the advantage to be computationally faster than the MLE. The estimated parameters of the realization shown in Fig. 1(a) using MLE are µ = 0.016, α = 0.024 and β = 0.11. They match the true parameter values (µ = 0.16, α = 0.024, β = 0.11).

The estimation using the regression estimator appears to be almost as accurate as the MLE: we obtain µ = 0.016, α = 0.021 and β = 0.010. In Fig. 1(b) we display the estimated and theoretical signature plots (Eq. ( 10)). One sees that the latter provides a good fit of the data.

3 The model in the bivariate case

Definition

The bivariate model is a natural extension of the former construction. We start from two processes X 1 (t) and X 2 (t) with t ∈ R, each constructed as in Section 2.1 (i.e. in dimension 1) and introduce a supplementary coupling on the intensities of the processes in order to create a dependence structure. More precisely, we consider four mutually exciting point processes {N i (t)} i=1,...,4 associated with the positive and negative variations of X 1 (t) and X 2 (t):

X 1 (t) = N 1 (t) -N 2 (t) X 2 (t) = N 3 (t) -N 4 (t) .
The joint law of the processes N i (t) is characterized by their intensities:

λ i (t) = µ i + 4 j=1 t -∞ ϕ ij (t -s) dN j (s) i = 1, . . . , 4
Let us note that, as previously, the kernels ϕ ij account for the mutual and cross excitations of positive/negative parts of the couple of assets. In the sequel, in accordance with the univariate case, we suppose that µ 1 = µ 2 and µ 3 = µ 4 (but we do not assume that µ 1 = µ 3 ). In order to account for mean reversion and cross coupling between the two assets, we do not consider all possible mutual and cross excitations. More precisely, we only want to consider Upward-X 1 -Upward-X 2 and Downward-X 1 -Downward-X 2 couplings. We ignore further possible couplings Upward-X 1 -Downward-X 2 and Downward-X 1 -Upward-X 2 between X 1 and X 2 . We thus choose the matrix Φ = {ϕ ij } 1≤i,j≤4 of the following form

Φ =     0 ϕ 12 ϕ 13 0 ϕ 12 0 0 ϕ 13 ϕ 31 0 0 ϕ 34 0 ϕ 31 ϕ 34 0     . ( 16
)
Let us consider the matrix Γ = {Γ ij } 1≤i,j≤4 with entries Γ ij = ||ϕ ij || 1 and Λ the vector of mean intensities:

Λ i = E dN i dt (17)
It is shown in Hawkes [17] that:

Λ = (Id -Γ) -1 µ (18)
The solution to this equation reads:

Λ =         - µ 1 Γ 34 -µ 1 -Γ 13 µ 3 Γ 12 Γ 34 -Γ 12 -Γ 34 +1-Γ 31 Γ 13 - µ 1 Γ 34 -µ 1 -Γ 13 µ 3 Γ 12 Γ 34 -Γ 12 -Γ 34 +1-Γ 31 Γ 13 -µ 3 Γ 12 +Γ 31 µ 1 +µ 3 Γ 12 Γ 34 -Γ 12 -Γ 34 +1-Γ 31 Γ 13 -µ 3 Γ 12 +Γ 31 µ 1 +µ 3 Γ 12 Γ 34 -Γ 12 -Γ 34 +1-Γ 31 Γ 13         (19)
As for the univariate case, we consider functions ϕ ij that are causal exponentials:

ϕ ij (x) = α ij e -β ij x 1 R + (x). (20)
In that case one simply obtains:

Γ ij = α ij β ij . ( 21
)
For some t 0 ∈ R and a scale τ , let us define the increments of a process X as ∆ τ X(t 0 ) = X(t 0 + τ ) -X(t 0 ).

If one wants to characterize multiscale self and cross-correlations of two asset prices X 1 (t) and X 2 (t) the quantity of interest is the covariance matrix C(t, τ ) = {C kl (t, τ )} 1≤k,l≤2 with entries

C kl (τ, t) = Cov [∆ τ X k (t 0 ), ∆ τ X l (t 0 + t)] . (22)
Thanks to the stationarity of the increments of X k , the matrix C does not depend on t 0 . In particular, we recover from C(t, τ ) the mean signature plots on the diagonal and the Epps effect off diagonal as τ varies. We also can estimate possible lead-lag effects across various time scales, that we can loosely define as the property that the function t → C kl (τ, t) is not symmetric around 0 for k = l (see below). In the following, we denote by C kl (τ ) = C kl (τ, t = 0) the correlations at lag t = 0.

