Dr L Schwartz
email: laurent.schwartz@polytechnique.fr

H Maitournam

C Stolz

J M Steayert

M C Ho

Ba Tho

B Halphen

MD, Service de Radiotherapie Hô Opital Piti

Growth and cellular differentiation: a physicobiochemical conundrum?The example of the hand

Currently, the predominant hypothesis explains cellular differentiation as an essentially genetic intracellular process. The goal of this paper is to suggest that cell growth and differentiation may be, simply, the result of physical and chemical constraints. Bone growth occurs at the level of cartilage conjunction (growth plate) in a zone of lesser constrain. It appears that this growth also induces muscle, tendon, nerve and skin elongation. This cartilage growth by itself seems to explain the elongation of the hand. Growth stops at puberty likely because of feed-back from an increasing muscle load. The ossification (that is differentiation of cartilage into bone) appears to result from the shear stress induced. The study of bone age, obtained by X-ray picture of the hand, shows that ossification of epiphyses is very precise both in time and space. Computer modelization suggests that this ossification occurs where shear stress is greatest. The cartilage which does not ossify (joint, nose, larynx, ear, bronchus, etc.) is not exposed to high shear. Shear stress induces the secretion of extracellular matrix and a change of the biochemical environment of the cell. Precipitation of calcium phosphate, as in ossification, seems related to the alkalosis induced by shear stress. To speak in more general terms, loss of cellular differentiation, as occurs with cancer, can result from a change in the physical-chemical environments.

INTRODUCTION

The growth and maturation of the hand is a highly predictable phenomena [START_REF] Graham | Assessment of bone maturation-methods and pitfall[END_REF]. During embryogenesis, bones of the diaphysis are formed on an initial cartilaginous model. Cartilage is later replaced by bone; this process is called enchondral ossification [START_REF] Cancedda | Development control of chondrogenesis and osteogenesis[END_REF]. The long bones of the hand end near the joint in a separate epiphysis, which is ossified during childhood. The epiphysis becomes fused with the shaft of the bone (diaphysis) at puberty when growth comes to an end [START_REF] Cancedda | Development control of chondrogenesis and osteogenesis[END_REF]. The epiphyses undergo a characteristic series of events: central calcification, absorption of cartilage and enchondral ossification [START_REF] Cancedda | Development control of chondrogenesis and osteogenesis[END_REF]. At birth, there is no limited detectable bone on X-ray of the epiphysis of the hand. Bone development assessed by a simple radiological picture of the right hand has been used to confirm the age of the infant [START_REF] Graham | Assessment of bone maturation-methods and pitfall[END_REF], assess its nutritional status if there is a delay in ossification because of an underlying disease [START_REF] Sutphen | Growth as a measure of the nutritional status[END_REF] or even estimate the age of the cadaver of an immature child [START_REF] Ubelaker | Estimating age of death from immature human skeletons: an overview[END_REF].

COORDINATE DEVELOPMENT OF THE HAND

The proper development of the musculoskeletal system requires the coordinated development of cartilage, bone, muscle, tendon and cartilage [START_REF] Kardon | Muscle and tendon morphogenesis in the avian hind limb[END_REF]. In the embryo, ossification of the cartilaginous anlagen of the metatarsus starts in parallel with active movement of the feet by muscle contraction [START_REF] Burger | Modulation of osteogenesis in fetal rudiments by mechanical stress[END_REF]. Mechanical stress resulting from muscle contraction seems to guide enchondral ossification patterns [START_REF] Burger | Modulation of osteogenesis in fetal rudiments by mechanical stress[END_REF][START_REF] Carter | Musculoskeletal ontogeny, phylogeny, and functional adaptation[END_REF][START_REF] Turner | Mechanotransduction and functional response of the skeleton to physical stress: the mechanism and mechanics of bone adaptation[END_REF].

Comparison of muscle and tendon reveals that much of their morphogenesis is temporally and spatially closely associated [START_REF] Kardon | Muscle and tendon morphogenesis in the avian hind limb[END_REF]. Reciprocal muscle-tendon interactions are necessary for correct muscle-tendon patterning and morphogenesis. Tendon development was examined in muscleless limbs produced by coelomic grafting of early limb buds and muscle development was analyzed in limbs where tendon had been surgically altered [START_REF] Kardon | Muscle and tendon morphogenesis in the avian hind limb[END_REF]. Subdivision of muscle masses and segregation of tendon primordia into individual tendons do require reciprocal interactions between muscle and tendon [START_REF] Kardon | Muscle and tendon morphogenesis in the avian hind limb[END_REF].

CARTILAGE DEFECTS IS RESPONSIBLE OF MULTIPLE MALFORMATIONS

Achondroplasia is a rare genetic disease. The cartilage shows morphological and biochemical abnormalities caused frequently by the substitution of arginine for glycine of the fibroblast growth factor receptor [START_REF] Miura | Congenital hand anomalies, and their association with other congenital abnormalities[END_REF][START_REF] Stanescu | Pathogenic mechanisms in osteochondrodysplasia[END_REF][START_REF] Hall | Intracellular and extracellular control of the differentiation of cartilage and bone[END_REF][START_REF] Matsui | Skeletal develpment of achondroplasia[END_REF]. This anomaly of the cartilage is responsible for a skeletal dysplasia. At birth, short stature and squared pelvis are not prominent [START_REF] Matsui | Skeletal develpment of achondroplasia[END_REF]. The deformity increases with age [START_REF] Matsui | Skeletal develpment of achondroplasia[END_REF] and is not limited to the cartilage but also to the bone, the muscles and the peripheral nerves. To speak in more general terms, when there is bone malformation, muscles and tendons do not develop normally; their development is linked to that of the bones, probably due to changes of constraint on muscles and tendons.

By contrast, muscular dystrophies appear mostly limited to the muscle, neurodegenerative disease to the nerves [START_REF] Roberts | Orthopedic aspects of neuromuscular disease in children[END_REF].

GROWTH PLATE ARREST STOPS THE GROWTH OF THE HAND

The use of rigid fixation for fracture of the extremity is common place [START_REF] Opperman | A pilot study of short and long term sequelae to rigid fixation across metaphyseal physes in a baboon model[END_REF]. Epiphyses plated for a year show increased bone differentiation, premature closure and growth arrest [START_REF] Opperman | A pilot study of short and long term sequelae to rigid fixation across metaphyseal physes in a baboon model[END_REF]. They also have shortened muscles, tendons and nerves. In young cancer patients, growth plate injury often results from surgery or radiation therapy. This translates into marked deformity, sclerotic metaphysal bands, muscular atrophy and growth arrest [START_REF] Roebuck | Skeletal complications in paediatric oncology[END_REF]. Here again, the muscles, tendons and nerves are shortened. Similar treatments delivered sparing the growth plate have markedly fewer side effects [START_REF] Roebuck | Skeletal complications in paediatric oncology[END_REF].

It is at puberty that growth plates are fused and growth stops. This may be due to a direct hormonal effect. But the normal growth plates are under increased pressure; the largest load comes from muscle [START_REF] Schiessl | Estrogen and bone-muscle strength and mass relationships[END_REF]. The muscle load increases around puberty. The most intense force development occurs between 12 and 15 years of age in boys, and earlier in girls [START_REF] Backman | Methods for measurement of muscle function. Methodological aspects, reference values for children, and clinical applications[END_REF]. Girls stop growing earlier than boys do. Treatment with testosterone of young boys prior to puberty increases muscle mass, induces puberty and induces growth cessation [START_REF] Soliman | Testosterone treatment in adolescent boys with constitutional delay of growth[END_REF]. Similarly, heavily trained gymnasts [START_REF] Caine | Does gymnastics training inhibit growth of females[END_REF] or swimmers [START_REF] Wawryniak | Biological age in children who practise swimming[END_REF] experience attenuated growth during their years of training followed by catch-up growth during reduced training schedule or the months following retirement.

GROWTH PLATE GROWTH MAY BE ENOUGH TO EXPLAIN THE EXPANSION OF THE HAND

Growth of long bones occurs at the growth plate, a cartilage structure that contains three main layers: the resting, proliferative and hypertrophied zones. Growth of long bones occurs at the growth plate, a layer of cartilage that separates the epiphysis from the metaphysis. Growth plates exhibit spatial polarity. Proliferative chondrocytes undergo terminal differentiation when they approach the metaphyseal, but not the epiphyseal, border of the growth plate [START_REF] Abad | Determinants of spatial polarity in the growth plate[END_REF]. By contrast, the elongation of the muscles, the tendons, the skin or the nerves appears secondary to bone growth.

Muscle is highly responsive to changes in functional demands. Overload leads to hypertrophy, while decreased load force generation and immobilization with the muscle in the shortened position leads to atrophy [START_REF] Goldspink | Changes in muscle mass and phenotype and the expression of autocrine and systemic growth factors by muscle in response to stretch and overload[END_REF].

Human tendon is elongated by the application of load at its extremity [START_REF] Maganaris | Load-elongation characteristics of in vivo human tendon and aponeurosis[END_REF]. Mechanical tension increases the number of fibroblasts [START_REF] Breen | Calcyclin gene expression is increased by mechanical strain in fibroblasts[END_REF].

Possessing viscous and elastic rheological properties, skin is viscoelastic. Mechanical creep, defined as the elongation of skin with a constant load over time beyond intrinsic extensibility, has been described as the vehicle harnessed for wound closure with intraoperative tissue expansion [START_REF] Wilhelmi | Creep vs. stretch: a review of the viscoelastic properties of skin[END_REF]. Prior to a difficult surgery, because of limited skin for wound closure, a skin-stretching device (an inflattable ballon) is inserted under the skin. After incremental traction the skin is elongated, the wound can be closed. The generation of new results from chronic stretching force confirms that biological creep plays a role in skin growth. Similarly, elongation of peripheral nerves increases the length of the nerves [START_REF] Hall | Peripheral nerve elongation with tissue expansion techniques[END_REF].

MECHANICAL REGULATION OF CELL PROLIFERATION

The development of an organ is constrained by internal and external limitations [START_REF] Davies | Mechanical stress mechanisms and the cell[END_REF]. The effect of shear stress on the endothelial cell is the best-studied example. Shear stress is responsible for vascular network formation, the fractal organization of the arterial and venous trees, as well as the unavoidable tropism of arteries towards capillaries and then veins [START_REF] Fleury | from shear stress as a simple model of vasculogenesis[END_REF]. At the cellular level, shear stress induces cell proliferation. The endothelium, however, is not unique in responding to external forces; virtually all cells are affected by the mechanical environment [START_REF] Davies | Mechanical stress mechanisms and the cell[END_REF]. For example, mechanical deformation of fetal rat lung cells simulating fetal respiratory movements increases cellular replication [START_REF] Liu | Inhibition of mechanical strain-induced fetal rat lung cell proliferation by gadolinium, a stretch-activated channel blocker[END_REF]. Rhythmic deformation also increases intestinal epithelial cell proliferation in a frequency dependent manner [START_REF] Murnin | Effects of glutamine isomers on human (Caco-2) intestinal epithelial proliferation, strain responsiveness and deformation[END_REF].

Proliferation is the acknowledged 'default state' of prokaryotes, unicellular eucaryotes or plant cells. The same is found in normal human cells [START_REF] Sonnenschein | Somatic mutation theory of carcinogenesis. Why it should be dropped and replaced?[END_REF]. Both, in vivo and in vitro, normal cells stop proliferating because of contact inhibition caused, in part, by mechanical force [START_REF] Fleury | Numerical investigation of loss of cellular polarity on squamous cell carcinoma invasivness and geometry[END_REF]. It is possible that cell proliferation is limited to the cartilage because of lesser physical constraints compared to the bone.

OSSIFICATION APPEARS TO RESULT FROM GROWTH PLATE GROWTH

In 1911, Gebhardt [START_REF] Wong | Theoretical stress analysis of organ culture osteogenesis[END_REF][START_REF] Tanck | Why does intermittent hydrostatic pressure enhance the mineralization process in fetal cartilage?[END_REF] suggested that the ossification of the chondroepiphysis started at a point where the accumulation of stress was the greatest. The ossification center of the epiphysis is formed by hypertrophied chondrocytes. In a short period of time, calcium phosphate is deposited in the matrix around these cells [START_REF] Tanck | Why does intermittent hydrostatic pressure enhance the mineralization process in fetal cartilage?[END_REF]. In vitro, mouse metatarsal are exposed to external load [START_REF] Tanck | Why does intermittent hydrostatic pressure enhance the mineralization process in fetal cartilage?[END_REF]. Intermittent ambient hydrostatic pressure increases the calcification [START_REF] Wong | Theoretical stress analysis of organ culture osteogenesis[END_REF].There is accelerated osteogenesis in the area of intermittent high shear [START_REF] Wong | Theoretical stress analysis of organ culture osteogenesis[END_REF][START_REF] Burger | Modulation of osteogenesis in fetal bone rudiments by mechanical stress in vitro[END_REF].

Recently, Maitournam et al. [START_REF] Maitournam | Shear stress and ossification[END_REF] has reviewed a data bank of X-ray pictures of the hand of children of various ages. Using a finite-element analysis he has shown that the localization and the pattern of growth of ossification takes place in areas of high hydrostatic pressure and shear stress.

The fact that the combination of shear stress and pressure is correlated to mineralization appears to be a general phenomenon. The ossification of the vertebral body starts at its center, where the shear stress is maximum. The early ossification of the diaphysis of the femur takes place at a time when the joints are flatter [START_REF] Maitournam | Shear stress and ossification[END_REF][START_REF] Shi-Jun | Morphometric study on development of fetal hip[END_REF] again in an area of high shear. The cartilage which does not ossify (joint, nose, larynx, ear, bronchus, etc.) is not exposed to high shear stress.

MECHANICAL STRESS AND CELLULAR DIFFERENTIATION

Mesenchymal stem cells are multipotent cells that can be induced to differentiate into a variety of mesenchymal tissues, including bone, cartilage, tendon, fat, bone marrow stroma and muscle [START_REF] Ubelaker | Estimating age of death from immature human skeletons: an overview[END_REF]. Several mesenchymal cells (mechanocytes) e.g. osteoblasts and fibroblasts as well as muscle cells are activated by mechanical strain [START_REF] Caterson | Application of mesenchymal stem cells in the regeneration of musculoskeletal tissue[END_REF]. In the past two decades, it has been well established that many cells are sensitive to mechanical forces and can change their phenotype and surrounding extracellular matrix (ECM) in response to mechanical environment [START_REF] Swartz | Mechanical stress is communicated between different celle types to elicit matrix remodelling[END_REF]. Traction appears to generate condensation and maturation of chondrocytes or feather, scale and hair formation [START_REF] Fleury | from shear stress as a simple model of vasculogenesis[END_REF][START_REF] Liu | Inhibition of mechanical strain-induced fetal rat lung cell proliferation by gadolinium, a stretch-activated channel blocker[END_REF][START_REF] Murnin | Effects of glutamine isomers on human (Caco-2) intestinal epithelial proliferation, strain responsiveness and deformation[END_REF](40)[START_REF] Oster | A model for chondrogenic condensations in the developing limb: the role of extracellular matrix and cell tractions[END_REF][START_REF] Googdman | Effects of mechanical stimulation on the differentiation of hard tissue[END_REF][START_REF] Carter | Mechanobiology of skeletal regeneration[END_REF].

Chondrocytes are known to sense and respond to the mechanical stimuli by multiple regulatory pathways: upstream signaling transcription, translation, posttranslational modifications and vesicular transport [START_REF] Franck | Tissue shear deformation stimulates proteoglycan and protein biosynthesis in bovine cartilage explants[END_REF]. Fluid induced shear causes chondrocytes to elongate and align [START_REF] Smith | Effects of fluid induced shear stress on articular chondrocytes morphology and metabolism in vitro[END_REF]. Chondrocytes respond to shear stress by an increased secretion of extra cellular matrix namely collagen and proteoglycan [START_REF] Franck | Tissue shear deformation stimulates proteoglycan and protein biosynthesis in bovine cartilage explants[END_REF], as well as the modification of metabolism [START_REF] Smith | Effects of fluid induced shear stress on articular chondrocytes morphology and metabolism in vitro[END_REF]. But primary adult bone cells do not appear to respond to fluid-flow induced shear stress in these physiological ranges [START_REF] Brand | Primary adult bone cells do not respond to tissue (continuum) level strains[END_REF].

External load also plays a critical role in determining muscle mass and its phenotype in myocytes [START_REF] Sadoshima | The cellular and molecular response of cardiac myocytes to mechanical stress[END_REF]. Myocytes have the ability to sense mechanical stretch and convert it into intracellular growth signals, which lead to hypertrophy. Stretch is, by itself, an important mechanical signal for the production of more actin and myosin filaments and the addition of new sarcomeres. This is preceded by upregulation of transcription of the appropriate genes, some of which, such as the myosin isoforms, markedly change muscle phenotype. Indeed, the switch in the expression induced by mechanical activity of myosin heavy chain genes, which encode different molecular motors, is a means via which the tissue adapts to a given type of physical activity [START_REF] Wilhelmi | Creep vs. stretch: a review of the viscoelastic properties of skin[END_REF]. Mechanical stretch of myocytes in vitro causes activation of multiple second messenger systems that are very similar to growth factor-induced cell signaling systems.

Similarly, traction induces the secretion of extracellular matrix by fibroblasts, distorts collagen gels and creat patterns similar to tendon [START_REF] Harris | Fibroblast traction as a mechanism for collagen morphogenesis[END_REF]. This morphogenetic rearrangement of extracellular matrix is the primary function of fibroblast traction and explains its excessive strength [START_REF] Harris | Fibroblast traction as a mechanism for collagen morphogenesis[END_REF].

Stem cells may be expanded in culture and subsequently permitted to differentiate into the desired lineage. This directed differentiation might be reached by the application of bioactive molecules, growth factors and signaling molecules [START_REF] Caterson | Application of mesenchymal stem cells in the regeneration of musculoskeletal tissue[END_REF]. It is known that physical stress induces the secretion of these growth factors and signaling molecules [START_REF] Fleury | from shear stress as a simple model of vasculogenesis[END_REF]. The question is whether mechanics alone during normal development are sufficient to explain the growth and differentiation of the tissues. If this hypothesis is proven correct, the next question is whether the effects of mechanical forces are being mediated or mimicked through the release of chemicals.