Computation of the signature plot and the Epps effect

In Appendix 2, we show that we obtain a closed form expression for the covariance matrix [START_REF] Joulin | Stock price jumps: news and volume play a minor role[END_REF]. The general expression where all parameters are arbitrary is cumbersome and we limit ourselves to a matrix Φ of the form [START_REF] Gloter | Diffusion with measurement errors. ii. optimal estimator[END_REF] with

ϕ ij (x) = α ij exp(-β ij x)1 R + (x)
and

β ij = β, i, j = 1, 3.
In that case we get the following proposition: Proposition 3.1. The expression of the covariance C kl (τ ) as a function of the time scale τ reads:

C11(τ ) τ = 2A11e -G 1 τ G 2 1 τ + 2B11e -G 2 τ G 2 2 τ + 2Λ1 + 2A11 G1 + 2B11 G2 - 2A11 G 2 1 τ - 2B11 G 2 2 τ , C21(τ ) τ = C12(τ ) τ = (A21 + A12)( e -G 1 τ -1 G 2 1 τ + 1 G1 ) + (B21 + B12)( e -G 2 τ -1 G 2 2 τ + 1 G2 ), C22(τ ) τ = 2A22e -G 1 τ G 2 1 τ + 2B22e -G 2 τ G 2 2 τ + 2Λ3 + 2A22 G1 + 2B22 G2 - 2A22 G 2 1 τ - 2B22 G 2 2 τ
, with explicit constants provided in Appendix 2.

It is interesting to discuss the fully symmetric case, i.e. when ϕ 12 = ϕ 34 , ϕ 13 = ϕ 31 and µ 1 = µ 3 = µ. In this case a direct computation from [START_REF] Hoffmann | Nonparamtreic estimation of the volatility under microstructure noise[END_REF]

leads to Λ = Λ 1 = Λ 2 = Λ 3 = Λ 4 = µ 1 -Γ 12 -Γ 13 (23)
If follows from the results of Appendix 2, that

Q 1 = Q 4 = -µ Γ 2 12 + Γ 12 -Γ 2 13 (Γ 12 + 1) 2 -Γ 2 13 (1 -Γ 12 -Γ 13 ) Q 2 = Q 3 = -µΓ 13 (Γ 12 + 1) 2 -Γ 2 13 (1 -Γ 12 -Γ 13 )
After some algebra, one obtains: Corollary 3.2. In the fully symmetric case, the covariance matrix has the following expression:

C 11 (τ ) τ =Λ + RC 1 2G 1 + RC 2 2G 2 + R C 2 G 1 2 e -τ G 2 -C 1 G 2 2 + Q 1 G 2 2 e -τ G 1 -C 2 G 1 2 2G 2 2 G 1 2 τ , and
C 12 (τ ) τ = -RC 1 2G 1 + RC 2 2G 2 + R C 1 λ 2 2 -C 2 G 1 2 -C 1 G 2 2 e -λ 1 τ + C 2 G 1 2 e -G 2 τ 2G 2 2 G 1 2 τ
where

R = βµ Γ 12 + Γ 13 -1 C 1 = (2 + Γ 12 + Γ 13 )(Γ 12 + Γ 13 ) 1 + Γ 12 + Γ 13 C 2 = (2 + Γ 12 -Γ 13 )(Γ 12 -Γ 13 ) 1 + Γ 12 -Γ 13 and G 1 = β(1 + Γ 12 + Γ 13 ), G 2 = β(1 + Γ 12 -Γ 13 ).
One can reproduce the Epps effect by evaluating the behavior of the correlation coefficient ρ(τ ) = C 12 (τ )/C 11 (τ ) as a function of τ . From Corollary (3.2) one can see that

ρ(τ ) = R(Q 2 -Q 1 ) 4Λ τ + O τ 2 as τ → 0, ρ(τ ) → R 2 -R 1 2Λ + R 1 + R 2 as τ → ∞,
where we have set R i = RQ i /λ i . If one considers the definition of each constant, one gets

ρ(τ ) → 2Γ 13 (1 + Γ 12 ) 1 + Γ 2 13 + 2Γ 12 + Γ 2 12 as τ → ∞. (24)
Notice that when the processes are not correlated, i.e. when Γ 13 = 0, we have

G 1 = G 2 = β(1+Γ 12 ) = β+α 12 = γ, Λ = µ/(1-Γ 12 ), R = βµ/(Γ 12 -1), Q = Q 1 = Q 2 = Γ 12 (Γ 12 + 2)/(1 + Γ 12 ). Thus RQ λ 1 = µ Γ 12 (Γ 12 + 2) (Γ 12 -1)(Γ 12 + 1) 2
and therefore

C 11 (τ ) τ = Λ + RQ G 1 + RQ G 2 1 τ (e -G 1 τ -1) = µ 1 -Γ 12 1 (1 + Γ 12 ) 2 + (1 - 1 (1 + Γ 12 ) 2 )
1 -e -γτ γτ .

We thus recover in that case the univariate result of Eq. ( 4).