Adult pathology can suggest the biological action mechanism of shear stress.

PATHOLOGIC CALCIFICATION AND OSSIFICATION CAN BE INDUCED BY HYPOXIA AND ALKALOSIS

Circumstantial evidence suggests that hypoxia and/or alkalosis also play a key role in pathologic calcification and ossification.

Cardiac ligature, subcutaneous implantation of glass diaphragms or exposure of a transplanted tendon to anoxia result in a transient chondrogenesis followed by enchondral ossification [START_REF] Danis | Soft tissue ossification: mechanism[END_REF].

Similarly, Busher et al. [START_REF] Buscher | Heterotopic ossification induced by hypoxia in a retrosternal gastric tube following transhiatal Oesophagectomy[END_REF] report the case of a patient who underwent gastric tube reconstruction. Following cardiac arrhythmia, he developed hypoxia and ischaemic necrosis over 5 cm of the proximal gastric tube. Three weeks later that area was ossified; trabecular bone was present along the entire length of the constricted gastric tube.

Ectopic ossification has been associated with several conditions in both neoplastic and not neoplastic tissue [START_REF] Buscher | Heterotopic ossification induced by hypoxia in a retrosternal gastric tube following transhiatal Oesophagectomy[END_REF][START_REF] Ackerman | Bone formation in the gastric mucosa following subtotal gastrectomy in rats[END_REF][START_REF] An | Heterotopic ossification and pseudoarthrosis in the shoulder following encephalitis[END_REF][START_REF] Asa | Myositis ossificans circumscripta: a complication of tetanus[END_REF]. Paraplegic patients frequently develop ectopic ossification [START_REF] Buscher | Heterotopic ossification induced by hypoxia in a retrosternal gastric tube following transhiatal Oesophagectomy[END_REF][START_REF] Izumi | Study of ectopic bone formation in experimental spinal cord injured rabbits[END_REF]. But there is no significant difference of ectopic bone formation between paraplegic rabbits and non-paraplegic rabbits under the same immobilization and passive movement of the posterior legs. For Izumi [START_REF] Izumi | Study of ectopic bone formation in experimental spinal cord injured rabbits[END_REF], the reason may simply be poor oxygenation because of blood stasis.

Calcium mineral deposition in the atherosclerotic plaque results, also, from hypoxia. The vascular cell, when exposed to hydrogen peroxide, differentiates into an osteoblastic cell [START_REF] Mody | Oxydative stress modulates differentiation of vascular and bone cells[END_REF] this effect was counteracted by anti oxydants. Oxidative stress modulates the differentiation of bone and vascular cells oppositely [START_REF] Mody | Oxydative stress modulates differentiation of vascular and bone cells[END_REF].

Metabolic acidosis increases calcium efflux from bone and hypercalciuria [START_REF] Bushinsky | Effects of pH on bone calcium and protons flux in vitro[END_REF]. Metabolic acidosis increases osteoposis and osteomalacia, and propensity to develop kidney stones [START_REF] Wiederkehr | Metabolic and endocrine effects of metabolic acidosis in human[END_REF].By contrast, alkalosis neutralize endogenous acid production and improve bone mineral accretion [START_REF] Bushinsky | Effects of pH on bone calcium and protons flux in vitro[END_REF][START_REF] Bushinsky | Metabolic alkalosis decreases bone calcium efflux by suppressing osteoclasts and stimulating osteoblasts[END_REF].

Calciphylaxis is a rapidly developing fatal process of vascular calcium deposition with prominent cutaneous manifestation. Metastatic pulmonary deposition is a complication of renal failure. The pathologic deposition of calcium is favored by alcalosis [START_REF] Matsuo | Acute respiratory failure to ''pulmonary calciphylaxis'' in a maintenance haemodialysis patient[END_REF][START_REF] Bloodworth | Localised pulmonary metastatic calcification associated with pulmonary artery obstruction[END_REF].

SHEAR STRESS AND PRESSURE MODIFY THE EXTRACELLULAR MATRIX AND CHANGE THE AVAILABILITY OF NUTRIENTS AND OXYGEN

How shear stress and pressure may induce cell differentiation is a matter of speculation. It may be a direct effect, the release of a differentiation factor (61) or decreased availability of oxygen because of limited diffusion due to increased extracellular matrix.

Collagen is a vital component of the extracellular matrix of both muscles and tendons [START_REF] Bishop | Regulation of cardiovascular collagen deposition by mechanical forces[END_REF]. It acts as a scaffold to maintain muscle shape and permit an even distribution of force, and plays a crucial role in the mechanical properties of the tendons. Under normal circumstances, collagen is continually being synthesized and degraded throughout life. Increased mechanical stress, which causes muscle hypertrophy stimulates collagen synthesis.The concentration of other extracellular molecules (laminin, fibronectin, vitronectin, etc.) is also modulated by mechanical stress [START_REF] Loeser | Chondrocyte integrin expression and function[END_REF].

Mechanical compression and tension generated by the condensing mesenchyme in the limb-bud [START_REF] Wilson | Development of avascularity during cartilage differentiation in the embryonic limb[END_REF] appears to be enough to constrict and close off the thinwalled undifferentiated blood vessels caught in the condensation foci [START_REF] Wilson | Development of avascularity during cartilage differentiation in the embryonic limb[END_REF], thus leading to avascular areas. The differentiation of muscle and cartilage has been interpreted in terms of vascular pre-pattern. Extracellular matrix and/or low oxygen tension differentiates stem cells into chondrocytes [START_REF] Der | Cartilage and cell differentiation:a review[END_REF]. Cartilage is practically avascular (nutrients are transported from the synovial fluid and vascularized subchondral bone) [START_REF] Murphy | Effect of oxygen tension on chondrocyte extracellular matrix accumulation[END_REF]. Oxygen tensions within cartilage are therefore significantly lower than vascularized tissues ranging from 2.7 to 7.5% oxygen [START_REF] Murphy | Effect of oxygen tension on chondrocyte extracellular matrix accumulation[END_REF][START_REF] Brighton | Oxygen tension in zone of the epiphyseal plate the metaphysis and the diaphysis[END_REF][START_REF] Ellis | EPR oxygen mapping of engineered cartilage in reactor grown in a hollow-fiber bioreactor[END_REF]. In Vivo, pregnant mice are exposed to hypobaric oxygen. The embryos suffer from major scoliosis. The malformations are present in the cartilaginous stage of development of the vertebral column [START_REF] Rivard | Moderate hypobaric hypoxia used as an inducer of congenital vertebral malformation in mouse's embryo[END_REF].

The effects of oxygen on cell differentiation are confirmed in vitro. Oxidative stress regulates cell function and proliferation [START_REF] Butler | Induction of the proliferative phenotype in differentiated myogenic cells by hypoxia[END_REF]. Low oxygen tension promotes and induces redifferentiation of dedifferentiated bovine articular chondrocytes [START_REF] Domm | Redifferentiation of dedifferentiated bovine articular chondrocytes in alginate culture under low oxygen tension[END_REF].

HYPOTHESIS: SHEAR STRESS INDUCES ALKALOSIS WHICH IS KNOWN TO INCREASE CALCIUM DEPOSITION

Stylophora pistillata is a scleractinian coral. Its calcification is a function of mechanisms which concentrate the CO 2 in coral cells [START_REF] Furla | Sources and mechanisms of inorganic carbon transport for coral calcification and photosynthesis[END_REF]. Pancreatites stones, a frequent complication of chronic pancreatitis are made of calcite(CaCO 3 ).The precipitation of CaCO 3 is a function of the pH and the availability of bicarbonates [START_REF] Moore | Pancreatic calcification:formation constants of CaHCO 3þ and CaCO3 (0) complexes determined with Ca 2þ electrodes[END_REF]. Similarly, extracellular alkalosis increases the calcium precipitation at the periphery of pancreatic B cells [START_REF] Rebolledo | Effect of extracellular alkalosis upon calcium distribution within B cells[END_REF].

The pH surrounding the osteoclast is highly acidic [START_REF] Baron | Molecular mechanisms of bone resorption by the osteoclast[END_REF]. The apical bone-resorbing compartment of the osteoclast is sealed off by the attachment of the osteoclast to the calcified matrix and is actively acidified by the osteoclast. In the low pH environment of the bone-resorbing lacuna produced by the osteoclast, the mineral phase dissolves, exposing the organic matrix to the action of the secreted enzymes. These observations are consistent with a scheme in which, in the low pH environment of the bone-resorbing lacuna produced by the osteoclast, the mineral phase dissolves, exposing the organic matrix to the action of the secreted enzymes. The activity of these enzymes is in turn presumably favored by the acidic milieu. All constituents of the matrix, whether mineral or organic, then would be reduced to their elemental forms (ions and amino acids) extracellularly.