In order to account for the existence of a possible lead-lag effect, one can introduce a measure of the asymmetry of the covariance matrix at scale τ as e.g.

∆(τ ) = C 12 (τ, τ ) -C 21 (τ, τ ) . ( 25
)
This coefficient measures the difference of the correlation of assets 1 and 2 returns at scale τ and lag τ when assets 1 is respectively in the past or in the future of asset 2. It is easy to see that if the matrix ( 16) is symmetric then ∆ = 0 and there is no lead-lag. In the general case, from the results (and within notations) of Appendix 2, if D 1 = A 12 -A 21 and D 2 = B 12 -B 21 then:

∆(τ ) = D 1 1 + e -2G 1 τ 2 -2e -G 1 τ G 2 1 + D 2 1 + e -2G 2 τ 2 -2e -G 2 τ G 2 2 (26)
The expressions for D 1 and D 2 are relatively heavy to handle in the general case. In the simple asymmetric case when α 13 = 0 one has:

D 1 = - 2α 12 α 31 βµ 1 (2β + α 12 ) (α 12 -α 34 )(2β + α 34 + α 12 )(β -α 12 ) (27)
The specific study of lead-lags effects within this approach will be the scope of a forthcoming paper.

Numerical simulations and parameter estimation

In this section, we focus on numerical simulations and parameters estimation of the bivariate model in the fully symmetric case (ϕ 12 = ϕ 34 , ϕ 13 = ϕ 31 and µ 1 = µ 3 = µ), in the case where all the functions ϕ ij are right sided exponential functions [START_REF] Jacod | Microstructure noise in the continuous case: The pre-averaging approach[END_REF] with β ij = β for all i, j. Thus, there are 4 parameters, namely : θ = (µ, α 1,2 , α 1,3 , β). A realization of this process bivariate (X 1 , X 2 ) over T = 20 hours is represented in Fig. 2 with α 12 = 0.23, α 13 = 0.05, β = 0.11 and µ = 0.015. (let us note that µ, α 12 , α 13 and β are all expressed in the same unit, namely seconds -1 ). According to Eq. ( 24), the asymptotic correlation between the (large scale) increments of X 1 and X 2 is ρ ≃ 0.15. Thus the two components are only moderately correlated (at a visual level, the graphs do not look alike). Fig. 3 shows the realization of the bivariate process (X 1 , X 2 ) in the case α 12 = 0.23, α 13 = 0.05, β = 0.11 and µ = 0.015. Eq. [START_REF] Munk | Lower bounds for volatility estimation in microstructure noise models[END_REF] shows that the correlation between the two components is quite strong: ρ ≃ 0.65. This can be clearly seen on the two graphs which look quite much alike.

In Fig. 4(b), the estimated and theoretical Epps effect (i.e. the function ρ(τ ) = C 12 (τ )/C 11 (τ )) are plotted for three sets of parameters, corresponding respectively to an asymptotic correlation coefficient of 0.15, 0.40 and 0.65.

As for the univariate case, one could perform maximum likelihood estimation of the parameters. Bivariate formula for the likelihood are easy to obtained following the computations in Section 2.3. In the bivariate model, the quantities of interest are not only the signature plots of both X 1 and X 2 but also the Epps effect between X 1 and X 2 . The regression estimation now takes the form

θ reg = Argmin θ a 1 | C 11 (τ ) -C 11 (τ )| 2 + a 2 | C 22 (τ ) -C 22 (τ )| 2 + a 12 | C 12 (τ ) -C 12 (τ ))| 2 ,
where C kl (τ ) = C kl (τ, 0) is defined as in [START_REF] Joulin | Stock price jumps: news and volume play a minor role[END_REF] and

C kl (τ ) = 1 T T /τ n=0 ∆ τ X k (nτ )∆ τ X l (n + 1)τ . ( 28
)
The constants a 1 ,a 2 and a 12 are constant weights that are used to fix the relative minimization error of each term. Both estimators ( θ MLE and θ reg ) lead to quite accurate results (with the same magnitude of errors in both cases).

Diffusive (large scale) limit of the model

It is interesting to discuss the large scale limit of the processes obtained by our construction. For that purpose, Let us define the following normalized processes for the general N -variate version of the model:

{X (T ) i (t)} i=1...N = { 1 √ T X i (tT )} i=1...N for t ∈ [0, 1]. ( 29
)
The question we would like to address concerns the existence and the properties of the macroscopic limit N -variate process {X (∞) (t)} i=1...N obtained by considering the large scale limit T → ∞. As far as the second order (correlation) properties of these processes are concerned, from Propositions 2.1 and 3.1, one sees that in both the univariate and the bivariate case, a diffusive limit exists. For instance, in the bivariate case, the limit process is characterized by the following covariance matrix:

C (2) =   2A 11 G 1 + 2B 11 G 2 A 12 +A 21 G 1 + B 12 +B 21 G 2 A 12 +A 21 G 1 + B 12 +B 21 G 2 2A 22 G 1 + 2B 22 G 2   , (30)
By using limit theorem for semimartingales, it is actually possible to rigorously obtain a limiting process which is a multivariate Brownian motion with the appropriate covariance matrix. We describe and prove this results in details in a forthcoming paper.

5 Comparison to empirical data

Presentation of the data

Let us recall that our goal is not to reproduce perfectly the features observed from empirical data since on the one hand, ultra high-frequency tick-by-tick series can be defined in various way (e.g. mid prices, transaction prices, ask or bid prices and so on), each of them having slightly different properties and representing an arbitrary projection of the complex order book dynamics. On the other hand, it is well know that market intraday fluctuations are not stationary and are characterized by strong long term heterogeneities. For computation and parsimony purposes, our model is somehow naive since the exponential shape of the kernels ϕ ij are somewhat arbitrary and the chosen symmetries are arbitrary and not necessarily suitable in practice. We rather aim at providing a first-brick model that can be considered as a tool to understand how the discrete nature of price variations at fine scales can be aggregated at coarser scales and lead to cross correlated Brownian diffusion processes. In that respect, comparisons to real data have to be interpreted rather at a qualitative level. The data that have been used in this paper consist in tick-by-tick last traded price time series, with, for each trade, the corresponding volume and a flag indicating whether the trade corresponds to a sell order or a buy order. The prices are either Eurex Euro-Bund futures contracts or Eurex Euro-Bobl futures contracts which correspond respectively to longterm (8.5-10.5 years) or medium-term (4.5 to 5.5 years) debt instrument issued by the Federal Republic of Germany. Euro-Bund and Euro-Boble are well known to have highly correlated price variations. So they are good candidates to study the Epps effect. Both open from (local-time) 8am to 10pm, but we shall restrict to the most liquid period : 8am to 5:15pm. However, the liquidity (and the volatility) is highly seasonal during the day. Our model does not account for such a seasonality, so we shall restrict the data to intraday periods for which the underlying stationarity assumption is reasonable. Based on rough empirical considerations, we select the time period 9am to 11am. Moreover all computations have been made on last traded prices of buy orders only. Choosing sell orders would not have change the results, however, taking into account in the same time-series both buy and sell orders would lead to a highly bouncing artefact that shall not be able to be captured by our modelling approach.

To summarize, the computations are made on two datasets : ). The signature plot displayed in Fig. 5(b) has been computed using the dataset I described at the end of Section 5.1. Each day is considered to be an independent and identically distributed realization of the same process. In solid line we have superimposed the signature plot obtained when we fit the mean theoretical signature plot by a regression model in the univariate version of the model. The estimates we obtain are µ = 0.016, α = 0.024 and β = 0.11. We see that the curve associated with the model fits the data relatively well. Notice that the MLE method also provides fairly good results (in that case, we obtain µ = 0.014, α = 0.030 and β = 0.08). However, as explained previously, one expects the MLE to be less stable with respect to "noise" than the regression method on the mean signature plot and we empirically observe that its performance are worse than for the regression estimator. On other instances, they could lead to dramatically bad results and this is the reason why we discard the MLE estimator further on.

Signature plots and Epps effect

Figure 6 shows the logarithm of the last traded price (only buy orders) of the Euro-Bund top) and Euro-Bobl (bottom) during a whole day 11/03/2009 (maturity 12/2009). One can directly observe the return correlations between the two assets. The large scale correlation coefficient we find is close to ρ = 0.77. In Figures 7(a) and 7(b) we plot the signature plots associated with the two assets while 7(c) displays the estimated Epps effect as measured by the correlation coefficient C 12 (τ ) at different scales. The computations were made using the dataset II described at the end of Section 5.1. The solid lines represent the fits according to the regression method of the bivariate model. One can see that although significant discrepencies between empirical and fitted data are observed, given the simplicity of the model, one can consider that it captures fairly well both variance and covariance features of assets from small to large time scales simultaneously.