The secretion of acid by proton pumps involves cation-independent mannose-6-phosphate receptors. These receptors bind to an enzyme-linked mannose-6phosphate [START_REF] Baron | Molecular mechanisms of bone resorption by the osteoclast[END_REF]. There is 'in vitro' evidence of the importance of pH in bone formation and resorpsion. Metabolic acidosis increases urinary calcium excretion [START_REF] Bushinsky | Metabolic alkalosis decreases bone calcium efflux by suppressing osteoclasts and stimulating osteoblasts[END_REF]. By contrast, alkalosis neutralizes endogenous acid production and improves bone mineral accretion [START_REF] Bushinsky | Metabolic alkalosis decreases bone calcium efflux by suppressing osteoclasts and stimulating osteoblasts[END_REF]. Alkalosis causes a decrease in osteoclastic b-glucuronidase release, an increase in osteoblastic collagen synthesis and calcium deposition on the bone cells [START_REF] Carano | Acid and base effects on avian osteoclast activity[END_REF].

The link between hypoxia, alkalosis and calcium precipitation is complex and beyond the reach of this paper. Chemical hypoxia induces hypocapnia alkalosis in primary culture [START_REF] Jorgensen | Increase in Ca 2þ and changes in intracellular pH during chemical anoxia in mouse neocortical neurons in primary culture[END_REF]. Similarly, ischaemia and hypoxia may induce alterations of ion homeostasis, including alkalosis [START_REF] Katsura | Coupling among changes in energy metabolism, acid-base homeostasis and ion fluxes in ischaemia[END_REF]. Hexokinase transforms mannose into mannose-6-phosphate. This key enzyme is known to be regulated by hypoxia [START_REF] Yoon | Identification of genes differentially induced by hypoxia in pancreatic cancer cells[END_REF][START_REF] Riddle | Hypoxia induces hexokinase II gene expression in human lung cell line A549[END_REF][START_REF] Niitsu | Exposure of cultured primary rat astrocytes to hypoxia results in intracellular glucose depletion and induction of glycolytic enzymes[END_REF].

CONCLUSION

Currently, the predominant hypothesis explains cellular differentiation as an essentially genetic intracellular process. However, it is possible or even probable, that this differentiation is the result of simple biochemical and physical events of extracellular events origin.

The loss of cellular polarity is enough to explain the cancer shape, the shift from the branchial dichotomus structure of normal glands to the invasive dendritic nature of cancer [START_REF] Fleury | Numerical investigation of loss of cellular polarity on squamous cell carcinoma invasivness and geometry[END_REF][START_REF] Schwartz | Cancer, the role of extracellular disease[END_REF].The carcinogenic process is associated with a loss of cellular differentiation [START_REF] Fleury | Numerical investigation of loss of cellular polarity on squamous cell carcinoma invasivness and geometry[END_REF][START_REF] Schwartz | Cancer, the role of extracellular disease[END_REF].This dedifferentiation can be reversed [START_REF] Schwartz | Cancer, the role of extracellular disease[END_REF][START_REF] Wennerberg | Regulation of the differentiation of diploid and aneuploid rat liver epithelial cells by the liver microenvironment[END_REF][START_REF] Illmensee | Totipotency and normal differentiation of single teratocarcinoma cell cloned by injection into blastocysts[END_REF].

It is possible that this dedifferentiation is simply due to a change in the cell's physico-chemical environment. The importance of the extracellular milieu could also explain how a tumor cell reinjected into a healthy blastocyst [START_REF] Illmensee | Totipotency and normal differentiation of single teratocarcinoma cell cloned by injection into blastocysts[END_REF] or tissue [START_REF] Wennerberg | Regulation of the differentiation of diploid and aneuploid rat liver epithelial cells by the liver microenvironment[END_REF], exposed to a normal environment can differentiate normally.

ACKNOWLEDGEMENTS

We want to acknowledge the constant help and advices of J. Leibowitch.


Marc Arnaudon
email: marc.arnaudon@math.univ-poitiers.fr

Jean-Christophe Breton
email: jean-christophe.breton@univ-lr.fr

Nicolas Privault
email: nprivaul@cityu.edu.hk

Convex ordering for random vectors using predictable representation

Keywords: Convex ordering, forward-backward stochastic calculus, deviation inequalities, Brownian motion, jump processes. 2000 MR Subject Classification : 60E15; 60H05, 60G44, 60G55

published or not. The documents may come

Convex ordering for random vectors using predictable representation

Introduction

Given two finite measures µ and ν on R d we say that µ is convex dominated by ν, and we write µ cx ν, if

R d φ(x)µ(dx) ≤ R d φ(x)ν(dx) (1.1)
for all sufficiently integrable convex functions φ : R d → R. In case µ and ν are the respective probability distributions of two random variables F and G, Relation (1.1) variables. We stress however that the arguments of [START_REF] Th | Convex concentration inequalities via forward/backward stochastic calculus[END_REF] are particular to the onedimensional case and in general they can not be applied to the vector valued setting considered in this paper, for which specific methods have to be developed.

Note also that by a classical argument, the application of (1.1) to φ(x) = exp(λ x ), λ > 0, entails the deviation bound

P ( F ≥ x) ≤ inf λ>0 E[e λ( F -x) 1 { F ≥x} ] ≤ inf λ>0 E[e λ( F -x) ] ≤ inf λ>0 E[e λ( G -x) ],
x > 0, hence the deviation probabilities for F can be estimated via the Laplace transform of G .

We will prove the following type of result. Let (W (t)) t∈R + and Z(t) = (Z 1 (t), . . . , Z n (t))

be respectively a standard n-dimensional Brownian motion and a vector of independent real point processes with compensator (λ 1 (t), . . . , λ n (t)) t∈R + generating a filtration F M . Let M d×n denote the set of d×n real matrices, with M d = M d×d . Consider F and G two random variables with the predictable representations

F = ∞ 0 A(t)dW (t) + ∞ 0 J(t)(dZ(t) -λ(t)dt)
where (A(t)) t∈R + , (J(t)) t∈R + are square-integrable M d×n -valued F M t -predictable processes, and

G = ∞ 0 Â(t)d W (t) + ∞ 0 Ĵ (t)(d Z(t) -λ(t)dt)
where ( Â(t)) t∈R + , ( Ĵ (t)) t∈R + are M d×n -valued square-integrable F M t -predictable processes and W (t) and Z(t) = ( Z1 (t), . . . , Zn (t)), t ∈ R + , are a n-dimensional Brownian motion and a vector of real point processes with respective intensities ( λi (t)) t∈R + , i = 1, . . . , n, independent of F M . In terms of the convex orders cxp, cxpi and psd introduced in Definitions 3.3 and 3.2 below, we have for example the following corollary of Theorem 4.2. In the sequel, the † symbol stands for matrix transposition.

Corollary 1.3. The convex concentration inequality (1.1) holds provided A † (t)A(t) ≤ psd  † (t) Â(t), dP dt -a.e.,

and for almost all t ∈ R + , we have either: i) n j=1 λ j (t)δ (J 1,j (t),...,J d,j (t)) cxp n j=1 λj (t)δ ( Ĵ1,j (t),..., Ĵd,j (t)) . or:

ii) J i,j (t) ≥ 0, Ĵi,j (t) ≥ 0, i = 1, . . . , d, j = 1, . . . , n, and n j=1 λ j (t)δ (J 1,j (t),...,J d,j (t)) cxpi n j=1 λj (t)δ ( Ĵ1,j (t),..., Ĵd,j (t)) .

Condition (ii) above will hold in particular if λ i (t) ≤ λi (t), and 0 ≤ J i,j (t) ≤ Ĵi,j (t),

1 ≤ i ≤ n, 1 ≤ j ≤ d, for dt-almost all t ∈ R + .
In Theorem 5.5, we provide a geometric interpretation of the convex ordering condition (i) for finitely supported measures.

In case F and G are Gaussian random vectors with covariance matrices Σ and Σ, we

recover (1.2) from Theorem 1.3 by taking λ(t) = λ(t) = 0, A(t) = 1 [0,T ] (t) Σ/T and Â(t) = 1 [0,T ] (t) Σ/T , t ∈ R + .
Note that related convex comparison results have also been obtained in [START_REF] Bergenthum | Comparison of option prices in semimartingale models[END_REF], [START_REF] Bergenthum | Comparison of semimartingales and Lévy processes[END_REF] for diffusions with jumps, under different hypotheses. Namely, it is assumed therein that G is given by the value at time T of a diffusion with jumps. Convex ordering then holds under similar assumptions on the process characteristics, provided the generator of this diffusion satisfies the propagation of convexity property.

This paper is organized as follows. In Section 2 we introduce the notation of multidimensional forward-backward stochastic calculus with jumps, which will be used in the next sections. In Section 3 we prove some convex ordering results for the sums of forward and backward martingales, and in Section 4 we apply those results to random variables given by their predictable representation in terms of a diffusion and a point process. Section 5 is devoted to a geometric interpretation of convex ordering for discrete measures on R d , which gives a better understanding of the conditions set of the jump heights and intensities of the considered point processes.

Notation

Let (Ω, F , P ) be a probability space equipped with an increasing filtration (F

∆M(t) = M(t) -M(t -), ∆ * M * (t) = M * (t) -M * (t + ),
their forward and backward jumps. The processes (M(t)) t∈R + and (M * (t)) t∈R + have jump measures

µ(dt, dx) = s>0 1 {∆M (s) =0} δ (s,∆M (s)) (dt, dx),
and

µ * (dt, dx) = s>0 1 {∆ * M * (s) =0} δ (s,∆ * M * (s)) (dt,
(M(t n k ) -M(t n k-1 ))(M(t n k ) -M(t n k-1 )) † ,
and

[M * , M * ] t = lim n→∞ n-1 k=0 (M * (t n k ) -M * (t n k+1 ))(M * (t n k ) -M * (t n k+1 )) † , for all refining sequences {0 = t n 0 ≤ t n 1 ≤ • • • ≤ t n n = t}, n ≥ 1, of partitions of [0, t] tending to the identity. We let M J (t) = M(t) -M c (t), M * J (t) = M * (t) -M * c (t), [M J , M J ] t = 0<s≤t ∆M(s)∆M(s) † , [M * J , M * J ] t = 0≤s<t ∆ * M * (s)(∆ * M * (s)) † , with M c , M c t = [M, M] t -[M J , M J ] t ,
and

M * c , M * c t = [M * , M * ] t -[M * J , M * J ] t , t ∈ R + .
Denote by ( M J , M J t ) t∈R + , ( M * J , M * J t ) t∈R + the conditional quadratic variations of (M J (t)) t∈R + and of (M * J (t)) t∈R + , with

d M J , M J t = R d xx † ν(dt, dx) and d M * J , M * J t = R d xx † ν * (dt, dx).
The conditional quadratic variations ( M, M t ) t∈R + , ( M * , M * t ) t∈R + of (M(t)) t∈R + and of (M * (t)) t∈R + satisfy

M, M t = M c , M c t + M J , M J t ,
+ d i=1 t s + ∂f ∂x i (M(u -), M * (u))dM i (u) + 1 2 d i,j=1 t s ∂ 2 f ∂x i ∂x j (M(u), M * (u))d M c i , M c j u + s<u≤t f (M(u), M * (u)) -f (M(u -), M * (u)) - d i=1 ∆M i (u) ∂f ∂x i (M(u -), M * (u)) - d i=1 t - s ∂f ∂y i (M(u), M * (u + ))d * M * i (u) - 1 2 d i,j=1 t s ∂ 2 f ∂y i ∂y j (M(u), M * (u))d M * c i , M * c j u - s≤u<t f (M(u), M * (u)) -f (M(u), M * (u + )) - d i=1 ∆ * M * i (u) ∂f ∂y i (M(u), M * (u + )) , 0 ≤ s ≤ t,
(M i (t n k ) -M i (t n k-1 )) ∂f ∂x i (M(t n k-1 ), M * (t n k-1 )) and n-1 k=0 (M * i (t n k ) -M * i (t n k+1 )) ∂f ∂y i (M(t n k+1 ), M * (t n k+1 )) for all refining sequences {s = t n 0 ≤ t n 1 ≤ • • • ≤ t n n = t}, n ≥ 1, of partitions of [s, t] tending to the identity.
Here,

t 0 η(u)dM i (u), resp. ∞ t η * (u)dM i (u)
, refer to the right, resp. left, continuous version of the indefinite stochastic integrals of the forward, resp. backward, adapted and sufficiently integrable processes (η(u)) u∈R + , resp. (η * (u)) u∈R + .

Convex ordering for martingales

We denote by •, • and • the usual Euclidean scalar product and norm on R d . Let M d be the space of real matrices with the scalar product

A, B := Tr (AB † ) = d i,j=1 A i,j B i,j , A, B ∈ M d ,
where we recall that A † stands for the transpose (A j,i ) 1≤i,j≤d of A = (A i,j ) 1≤i,j≤d , and let M d + be the subset of M d made of positive semidefinite matrices. x, Ax = Tr (AB † ) ≥ 0.

Definition 3.2. Given A, B ∈ M d , we will write A ≤ psd B if B -A is positive semidefinite, i.e. x, Ax ≤ x, Bx , x ∈ R d .
In the sequel, a function f : R d → R will be said to be non-decreasing if

f (x 1 , . . . , x d ) ≤ f (y 1 , . . . , y d ) for all x 1 , . . . , x d ∈ R and y 1 , . . . , y d ∈ R such that x i ≤ y i , i = 1, . . . , d.
In the sequel, we will need the following orders between positive measures µ, ν on R d .

Definition 3.3. i) We say that µ cxp ν if R d φ(x)µ(dx) ≤ R d φ(x)ν(dx)
for all non-negative convex functions φ :

R d → R + . ii) We say that µ cxpi ν if R d φ(x)µ(dx) ≤ R d φ(x)ν(dx)
for all non-negative and non-decreasing convex functions φ : R d → R + .

If µ and ν are finite measures on R d , then both µ cxp ν and µ cxpi ν imply

µ(R d ) ≤ ν(R d ).
More precisely we have the following result.

Proposition 3.4. Assume that µ and ν are finite measures on R d . Then

µ cx ν is equivalent to µ cxp ν and µ(R d ) = ν(R d ).
Proof. Assume that µ cxp ν and µ(R d ) = ν(R d ), and let φ ∈ L 1 (µ) L 1 (ν). For all a ∈ R we have

R d φ(x)µ(dx) - {φ<a} φ(x)µ(dx) + aµ({φ < a}) = {φ≥a} (φ(x) -a) + µ(dx) + aµ(R d ) ≤ {φ≥a} (φ(x) -a) + ν(dx) + aν(R d ) = R d φ(x)ν(dx) - {φ<a} φ(x)ν(dx) + aν({φ < a}),
and for a ≤ 0,

{φ<a} φ(x)µ(dx) ≤ aµ({φ < a}) ≤ 0, {φ<a} φ(x)ν(dx) ≤ aν({φ < a}) ≤ 0, hence letting a tend to -∞ yields R d φ(x)µ(dx) ≤ R d φ(x)ν(dx).
Conversely we note that µ cxp ν clearly implies µ cx ν, and we recover the identity

µ(R d ) = ν(R d
) by applying the property µ cx ν successively with φ = 1 and φ = -1.

Consequently, µ cxp ν implies

R d x i µ(dx) ≤ R d
x i ν(dx), and -

R d x i µ(dx) ≤ - R d x i ν(dx), i = 1, . . . , d,
i.e. µ and ν have same barycenter, provided µ(R d ) = ν(R d ) and µ, ν are integrable.

This also holds when Then we have

µ cxpi ν, µ(R d ) = ν(R d )
E R d ×R + x ν t (dx)dt < ∞, E R d ×R + x ν * t (dx)dt < ∞. ( 3
E[φ(M(s) + M * (s))] ≥ E[φ(M(t) + M * (t))], 0 ≤ s ≤ t, (3.9)
for all convex functions φ : R d → R.

Proof. We start by assuming that φ is a C 2 , convex Lipschitz function and we apply Itô's formula (2.1) for forward-backward martingales to f (x, y) = φ(x + y). Taking expectations on both sides of Itô's formula we get

E[φ(M(t) + M * (t))] = E[φ(M(s) + M * (s))] + 1 2 d i,j=1 E t s φ ′′ i,j (M(u) + M * (u))d( M c i , M c j u -M * c i , M * c j u ) 10 + E t s R d (φ(M(u) + M * (u) + x) -φ(M(u) + M * (u)) -x, ∇φ(M(u) + M * (u)) )ν u (dx)du -E t s R d (φ(M(u) + M * (u) + x) -φ(M(u) + M * (u)) -x, ∇φ(M(u) + M * (u)) )ν * u (dx)du = E[φ(M(s) + M * (s))] + 1 2 E t s ∇ 2 φ(M(u) + M * (u)), H(u) -H * (u) du (3.10) + E t s R d Ψ(x, M(u) + M * (u))(ν u (dx) -ν * u (dx))du , where Ψ(x, y) = φ(x + y) -φ(y) - d i=1 x i ∂φ ∂y i (y), x, y ∈ R d .
Due to the convexity of φ, the Hessian ∇ 2 φ is positive semidefinite hence Lemma 3.1 yields

E[φ(M(t) + M * (t))] ≤ E[φ(M(s) + M * (s))] (3.11) +E t s R d Ψ(x, M(u) + M * (u))(ν u (dx) -ν * u (dx))du , since H * (u) -H(u) is positive semidefinite for fixed (ω, u) ∈ Ω × R + .
Finally we examine the consequences of hypotheses (i) and (ii) on (3.11).

i) By convexity of φ, x → Ψ(x, y) is non-negative and convex on R d for all fixed y ∈ R d , hence the second term in (3.11) is non-positive.

ii) When ν u and ν * u are supported by R d + , (3.11) is also non-positive since for all y, x → Ψ(x, y) is non-decreasing in x ∈ R d + .

The extension to convex non C 2 functions φ follows by approximation of φ by an increasing sequence of C 2 convex Lipschitz functions, and by application of the monotone convergence theorem.