Conclusion and prospects

We have proposed in this paper a simple bivariate tick-by-tick price model based on Hawkes (self and mutually exciting) point processes. We have shown that closed form expressions can be obtained for its second order properties at all time scales. This allows one to recover major high frequency stylized facts, namely the signature plot behavior and the Epps effect. We have shown that the model is easy to simulate and can be estimated with a MLE method or using a moment method. When compared to real data, we have seen that the 2D model hardly accounts for the exact behavior of signature plots and correlation functions. However, as emphasized in section 5, the approach introduced in this paper has to be considered as a simple framework that allows one to address issues related to the relationship between fine and coarse scale properties of market dynamics. It can also be used as a simple tool in order to investigate intraday market features using few parameters that are easy to interpret. For instance, according to the univariate version of the model, the variance associated with the microstructure is Λ = 2µ 1 -||ϕ|| 1 while the large scale diffuse volatility is

Λκ 2 = 2µ 1 -||ϕ|| 1 1 (1 + ||ϕ|| 1 ) 2 .
One sees that the mean reversion (as measured by 0 < ||ϕ|| 1 < 1) is softened at large scale thanks to the diffusion. however, the influence of ϕ does not completely disappear. From the above tow expressions of the microscopic and the macroscopic variance, the large scale effect of mean reversion can be quantified by the function:

||ϕ|| 1 = x ∈ [0, 1) → 1 (1 -x)(1 + x) 2 (31)
which has a minimum around x = 1/3. In Fig. 8 we plot the histograms of estimated values of ||ϕ|| 1 during a period of 6 months for Bund and Bobl front month contracts (from 05/14/2009 to 12/31/2009). It is striking to observe that both distributions are peaked (with rather large deviations however) around x = 1/3. This suggests that the market chooses the strength of microstructural mean reversion in order to minimize its long term volatility.

In a future work, we will address the questions related to lead-lag effects that can be easily accounted, for each time scale, within our model. It will also be interesting to consider non parametric estimates of the kernel shapes (ϕ ij ) along the line of the work of [START_REF] Reynaud-Bouret | Adaptive estimation for hawkes processes; application to genome analysis[END_REF] and to consider various questions related to the well known non stationarity and long-range correlations of the volatility. Let us stress that it is also tempting to bridge the parametric approach advocated in this paper to the recent work of Joulin et al. [START_REF] Joulin | Stock price jumps: news and volume play a minor role[END_REF] where the authors studied and quantified the effects of exogeneous news with respect to the endogeneous noise on the jumps of stock prices.

Let M ij (t) denotes the continuous part of the covariance of dN i (t) and dN j (t) and M (t) = M 11 (t) -M 12 (t). From the definition of N 1 (t) and N 2 (t), we then have:

E N 1 (τ ) 2 = Λt + Λ2 τ 2 + τ 0 τ 0 M 11 (u -v)dudv E [N 1 (τ )N 2 (τ )] = Λ2 τ 2 + τ 0 τ 0 M 12 (u -v)dudv and C(τ ) = 2 Λ + τ -1 τ 0 τ 0 M 12 (u -v)dudv . (35)
In order to compute M ij (τ ) and M (τ ), one can first estimate the conditional mean with respect to F t+τ and then perform unconditional averages. Using this trick, from the definition of λ 1 (t) and λ 2 (t), one finds:

M 11 (τ ) = τ -∞ ϕ(τ -u)M 12 (u) du M 12 (τ ) = Λϕ(τ ) + τ -∞ ϕ(τ -u)M 11 (u)du
and consequently, M (τ ) satisfies:

M (τ ) = -ϕ(τ ) Λ + τ -∞ ϕ(τ -u)M (u) du . (36)
If one seeks for an exponential solution M (x) = ae -bx to this equation, one finds:

b = -(α + β) a = - Λα(α + 2β) 2(α + β)
and finally, from [START_REF] Zhang | A tale of two time scales: Determining integrated volatility with noisy high-frequency data[END_REF],

C(τ ) = 2 Λ 1 (1 + α β ) 2 + 1 - 1 (1 + α β ) 2 1 -e -(α+β)τ (α + β)τ . (37)
It is straightforward to show that C can be obtained in terms of M as:

C ρν (τ, t) = τ 0 du t+τ t dv K ρν (v -u) , (38)
where the matrix K is

K 11 (z) = (Λ 1 + Λ 2 )δ(z) + J 11 (z) K 12 (z) = J 12 (z) K 21 (z) = J 21 (z) (39) K 22 (z) = (Λ 3 + Λ 4 )δ(z) + J 22 (z)
with J defined as:

J 11 (z) = M 11 (z) + M 22 (z) -M 12 (z) -M 21 (z) J 12 (z) = M 13 (z) + M 24 (z) -M 23 (z) -M 14 (z) J 21 (z) = M 31 (z) + M 42 (z) -M 41 (z) -M 32 (z) (40) J 22 (z) = M 33 (z) + M 44 (z) -M 34 (z) -M 43
Along the same line as in [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF], one can show that the matrix M satisfies the following integral equation:

M(u) = Φ(u) diag (Λ) + u -∞ Φ(u -v)M(v) dv for u > 0. (41)
The values for u < 0 can be obtained thanks to the equality M ij (-v) = M ji (v) that is a direct consequence of the definition of M.