Remark 3.12. When φ ∈ C 2 , the hypothesis on the diffusion part and on the jump part can be mixed together. Indeed, in order for the conclusion of Theorem 3.8 to hold it suffices that

Tr (∇ 2 φ(y)H t ) + R d Tr (∇ 2 φ(y + τ x)xx t )ν t (dx) ≤ Tr (∇ 2 φ(y)H * t ) + R d Tr (∇ 2 φ(y + τ x)xx t )ν * t (dx), (3.13) y ∈ R d , 1 [0,1] (τ )dτ dt-a.e.
Proof. Using the following version of Taylor's formula

φ(y + x) = φ(y) + d i=1 x i φ ′ i (y) + 1 0 (1 -τ ) d i,j=1 x i x j φ ′′ i,j (y + τ x)dτ, x, y ∈ R d , we have Ψ(x, y) = 1 0 (1 -τ ) ∇ 2 φ(y + τ x)x, x dτ = 1 0 (1 -τ )Tr (∇ 2 φ(y + τ x)xx † )dτ
and (3.10) rewrites as

E[φ(M t + M * t )] -E[φ(M s + M * s )] = 1 2 E t s (Tr (∇ 2 φ(M u + M * u )H u ) -Tr (∇ 2 φ(M u + M * u )H * u ))du +E 1 0 t s R d (1 -τ )Tr (∇ 2 φ(M u + M * u + τ x)xx t )(ν u (dx) -ν * u (dx))dudτ
which is non-positive from (3.13).

Let now (F M t ) t∈R + and (F M * t ) t∈R + , denote the forward and backward filtrations generated by (M(t)) t∈R + and by (M * (t)) t∈R + . The proof of the following corollary of Theorem 3.8 is identical to that of Corollary 3.7 in [START_REF] Th | Convex concentration inequalities via forward/backward stochastic calculus[END_REF].

Corollary 3.14. If (3.9) holds and if in addition

E[M * (t) | F M t ] = 0, t ∈ R + , then E[φ(M(s) + M * (s))] ≥ E[φ(M(t))], 0 ≤ s ≤ t.
In particular, if

M 0 = E[M(t)] is deterministic (or if F M 0 is the trivial σ-field), Corol- lary 3.14 shows that M(t) -E[M(t)] is more convex concentrated than M * 0 , i.e.: E[φ(M(t) -E[M(t)])] ≤ E[φ(M * 0 )], t ≥ 0,
for all sufficiently integrable convex functions φ on R d . In applications to convex concentration inequalities the independence of (M(t)) t∈R + with (M * (t)) t∈R + will not be required, see Section 4.

Note that in case ν * (dt, dx) has the form

ν * (dt, dx) = λ * (t)δ k (dx)dt,
where k ∈ R d and (λ * (t)) t∈R + is a positive F * t -predictable process, then condition (i) (resp. (ii)) of Theorem 3.8 is equivalent to: We can take

ν t = λ(t)δ k and λ(t) ≤ λ * (t) resp. to: k ∈ (R + ) d , ν t (R d ) ≤ λ * (t) and ν t R d \ d i=1 ] -∞, k i ] = 0, i.e.
M(t) = M 0 + t 0 A(s)dW (s) + t 0 J(s)(dZ(s) -λ(s)ds), t ∈ R + , and
M * (t) = +∞ t A * (s)d * W * (s) + +∞ t J * (s)(d * Z * (s) -λ * (s)ds), t ∈ R + ,
where (A(t)) t∈R + , (J(t)) t∈R + , resp. (A * (t)) t∈R + , (J * (t)) t∈R + are M d×n -valued and predictable with respect to

F M t := σ(W (s), Z(s) : s ≤ t), resp. F M * t := σ(W * (s), Z * (s) : s ≥ t), t ∈ R + , i.e. M i (t) = M i (0) + n j=1 t 0 A i,j (s)dW j (s) + n j=1 t 0 J i,j (s)(dZ j (s) -λ j (s)ds)
and More details will be given in Section 5 on the meaning of conditions (i) and (ii) of Theorem 3.8 imposed on ν t and ν * t defined in (3.15) and (3.16) for the order cxp .

M * i (t) = M * i (0) + n j=1 ∞ t A i,j (s)dW * j (s) + n j=1 ∞ t J i,j (s)(d * Z * j (s) -λ j (s)ds), t ∈ R + , i = 1, . . . ,
Conditions (3.5) (3.6) will hold in particular when

F t = F M t ∨ F M * 0 and F * t = F M ∞ ∨ F M * t , t ∈ R + , see Section 4.
→ Bt (x) two R d -valued F M t -predictable pro- cesses in L 1 (Ω × R d × R + , dP ν t (dx)dt).
Theorem 4.2. Let ( W (t)) t∈R + be an n-dimensional Brownian motion and μ(dx, dt) be a jump measure with jump characteristic of the form ν(dt, dx) = νt (dx)dt, both independent of F M , and consider

F = ∞ 0 A(t)dW (t) + ∞ 0 R n B t (x)(µ(dt, dx) -ν t (dx)dt)
and

G = ∞ 0 Â(t)d W (t) + ∞ 0 R n Bt (x)(μ(dt, dx) -νt (dx)dt).
Assume that

A † (t)A(t) ≤ psd  † (t) Â(t), dP dt -a.e.,
and that for almost all t ∈ R + , we have either:

i) ν t • B -1 t cxp νt • B-1 t , P -a.s., or :
ii) B t and Bt are non-negative and

ν t • B -1 t cxpi νt • B-1 t , P -a.s.
Then for all convex functions φ : R d → R we have

E[φ(F )] ≤ E[φ(G)]. (4.3)
Proof. Again we start by assuming that φ is a Lipschitz convex function. Let (M(t)) t∈R + denote the forward martingale defined as

M(t) = t 0 A(s)dW (s) + t 0 R n B s (x)(µ(ds, dx) -ν s (dx)ds), t ∈ R + , let (F M t )
t∈R + denote the backward filtration generated by { W (t), μ(dx, dt)}, and let

F t = F M t ∨ F M ∞ and F * t = F M ∞ ∨ F M t ,
so that (M(t)) t∈R + is an F t -forward martingale. Since ( Â(t)) t∈R + , ( Bt ) t∈R + are F M tpredictable, the processes ( Â(t)) t∈R + and ( Bt ) t∈R + are independent of ( Wt ) t∈R + and of μ(dt, dx). In this case, the forward and backward differentials coincide and the process (M * (t)) t∈R + defined as

M * (t) = E[G | F * t ] = ∞ t Â(s)d W (s) + ∞ t R n Bs (x)(μ(ds, dx) -νs (dx)ds), t ∈ R + , is an F * t -backward martingale with M * (0) = G. Moreover the jump charac- teristics of (M t ) t∈R + and of (M * t ) t∈R + are ν M (dx) = 1 R d \{0} (x)ν t • B -1 t (dx) and ν M * (dx) = 1 R d \{0} (x)ν t • B-1 t (dx).
Applying Theorem 3.8 to the forward and backward martingales (M(t)) t∈R + and

(M * (t)) t∈R + yields E[φ(M(t) + M * (t))] ≤ E[φ(M(s) + M * (s))], 0 ≤ s ≤ t,
for all convex functions φ and for 0 ≤ s ≤ t. Since M(0) = 0, M * (0) = G and lim t→∞ M * (t) = 0 in L 2 (Ω), we obtain (4.3) for convex Lipschitz function φ by taking s = 0 and letting t go to infinity. Finally we extend the formula to all convex integrable functions φ by considering an increasing sequence of Lipschitz convex functions φ n converging pointwise to φ. Applying the monotone convergence theorem to the nonnegative sequence φ n (F ) -φ 0 (F ), we have

E[φ(F ) -φ 0 (F )] = lim n→∞ E[φ n (F ) -φ 0 (F )],
which yields E[φ(F )] = lim n→∞ E[φ n (F )] since φ 0 (F ) is integrable. We proceed similarly for φ(G), allowing us to extend (4.3) to the general case.

Note that if ( Â(t)) t∈R + and ( Bt ) t∈R + are deterministic then ( W (t)) t∈R + and μ(dx, dt) can be taken equal to (W (t)) t∈R + and µ(dx, dt) respectively.

Example: point processes

Let (A(t)) t∈R + , ( Â(t)) t∈R + , (W (t)) t∈R + and ( W (t)) t∈R + be as in Theorem 4.2 above and consider

Z(t) = (Z 1 (t), . . . , Z n (t)) and Z(t) = ( Z1 (t), . . . , Zn (t))
to be two independent point processes in R n with compensators

n i=1 λ i (t)δ e i and n i=1 λi (t)δ e i ,
where e i = (0, . . . , 1, . . . , 0), i = 1, . . . , n, denotes the canonical basis in R n , and let

F M t = σ(W (s), Z(s) : 0 ≤ s ≤ t), t ∈ R + . Corollary 4.4. Given J(t) and Ĵ(t) two M d×n -valued integrable F M t -predictable pro- cesses, let F = ∞ 0 A(t)dW (t) + ∞ 0 J(t)(dZ(t) -λ(t)dt), and
G = ∞ 0 Â(t)d W (t) + ∞ 0 Ĵ(t)(d Z(t) -λ(t)dt).

Assume that

A † (t)A(t) ≤ psd  † (t) Â(t), dP dt -a.e.,
and that for almost all t ∈ R + , we have either: i) n j=1 λ j (t)δ (J 1,j (t),...,J d,j (t)) cxp n j=1 λj (t)δ ( Ĵ1,j (t),..., Ĵd,j (t)) , P -a.s., or:

ii) J i,j (t) ≥ 0, and Ĵi,j (t) ≥ 0, i = 1, . . . , d, j = 1, . . . , n, and n j=1 λ j (t)δ (J 1,j (t),...,J d,j (t)) cxpi n j=1 λj (t)δ ( Ĵ1,j (t),..., Ĵd,j (t)) , P -a.s.

Then for all convex functions φ : R d → R we have

E[φ(F )] ≤ E[φ(G)].
Proof. We apply Theorem 4.2 with B t (x) := J(t)x and Bt (x) := Ĵ(t)x, t ∈ R + , and

ν t (dx) = 1 R d \{0} (x) n j=1 λ j (t)δ (J 1,j (t),...,J d,j (t)) • B -1 t (dx) (4.5)
and

νt (dx) = 1 R d \{0} (x) n j=1 λj (t)δ ( Ĵ1,j (t),..., Ĵd,j (t)) • B-1 t (dx). (4.6)
Note that ν t cxp νt if and only if λ j (t) ≤ λj (t), j = 1, . . . , n, since Supp (ν t ) = Supp (µ t ) = {e 1 , . . . , e n }. In Section 5 we will give a geometric interpretation of the convex ordering cxp , with application to the conditions imposed on ν t

F M t = σ(W (s), ω([0, s] × A) : 0 ≤ s ≤ t, A ∈ B b (R n )), t ∈ R + , where B b (R n ) = {A ∈ B(R n ) : σ(A) < ∞}.
Corollary 4.7. Let (J t,x ) (t,x)∈R + ×R n be a R d -valued F M t -predictable process, integrable with respect to dP dtσ(dx), and let ( Ĵt,x ) (t,x)∈R + ×R n be an R d -valued deterministic function, integrable with respect to dtσ(dx). Consider the random variables

F = ∞ 0 A(t)dW (t) + ∞ 0 R n J t -,x (ω(dt, dx) -σ(dx)dt) (4.8)
and

G = ∞ 0 Â(t)d W (t) + ∞ 0 R n Ĵt -,x (ω(dt, dx) -σ(dx)dt).
Assume that

A † (t)A(t) ≤ psd  † (t) Â(t), dP dt-a.e.,
and that for almost all t ∈ R + , we have either:

i) σ • J -1 t -,• cxp σ • Ĵ-1 t -,
• , P -a.s., or:

ii) Ĵt -,x ≥ 0, σ(dx)-a.e., and

σ • J -1 t -,• cxpi σ • Ĵ-1 t -,• , P -a.s.
Then for all convex functions φ : R d → R we have

E[φ(F )] ≤ E[φ(G)].
Proof. We apply Theorem 4.2 with the jump characteristics

ν t (dx) = 1 R d \{0} (x)σ • J -1 t -,• (dx) and νt (dx) = 1 R d \{0} (x)σ • Ĵ-1 t -,• (dx),
and

B t (x) = Bt (x) = x, x ∈ R n .
Condition (i), resp. (ii) in Corollary 4.7 can be written as

R n f (J t -,x )σ(dx) ≤ R n f ( Ĵt -,x )σ(dx)
for all non-negative convex functions f : R d → R, resp. for all non-negative and non-decreasing convex functions f : R d → R. In particular, Corollary 4.7-ii) holds if we have σ cxpi σ and J t,x ≤ Ĵt,x , dtσ(dx)dP -a.e., and if x → J t,x , x → Ĵt,x are non-decreasing and convex on R n for all t ∈ R + .

We may also apply Theorem 4.2 to F as in (4.8) with

G = ∞ 0 Â(t)d W (t) + ∞ 0 Ĵ (t)(d Z(t) -λ(t)dt)
where ( Â(t)) t∈R + and ( Ĵ(t)) t∈R + are F M t -predictable M d×n -valued processes and ( Z1 , . . . , Zn ) is a R n -valued point process independent of F M , with

ν t = σ • J -1 t -,• and νt = n i=1 λi (t)δ e i .
In case x → J t,x is convex (resp. non-negative, non-decreasing and convex) on R n for all 0 ≤ t ≤ T , condition (i) resp. (ii), of Theorem 4.2 is satisfied provided

σ cxp n i=1 λi (t)δ e i , resp. σ cxpi n i=1 λi (t)δ e i . (4.9)

A geometric interpretation for discrete measures

The next lemma provides a first interpretation of the order cxp . However the necessary condition (5.2) is clearly not sufficient to ensure the convex ordering of µ and ν. Our aim in this section is to find a more precise geometric interpretation of µ cxp ν in the case of finite supports, with the aim of applying this criterion to the jump measures defined in (3.15), (3.16), (4.5), (4.6) and (4.9).

For all u ∈ S d-1 the unit sphere in R d , let

µ u = µ • u, • -1 (resp. ν u = ν • u, • -1 )
denote the image of µ, resp. ν, on R by the mapping x → u, x . We have µ u cxp ν u and the survival function φ µ,u associated with µ u , defined by

φ µ,u (a) = R (y -a) + dµ u (y) = R d ( y, u -a) + dµ(y),
is a convex function with φ µ,u ≤ φ ν,u for all u ∈ S d-1 . Moreover for all a ∈ R such that a is sufficiently large we have φ µ,u (a) = φ ν,u (a) = 0.

For every x ∈ R d and u ∈ S d-1 let

a x,u = inf {b ∈ R : b ≥ u, x , φ µ,u (b) = φ ν,u (b)} (5.3) and D x,u = {y ∈ R d : u, y ≤ a x,u },
which is a closed half-space containing x. Finally we let

C x := u∈S d-1 D x,u
which is a compact convex set containing x. On the other hand, letting

Du = {z ∈ R d : u, z ≤ ãu }, x ∈ R d , u ∈ S d-1 , where ãu = inf {b ∈ R : b ≥ u, y , ∀y ∈ Supp (ν)} , we have C (Supp (ν)) = u∈S d-1
Du . In order to see this, take u = 1 and then u = -1, and note that C -1 is contained in the vertical line passing through -1 and apply Theorem 5.5. Similarly we have

Note that we have

C 1 = [1 -i, 1 + i]. Next, consider µ = 1 3 δ 1/2 + 2 3 δ -1/4 and ν = 1 3 (δ 1 + δ j + δ j 2 ),
where j = e 2iπ/3 . Then Theorem 5.5 shows that

C 1/2 = C -1/4 = C (Supp (ν)),
since {y 1 , y 2 , y 3 } is necessarily equal to Supp (ν), as illustrated below: Proof. From the definition of C E it is clear that if x ∈ E then C x ⊂ C E . Consequently applying Theorem 5.5 to every x ∈ E gives the result.

When µ and ν are probability measures, the existence of the admissible kernel K such that µK = ν, used in the proof of Theorem 5.5, is also known as Strassen's theorem [START_REF] Strassen | The existence of probability measures with given marginals[END_REF], and it is equivalent to the existence of two random variables F, G with respective laws µ and ν, and such that F = E[G|F ]. Here we used Theorem 40 of [START_REF] Dellacherie | Probabilités et Potentiel[END_REF] which relies on the Hahn-Banach theorem. In dimension one this result has been recovered via a constructive proof in [START_REF] Müller | On the optimal stopping values induced by general dependence structures[END_REF]. We close this paper with the following remark which concerns the cx ordering.

Remark 5.10. The conclusion of Theorem 5.5, associated to the condition µ cx ν, implies the existence of an admissible kernel K such that µK = ν.

Proof. We use the notation of Theorem 5.5. First we show that if µ({x}) > ν({x})

then there exists a kernel K x such that K x (x, dy) is supported by {x, y 1 , . . . , y k } and is not equal to δ x , K x (x ′ , dy) is equal to δ x ′ if x ′ = x, and µ cx µK x cx ν. Indeed we can take

K x (x, dy) = (1 -ε)δ x (dy) + ε k i=1 a i δ y i (dy)
where the a i 's are positive, k i=1 a i = 1, x = k i=1 a i y i and ε > 0 is sufficiently smallthe existence of ε follows from the fact that the functions φ µ,u and φ ν,u are continuous in u, together with the compactness of S d-1 . Now let K be a maximal2 admissible kernel such that µK cx ν and the support of µK is included in Supp (µ)∪Supp (ν). If µK = ν then we can apply the argument above to µK, ν and x such that µK({x}) > ν({x}), and find a non trivial kernel K x such that µKK x cx ν, contradicting the maximality of K. So we conclude that µK = ν.

Thus an independent proof of Theorem 5.5, not relying on Theorem 40 of [START_REF] Dellacherie | Probabilités et Potentiel[END_REF], would provide a direct construction the admissible kernel K, extending the result of [START_REF] Müller | On the optimal stopping values induced by general dependence structures[END_REF] to higher dimensions.

Lemma 3 . 1 .

 31 Let A be a symmetric d × d matrix. Then the following statements are equivalent: i) A is positive semidefinite, ii) for all positive semidefinite matrices B we have A, B ≥ 0, Proof. Since A is symmetric, if it is positive semidefinite then its spectral decomposition is given as A = d k=1 λ k e k e † k , where the eigenvalues (λ k ) k=1,...,d of A are non-negative and (e 1 , . . . , e d ) denote the eigenvectors of A. Hence we have Tr (AB † ) = d k=1 λ k e k , Be k ≥ 0 if B is positive semidefinite. The converse follows by choosing B = x † x, x ∈ R d , and noting that

Lemma 5 . 1 .