Let us denote by F (s) the unilateral Laplace transform of F (u). By taking the Laplace transform of Eq. (41), one gets the following set of coupled linear equation of M(s):

Mij (s) = Λ j φij (s) + k φik (s) Mkj (s) + Mjk (β ik ) . (42)
By using Eqs. (40), it follows that the Laplace transform of J ρν satisfies the following linear system:

J11 (s) = -φ12 (s) J11 (s) + φ13 (s) J21 (s) -φ12 (s) Λ 1 + Λ 2 + J11 (β 12 ) + φ13 (s) J12 (β 13 ) J12 (s) = -φ12 (s) J12 (s) + φ13 (s) J22 (s) + φ13 (s) Λ 3 + Λ 4 + J22 (β 13 ) -φ12 (s) J21 (β 12 ) J21 (s) = -φ34 (s) J21 (s) + φ31 (s) J11 (s) + φ31 (s) Λ 1 + Λ 2 + J11 (β 31 ) -φ34 (s) J12 (β 34 ) J22 (s) = -φ34 (s) J22 (s) + φ31 (s) J12 (s) -φ34 (s) Λ 3 + Λ 4 + J22 (β 34 ) + φ31 (s) J21 (β 31 ).
The solution to this system reads:

J(s) =         v 1 +v
        , (43)
where we have denoted by v the vector

v =         -φ12 (s) Λ 1 + Λ 2 + J11 (β 12 ) + φ13 (s) J12 (β 13 ) φ13 (s) Λ 3 + Λ 4 + J22 (β 13 ) -φ12 (s) J21 (β 12 ) φ31 (s) Λ 1 + Λ 2 + J11 (β 31 ) -φ34 (s) J12 (β 34 ) φ34 (s) Λ 3 + Λ 4 + J22 (β 34 ) + φ31 (s) J21 (β 31 )         . ( 44
)
This vector can be computed if by evaluating J(s) at s = β 12 , . . . , β 34 . If one now considers that the Laplace transform of φij (x) is

φij (s) = α ij s + β ij , (45)
it is possible to compute the expression of each component J αβ and, by the inverse Laplace transform, to obtain the correlation matrix C αβ (t, τ ).

Accounting for the symmetries in Eq. ( 16), in Eq. (44), we now have 4 constants to determine: Q 1 = J11 (β), Q 2 = J12 (β), Q 3 = J21 (β) and Q 4 = J22 (β). If we solve the linear system obtained by evaluating (43) in s = β, we obtain, where the expressions for the Λ i are provided in Eq. [START_REF] Hoffmann | Nonparamtreic estimation of the volatility under microstructure noise[END_REF]. One can now inverse the Laplace transforms in Eq. ( 43) and and, from Eq. (39) the expressions of the functions K ρν (t). This leads to By performing the double integral (38), one finally obtains:

Q 1 = - Λ 1 "
K 11 (t) = 2Λ
C11(τ ) τ = 2A11e -G 1 τ G 2 1 τ + 2B11e -G 2 τ G 2 2 τ + 2Λ1 + 2A11 G1 + 2B11 G2 - 2A11 G 2 1 τ - 2B11 G 2 2 τ C21(τ ) τ = C12(τ ) τ = (A21 + A12)( e -G 1 τ -1 G 2 1 τ + 1 G1 ) + (B21 + B12)( e -G 2 τ -1 G 2 2 τ + 1 G2 ) C22(τ ) τ = 2A22e -G 1 τ G 2 1 τ + 2B22e -G 2 τ G 2 2 τ + 2Λ3 + 2A22 G1 + 2B22 G2 - 2A22 G 2 1 τ - 2B22 G 2 2 τ
.

Figure 1 :

 1 Figure 1: Numerical simulation of a 1D price model using mutually exciting Hawkes processes with parameters µ = 0.16, α = 0.024, β = 0.11 (Let us note that these values were chosen to match the estimated parameters on real data as shown in Section 5.2). (a) Simulated sample path X(t) (42 hours long) (b) Estimated signature plot C(τ ) with the corresponding expected analytical shape according to Eq. (10). τ is expressed in seconds.

Figure 2 :

 2 Figure 2: Numerical simulation of a 2D symmetric price model using Hawkes processes with parameters α 12 = 0.23, α 13 = 0.01, β = 0.11 and µ = 0.015. (a) Sample path of X 1 (t) (b) Sample path of X 2 (t). The asymptotic correlation between large scale increments is ρ = 0.15.

Figure 3 :

 3 Figure 3: Numerical simulation of a 2D symmetric price model using Hawkes processes with parameters α 12 = 0.23, α 13 = 0.05, β = 0.11 and µ = 0.015.(a) Sample path of X 1 (t) (b) Sample path of X 2 (t). The processes appear more correlated then in Fig.2because the asymptotic correlation of large scale increments is ρ = 0.65.

Figure 4 :

 4 Figure 4: Epps effect for simulation samples. The estimated correlation coefficients are compared to the expected analytical curves (solid lines). From top top bottom the cross correlation between the two asset returns increases from 0.15 to 0.65.

Figure 5 (

 5 Figure5(a) shows the logarithm of the last traded price (only buy orders) of the Euro-Bund during a whole day 11/03/2009 (the contract maturity is 12/2009). The signature plot displayed in Fig.5(b) has been computed using the dataset I described at the end of Section 5.1. Each day is considered to be an independent and identically distributed realization of the same process. In solid line we have superimposed the signature plot obtained when we fit the mean theoretical signature plot by a regression model in the

Figure 5 :

 5 Figure 5: (a) Last traded (only buy market orders) prices path of the Euro-Bund contract on the 3rd of November 2009 (contract maturity 12/2009) from 8am to 10pm. (b) Associated mean daily signature plot (computed on the dataset I described at the end of Section 5.1). The solid line represents the fit using the regression estimator θ reg for the 1D Hawkes model (Section 2.3).

Figure 6 :

 6 Figure 6: (a) Last traded (only buy orders) prices path of the Euro-Bobl contract on the 3rd of November 2009 (contract maturity 12/2009) from 8am to 10pm. (b) Same as (a) but for the Euro-Bund.

Figure 7 :

 7 Figure 7: Self and cross correlation between Euro-Bund et Euro-Bobl returns as a function of the time scale τ . The computations have been made using the on the dataset II described at the end of Section 5.1. (a) Signature plot associated with the Euro-Bund (b) Same as in (a) for the Euro-Bobl. (c) Epps effect between Euro-Bund and Euro-Bobl as measured by the covariance C 12 τ between the two asset returns at different scales τ . In (a), (b) and (c) the heavy lines represent a fit of the empirical curves according to the 2D Hawkes model (Proposition 3.1).

Figure 8 :

 8 Figure 8: (a) Histogram of values of x (as defined in Eq. (31)) for the Bund front month contract from 5/14/2009 to 12/31/2009. (c) Same plot for the Bobl over the same period. The mean values for the two assets are respectively x = 0.29 and x = 0.36.

  ). The bivariate process N 1 (t), N 2 (t) is a linear Hawkes process if N 1 (t) and N 2 (t) have no common jumps and if there exist four nonnegative functions {ϕ ij } i,j=1,2 such that

  1 ϕ 34 +ϕ 13 v 3 1+ϕ 12 +ϕ 34 +ϕ 34 ϕ 12 -ϕ 31 ϕ 13 v 2 +v 2 ϕ 34 +ϕ 13 v 4 1+ϕ 12 +ϕ 34 +ϕ 34 ϕ 12 -ϕ 31 ϕ 13 ϕ 31 v 1 +v 3 +v 3 ϕ 12 1+ϕ 12 +ϕ 34 +ϕ 34 ϕ 12 -ϕ 31 ϕ 13 ϕ 31 v 2 +v 4 +v 4 ϕ 12 1+ϕ 12 +ϕ 34 +ϕ 34 ϕ 12 -ϕ 31 ϕ 13

  α 31 α 13 (1 + α 34 + β + α 12 ) -3βα 34 α 12 -βα 2 12 -α 2 34 α 12 -2α 12 β 2 -α 2 12 α 34 " + βΛ 3 α 2 13 α 13 α 31 (2β + α 12 + α 34 ) -3β 2 α 12 -α 2 12 α 34 -α 2 34 α 12 -3β 2 α 34 -2β 3 -4βα 34 α 12 -βα 2 34 -α 2 12 β Q 2 = -Λ 3 (2β 2 α 13 -α 31 α 2 13 + 2α 12 βα 13 + α 34 α 12 α 13 + α 34 βα 13 ) + Λ 1 (α 13 α 2 31 -α 12 βα 31 -α 31 α 34 α 12 ) α 13 α 31 (2β + α 12 + α 34 ) -3β 2 α 12 -α 2 12 α 34 -α 2 34 α 12 -3β 2 α 34 -2β 3 -4βα 34 α 12 -βα 2 34 -α 2 12 β Q 3 = -Λ 1 (2β 2 α 31 -α 31 α 2 13 + 2α 34 βα 31 + α 34 α 12 α 31 + α 12 βα 31 ) + Λ 3 (α 13 α 2 31 -α 34 βα 13 -α 13 α 34 α 12 ) α 13 α 31 (2β + α 12 + α 34 ) -3β 2 α 12 -α 2 12 α 34 -α 2 34 α 12 -3β 2 α 34 -2β 3 -4βα 34 α 12 -βα 2 34 -α 2 12 β " α 31 α 13 (1 + α 12 + β + α 34 ) -3βα 34 α 12 -βα 2 34 -α 2 12 α 34 -2α 34 β 2 -α 2 34 α 12 " + βΛ 1 α 2 34 α 13 α 31 (2β + α 12 + α 34 ) -3β 2 α 12 -α 2 12 α 34 -α 2 34 α 12 -3β 2 α 34 -2β 3 -4βα 34 α 12 -βα 2 34 -α 2 12 β .