 51 If µ and ν are two measures on R d with finite supports, then µ cxp ν impliesC (Supp (µ)) ⊂ C (Supp (ν)),(5.2)where C (A) denote the convex hull of any subset A of R d .Proof. Let H be any half-space of R d such that Supp (ν) ⊂ H. For any convex function φ such that {φ ≤ 0} = H and φ |∂H = 0 we have R d φ + ν(dx) = 0, henceR d φ + µ(dx) = 0 since µ cxp ν, which implies Supp (µ) ⊂ H.The conclusion follows from the characterization of the convex hull C (Supp (µ)), resp. C (Supp (ν)), as the intersections of all half-spaces containing it.

  figure:

C E := u∈S d- 1 D

 1 that one can extend the result of Theorem 5.5 to any non empty subsetE of R d such that x ∈ E implies µ(x) > ν(x). Letting a E,u = inf b ∈ sup x∈E u, x , ∞ , φ µ,u (b) = φ ν,u (b) for u ∈ S d-1 , with a E,u = sup x∈E a x,u since E is finite, D E,u = {y ∈ R d : u, y ≤ a E,u },and defining E,u which is a compact convex set containing E, we have C E ⊂ Supp (ν) (since as previously a E,u ≤ ãu ) and: Corollary 5.9. Assume that µ and ν have finite supports and that µ cxp ν. Then for all non empty subset E of R d such that x ∈ E implies µ(x) > ν(x), there exists k ∈ {2, . . . , (d + 1)card(E)} and k elements y 1 , . . . , y k ∈ Supp (ν) distinct from x, such that E ⊂ C ({y 1 , . . . , y k }) ⊂ C E .

  t ) t∈R + and a decreasing filtration (F * t ) t∈R + . Consider (M(t)) t∈R + = (M 1 (t), . . . , M d (t)) t∈R + a d-dimensional F t -forward martingale and (M * (t)) t∈R + = (M * 1 (t), . . . , M * d (t)) t∈R + a d-dimensional F * t -backward martingale, such that (M(t)) t∈R + has right-continuous paths with left limits, and (M * (t)) t∈R + has left-continuous paths with right limits. Denote respectively by (M c (t)) t∈R + and (M * c (t)) t∈R + the continuous parts of (M(t)) t∈R + and of (M * (t)) t∈R + , and by

are the M d -valued processes defined as the limits in uniform convergence in probability [M, M] t = lim n→∞ n k=1

  The quadratic variations ([M, M] t ) t∈R + , ([M * , M * ] t ) t∈R +

dx), where δ (s,x) denotes the Dirac measure at (s, x) ∈ R + × R d . Denote by ν(dt, dx) and ν * (dt, dx) the (F t ) t∈R + and (F * t ) t∈R + -dual predictable projections of µ(dt, dx) and of µ * (dt, dx).

  where d and d * denote the forward and backward Itô differential, respec-

	tively defined as the limits of the Riemann sums
	n
	k=1

  * i,j (t)) 1≤i,j≤d are M d -valued, t ∈ R + , and predictable respectively with respect to (F t ) t∈R + and to (F * t ) t∈R + . In the sequel, we will also assume that (H(t)) t∈R + , (H * (t)) t∈R + ∈ L 2 (Ω × R + ), and that

	and
	d M c , M c
	Let now
	(M(t)) t∈R + be an F * t -adapted, F t -forward martingale,	(3.5)
	and
	(M * (t)) t∈R + be an F t -adapted, F * t -backward martingale,	(3.6)
	with characteristics of the form
	ν(dt, dx) = ν t (dx)dt	and	ν * (dt, dx) = ν * t (dx)dt,

and µ, ν are supported by R d + . t = H(t)dt, and d M * c , M * c t = H * (t)dt, where H(t) = (H i,j (t)) 1≤i,j≤d and H * (t) = (H

  .7) The hypotheses on (H(t)) t∈R + and (H * (t)) t∈R + imply that M c t and M * c t are in L 2 (Ω), t ∈ R + , and Condition 3.7 is a technical integrability assumption.

	Theorem 3.8. Assume that
	H(t) ≤ psd H

* (t), dP dt -a.e. and that for almost all t ∈ R + we have either: i) ν t cxp ν * t , or: ii) ν t cxpi ν * t and ν t , ν * t are supported by (R + ) d .

4

  Convex ordering and predictable representationLet (W (t)) t∈R + be a n-dimensional Brownian motions and µ(dx, dt) be a jump measure t∈R + and ( Â(t)) t∈R + two M d×n -valued, F M t -predictable square-integrable processes, and (t, x) → B t (x) and (t, x)

	with jump characteristics of the form
	ν(dt, dx) = ν t (dx)dt,	(4.1)
	generating a filtration (F M t ) t∈R + .
	Consider (A(t))

  ( ti ,x i ) (dt, dx) with respective intensities σ(dx)dt and σ(dx)dt on R n × R + under P . Let also (W (t)) t∈R + and ( W (t)) t∈R + be independent n-dimensional standard Brownian motions, independent of ω(dt, dx) under P and let (A(t)) t∈R + , ( Â(t)) t∈R + be as in Theo-

	(|x| 2 ∧ 1)σ(dx) < ∞,
	R n
	and two Poisson random measures
	ω(dt, dx) =
	rem 4.2 above, with

• B -1 t and νt • B-1 t in (i) and (ii) above. Example: Poisson random measures Consider σ, σ two atomless Radon measures on R n with R n (|x| 2 ∧ 1)σ(dx) < ∞, and i∈N δ (t i ,x i ) (dt, dx) and ω(dt, dx) = i∈N δ

Here, K ≤ K ′ means that there exists an admissible K ′′ such that K • K ′′ = K ′ .

since for all u ∈ S d-1 there exists z ∈ Supp (ν) such that u, z = ãu and for all b ∈ ( u, x , ãu ] we have φ ν,u (b) ≥ (ã u -b)ν({z}) > 0 and φ µ,u (b) = 0, implying a x,u = ãu and D x,u = Du . Note that in Theorem 5.5 below the existence of x ∈ R d such that µ({x}) > ν({x}) is always satisfied when µ cx ν and µ = ν. Theorem 5.5. Assume that µ and ν have finite supports and that µ cxp ν. Then for all x ∈ R d such that µ({x}) > ν({x}) there exists k ∈ {2, . . . , d + 1} and k elements

(5.6)

Proof. We only need to prove that x belongs to the convex hull of (C x \{x})∩ Supp (ν).

Indeed, if x ∈ C ((C x \{x}) ∩ Supp (ν)) then there exists k points y 1 , . . . , y k in this set, k ≥ 2, such that x is the convex barycenter of y 1 , . . . , y k , and the Caratheodory theorem (see e.g. [START_REF] Rockafeller | Convex Analysis[END_REF], Theorem 17.1) shows that the conclusion holds for some k ∈ {2, . . . , d + 1}.

Assume now that the assertion of the theorem is true when µ and ν have disjoint supports, and let µ and ν be any measures with finite supports, such that µ cxp ν and

These sets are not empty since µ = ν. Let µ ′ and ν ′ the measures defined by

which implies that µ cxp ν if and only if µ ′ cxp ν ′ . It also implies that for all u ∈ S d-1 and b ∈ R,

From this together with the fact that φ µ ′ ,u ≤ φ ν ′ ,u , we conclude that if D ′

x,u is defined as D x,u but with (µ, ν) replaced by (µ ′ , ν ′ ), then D ′

x,u = D x,u . Finally remarking that the support of ν ′ is included in the support of ν, we proved that it is sufficient to do the proof with µ ′ and ν ′ .

So in the sequel we assume that µ and ν have disjoint supports. As a consequence of Theorem 40 in [START_REF] Dellacherie | Probabilités et Potentiel[END_REF] applied to the cone of non-negative convex functions, there exists an admissible 1 kernel K such that µK = ν.

is included in the support of ν, and by (5.4), x is in the convex hull of the support of K(x, dy). Finally we are left to prove that the support of K(x, dy) is included in C x .

For this we let µ x be the measure defined by

Then µ cxp µ x and µ x cxp ν, which is easily proved by the existence of admissible kernels P and P ′ such that µP = µ x and µ x P ′ = ν. More precisely they are given by

x ′ ∈ Supp (µ)\{x}.

1 Admissible means here that for every x ∈ R d we have δ x cxp K(x, dy).

23

So for every u ∈ S d-1 , we have φ µ,u ≤ φ µ x ,u ≤ φ ν,u . Let us prove that this inequality implies that any point of the support of K(x, dy) belongs to D x,u . Assume that a point z of the support of K(x, dy) does not. An easy calculation shows that the right

From the definition of µ x we see that for t = u, x , µ x u ({t}) ≥ µ u ({t}). This implies that for t ≥ u, x ,

and this implies that φ µ,u (a x,u ) < φ µ x ,u (a x,u ) where a x,u is defined in (5.3). Since φ µ,u ≤ φ µ x ,u ≤ φ ν,u , we obtain φ µ,u (a x,u ) < φ ν,u (a x,u ), contradicting (5.3).

We proved that for every u, any point of the support of K(x, dy) belongs to D x,u .

This implies that the support of K(x, dy) is included in D x,u , achieving the proof.