			Λ 3
	Q 1	=	-

  1 δ(t) + A 11 e -G 1 t + B 11 e -G 2 t K 12 (t) = A 12 e -G 1 t + B 12 e -G 2 t K 21 (t) = A 21 e -G 1 t + B 21 e -G 2 t K 22 (t) = 2Λ 3 δ(t) + A 22 e -G 1 t + B 22 e -G 2 t 12 -α 34 ) 2 + 4α 13 α 31and the expressions of constants A ij , B ij read:A 11 = -Q 1 (2α13 α 31 + α 2 12 + 2Zα 12 -α 34 α 12 ) -Q 2 (α 12 α 13 + 2Zα 13 + α 34 α 13 ) +Λ 1 (4α 13 α 31 + 2α 2 12 + 4Zα 12 -2α 34 α 12 ) /(4Z) B 11 = -Q 1 (α 34 α 12 -2α 13 α 31 -α 2 12 + 2Zα 12 ) +Q 2 (α 13 α 12 -2Zα 13 + α 34 α 13 ) + +Λ 1 (2α 34 α 12 + 4Zα 12 -2α 2 12 -4α 13 α 31 ) /(4Z) A 12 = -Q 3 (2α 13 α 31 + 2Zα 12 + α 2 12 -α 34 α 12 ) -Q 4 (α 13 α 12 + 2Zα 13 + α 34 α 13 ) -Λ 3 (4Zα 13 + 2α 34 α 13 + 2α 12 α 13 ) /(4Z) B 12 = -Q 3 (α 34 α 12 -2α 13 α 31 + 2Zα 12 -α 2 12 ) +Q 4 (α 34 α 13 + α 12 α 13 -2Zα 13 ) +Λ 3 (2α 13 α 12 -4Zα 13 + 2α 13 α 34 ) /(4Z) A 21 = Q 1 (α 12 α 31 + 2Zα 31 + α 34 α 31 ) +Q 2 (α 12 α 34 -α 2 34 -2α 13 α 31 -2Zα 34 ) +Λ 1 (2α 12 α 31 + 2α 34 α 31 + 4Zα 31 ) /(4Z) B 21 = Q 2 (α 2 34 -α 12 α 34 + 2α 13 α 31 -2Zα 34 ) -Q 1 (α 12 α 31 -α 34 α 31 + 2Zα 31 ) +Λ 1 (4Zα 31 -2α 12 α 31 -2α 34 α 31 ) /(4Z) A 22 = Q 4 (α 12 α 34 -2α 13 α 31 -2Zα 34 -α 2 34 ) +Q 3 (α 31 α 34 + 2Zα 31 + α 31 α 12 ) +Λ 3 (2α 12 α 34 -2α 2 34 -4α 13 α 31 -4Zα 34 ) /(4Z) B 22 = Q 4 (2α 13 α 31 -α 12 α 34 + α 2 34 -2Zα 34 ) +Q 3 (2Zα 31 -α 12 α 31 -α 34 α 31 ) +Λ 3 (4α 13 α 31 -4Zα 34 + 2α 2 34 -2α 12 α 34 ) /(4Z).

	where
	G 1 = Y + Z
	G 2 = Y -Z Y = β + 1 2 (α 12 + α 34 )
	Z =	1 2	(α

Indeed, since only the increments of the processes Ni(t) come into play, we may (and will) assume that N1(t) = N2(t) = 0 hence X(0) = 0.

Acknowledgements. We thank Mathieu Rosenbaum for helpful discussions. M.H. wishes to thank Marek Musiela and the Electronic Trading Group in the FIRST team of BNP-Paribas for stimulating discussions on modelling dependence structures using point processes.

This research is part of the Chair Financial Risks of the Risk Foundation, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française and the Chair Finance and Sustainable Development sponsored by EDF and Calyon. The financial data used in this paper have been provided by the company Quant-House EUROPE/ASIA, http://www.quanthouse.com.

Appendix 1: Signature plot in the univariate case

In the univariate case the shape of the signature plot can be directly computed for an exponential kernel ϕ as given by Eq. [START_REF] Bouchaud | Theory of Financial Risk and Derivative Pricing[END_REF]. Indeed, if one defines

it directly results from Eqs ( 6) and ( 7) that

From the definition of the signature plot and by symmetry between the processes N 1 (t) and N 2 (t), we have, for τ > 0:

Appendix 2: Correlation function in the multivariate case

Let us define M(τ ) as the (continuous part of the) covariance matrix of individual point densities and Λ the mean intensity vector: