Claire Lacour

Benjamin Jourdain

Edward Waymire

Charlotte Dion

Magda- Lena Kobylanski

Soledad Torres

Olivier Bardou

Christophe Denis

Joaquin Fontbona

Arnaud Gloter

Dan Goreac

Antoine Lejay

Jaime San Mar- Tin

Sylvain Rubenthaler

Etienne Tanr

chacun d'entre eux.

Introduction

The works presented in these notes take their source from the study of processes related to the Skew Brownian Motion (SBM in short), which constitutes a core in my research. Up to my knowledge, historically, the first mention of the SBM can be found in the monograph [START_REF] Itô | Diffusion processes and their sample paths[END_REF], where it appears as a natural generalization of the standard Brownian motion : it is a process that behaves like a Brownian motion except that the sign of each excursion away from zero is chosen using an independent Bernoulli random variable of parameter α ∈ (0, 1) (the standard Brownian motion is retrieved as a particular case by choosing α = 1/2).

Afterwards, the SBM was presented as some kind of curiosity to highlight the theory of semimartingale local times since it is the first historical example of a diffusion possessing a local time discontinuous in the space variable (see [START_REF] Walsh | A diffusion with discontinuous local time[END_REF]). At the beginning of the 80's, Harrison and Shepp [START_REF] Harrison | On skew Brownian motion[END_REF] construct the Skew Brownian motion as the rescaled limit of a simple random walk perturbed at the origin : at each time it reaches zero the random walk chooses to go up with probability α independently of anything else. They show that the limiting process is in fact the solution of the simplest Stochastic Differential Equation involving the local time of the unknown process, namely

X x t = x + B t + βL 0 t (X x ), (1)
where β = 2α -1 ∈ (-1, 1) and (L 0 t (X x )) t≥0 stands for the (symmetric) local time at 0 of the unknown process (X x t ) t≥0 . Any solution of (1) is to be called a skew Brownian motion (of parameter β).

For an account on the SBM and its extensions, up to my knowledge one the best reference is surely the survey by A. Lejay [START_REF]On the constructions of the skew Brownian motion[END_REF], where several constructions for the solution Put(0,-845.04684) of (1) are presented. Note that this survey includes an original construction from the late 70's by N. Portenko who managed to extend the parametrix method in order to construct what must somewhat correspond to fundamental solutions of the semi-group associated to (1) but generalized in a multidimensional context (see [START_REF] Portenko | Diffusion processes with a generalized drift coefficient[END_REF][START_REF]Stochastic differential equations with a generalized drift vector[END_REF]) (see also [START_REF] Mastrangelo | Mouvements browniens asymétriques modifiés en dimension finie et opérateurs différentiels à coefficients discontinus[END_REF]).

A very natural generalization of (1) -that does not appear in the survey [START_REF]On the constructions of the skew Brownian motion[END_REF] -arises when, roughly speaking, the sign of an excursion away from zero is chosen using a different independent Bernoulli random variable at each time such an excursion occurs. Such a construction gives rise quite naturally to the following equation, which is a generalization of (1)

X x t = x + B t + t 0 β(s)dL 0 s (X x ). (2)
Here s → β(s) ∈ (-1, 1) stands for some measurable function that determines the law of the sign for each excursion. This issue was first investigated by S. Weinryb in [START_REF] Weinryb | étude d'une équation différentielle stochastique avec temps local, Seminar on probability[END_REF],

where a pathwise uniqueness property for solutions of ( 2) is proved. However, concerning existence of weak solutions to (2) (to be called Inhomogeneous Skew Brownian Motion (ISBM in short)) and though [START_REF] Weinryb | étude d'une équation différentielle stochastique avec temps local, Seminar on probability[END_REF] was published in 1983, in year 2012 we could not find a satisfactory reference with a proof for an existence result for the ISBM solution to (2).

Indeed, in the reissue of the reference book by D. Revuz and M. Yor [130, Chapter VI Exercise 2.24], we find "Let X be a continuous semimartingale, if it exists, such that (2) holds". So still by 1999 there existed a reasonable doubt concerning the existence of the ISBM.

In chapter 1, we present the results contained in [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] concerning the existence of solutions to (2). Note that since [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] was published, now we can also mention [START_REF] Bouhadou | On the time inhomogeneous skew Brownian motion[END_REF] where the authors propose another construction for the ISBM using step functions and the balayage formula.

A quite intriguing fact concerning the trajectories of the (homogeneous) SBM was discovered by the beginning of this century in [START_REF]Coalescence of skew Brownian motions[END_REF] : the authors prove the almost sure coalescence of two skew Brownian motions starting from different initial points, assuming that they are driven by the same Brownian motion. More precisely, for any x and y there exists a random time t ≥ 0, such that X x t = X y t . This simple fact was the starting point for a study of the skew Brownian flow, depending either on the starting point x or on the 10 Put(0,-845.04684) Put(0,-845.04684) skewness parameter β.

One of the difficulties arising when one starts studying the skew Brownian flow comes from the fact that, due to the presence of the local time in the equation, we cannot ensure that there is existence of the family of SBM (driven by the same Brownian motion) (X x t ) t≥0 for all x ∈ R simultaneously. For works related to the comprehension of the skew Brownian flow, we may mention [START_REF] Barlow | Variably skewed Brownian motion[END_REF][START_REF]Coalescence of skew Brownian motions[END_REF][START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] culminating with [START_REF] Burdzy | Lenses in skew Brownian flow[END_REF].

In the work [START_REF] Gloter | Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time[END_REF], taking as a basis the results of Burdzy and Chen [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF], we investigate the distance process between two skew Brownian motions with different individual skewness parameters. We manage to derive a stochastic differential equation which describes the evolution of this distance process up to its first reach of zero in some proper local time scale. The effect of this local time scale is to erase the portion of the time axis where the distance process remains constant (when both motions stay parallel), so that the resulting time-change distance process has a negative drift and positive jumps.

The jump dynamics is naturally driven by the excursions of one of the skew Brownian motions. This representation provides an access to the law of the local time at the first hitting time of the skew Brownian motions, which is seen to be related to the beta distribution. However, the jump dynamics degenerates as the distance gets closer to zero (when the skew Brownian motions are about to meet) and it remained to try to describe the behavior of the distance process after the first hit of zero. In [START_REF]Bouncing skew brownian motions[END_REF], we manage to circumvent this problem of the degeneracy of the dynamics of the jump SDE governing the distance process by observing that the distance process is in fact a self-similar process (possessing a scaling property). This leads us in [START_REF]Bouncing skew brownian motions[END_REF] to plunge the problem as a particular (explicit) case of the general theory of positive recurrent Markov extensions of self-similar processes (for an account on this beautiful theory originated by Lamperti in [START_REF] Lamperti | Semi-stable Markov processes. I[END_REF], we may distinguish [START_REF] Fitzsimmons | On the existence of recurrent extensions of self-similar Markov processes[END_REF][START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF]132] among many other references). Starting from this observation, the problem we faced was to investigate which one of the possible extensions we were dealing with. In [START_REF]Bouncing skew brownian motions[END_REF] we actually prove that the distance process corresponds to the only continuous extension at zero and not to a jumpin' in extension. Once this difficult and crucial point is cleared out, we manage to compute the law of the distance of the two skew Brownian motions at any time in the local time scale, when both original skew Brownian motions start from zero. As a by product, we can also study the Markovian Put(0,-845.04684) dependence on the skewness parameter β and give a partial answer to an open question formulated initially by C. Burdzy and Z.Q. Chen in [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF]. We present the main results of these companion papers [START_REF] Gloter | Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time[END_REF][START_REF]Bouncing skew brownian motions[END_REF] in Chapter 2.

Since the first appearance of the skew Brownian motion, various generalizations of (1) have been investigated. Indeed, the equation ( 1) can be viewed as the simplest case in the class of general Stochastic Differential Equations involving the local time of the unknown process (SDELTs). Existence and uniqueness results of possible solutions under fairly general conditions were obtained by Le Gall [START_REF] Gall | Applications du temps local aux équations différentielles stochastiques unidimensionnelles[END_REF][START_REF]One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF] at the beginning of the 80's.

The general case of a (homogeneous) SDELT writes

dX t = σ(X t )dB t + b(X t )dt + R L x t ν(dx) (3)
where ν(dx) is a signed measure satisfying that |ν{x}| < 1.

Such diffusion processes with discontinuous coefficients and involving the local time of the unknown process have been found to have many potential applications in many fields of Physics and Biology. Without being exhaustive, we mention for example Astrophysics [START_REF] Zhang | Calculation of diffusive shock acceleration of charged particles by skew brownian motion[END_REF], Geophysics and study of heterogeneous media [START_REF] Lang | Effective conductivity and skew Brownian motion[END_REF][START_REF] Lejay | Monte Carlo methods for fissured porous media : a gridless approach[END_REF][START_REF]A probabilistic representation for the solution to one problem of mathematical physics[END_REF][START_REF] Ramirez | A generalized Taylor-Aris formula and skew diffusion[END_REF]151], modeling of river networks with the pioneering works of J. Ramirez [START_REF] Ramirez | Multi-skewed Brownian motion and diffusion in layered media[END_REF]126], perturbed Hamiltonian systems [START_REF] Freidlin | Random perturbations of Hamiltonian systems[END_REF], Ecology (the dynamic of populations moving between different refuges in studied in [START_REF] Cantrelln | Diffusion models for population dynamics incorporating individual behavior at boundaries : applications to refuge design[END_REF]), finance and actuarial sciences [START_REF] Decamps | Asymmetric skew Bessel processes and their applications to finance[END_REF].

In Chapter 3, we examine a very simple equation of type (3), namely

dX β t = dB t + b(X β t )dt + βdL 0 t (X β ) (4)
We present the results of the articles [START_REF]Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process[END_REF][START_REF]Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift[END_REF] that provide algorithms for the exact simulation for solutions of (4) in the case where the drift b is allowed to possess a discontinuity at 0, the point where the local time perturbs the solution. We adapt the methodology of Beskos et al. [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF][START_REF] Beskos | Exact simulation of diffusions[END_REF] to this case and show that it is possible to simulate exactly skeleton paths of X β . One of the major difficulties we face in doing so is that, due to the presence of the local time in the equation, the measure induced by the solution of (4) on the path space (C([0, T ]), C) is no longer absolutely continuous w.r.t. the Wiener measure. Still, Put(0,-845.04684) using simple but crafty transformations, we manage to circumvent this problem and to adapt the methodology of Beskos et al. to our case.

One of the main interests of SDELTs is that the generator of the solution of (3) may be identified with a second order parabolic operator A in generalized divergence form

A = ρ 2 d dx a d dx + b d dx (5)
where ρ and a may be discontinuous. The measure ν(dx) in (3) captures the discontinuities of ρ and a by putting a mass at these points where a and ρ are discontinuous.

Assuming that the coefficients σ and b are smooth outside their discontinuity points, one may establish, via a Feynman-Kac formula, a link between solution of (3) and the classical solution of a parabolic Partial Differential Equation (PDE) with transmission conditions involving A. For results stating the link between solutions of (3) and the solutions of parabolic PDEs with transmission conditions involving the operator A given by ( 5) (in this one-dimensional context), one may refer to the overview [START_REF]On the constructions of the skew Brownian motion[END_REF], and the series of works [START_REF] Martinez | Inbterprétations probabilistes d'opérateurs sous forme divergence et analyse des méthodes numériques probabilistes associées[END_REF][START_REF] Martinez | Discrétisation d'équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu[END_REF][START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF], [START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF], and [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Étoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF] where numerical schemes are presented.

This interpretation leads naturally to address various theoretical and applied probability problems which are not restricted to the one-dimensional case but where divergence form operators with discontinuous coefficients play a crucial role and where stochastic simulations algorithms may be used : for example, the numerical resolution of solute transport equation in Geophysics (see, e.g., Salomon et al. [START_REF] Salomon | A review and numerical assessment of the random walk particle tracking method[END_REF] and references therein), the numerical resolution of the Poisson-Boltzmann equation in Molecular Dynamics (see, e.g., [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] and references therein) ; another motivation comes from Neuro-sciences, more precisely from an algorithm of identification of the magnetic permittivity around the brain (see [START_REF] Clerc | Comparison of bem and fem methods for the e/meg problem[END_REF][START_REF] Clerc | The fast multipole method for the direct e/meg problem[END_REF]).

In Chapter 4, we briefly present new results obtained with P. Etoré in the preprint [START_REF]A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF].

These results extend those previously obtained in the one-dimensional context with my PhD thesis director D. Talay in [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF] to the multidimensional case. The ideas of [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF] have also been extended in the recent work [START_REF] Bokil | An Euler-Maruyama method for diffusion equations with discontinuous coefficients and a family of interface conditions[END_REF] by Bokil & al. where the Authors present an Euler-Maruyama method for diffusion equations with discontinuous coefficients when Put(0,-845.04684) a family of interface conditions comes into play. Parabolic equations involving such L have been a major preoccupation for mathematicians in the fifties and the sixties. We may cite the pioneering works of J.Nash [START_REF] Nash | Parabolic equations[END_REF][START_REF]Continuity of solutions of parabolic and elliptic equations[END_REF], E. De Giorgi [START_REF] Giorgi | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF], and J. Moser [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF][START_REF]A Harnack inequality for parabolic differential equations[END_REF][START_REF]A Harnack inequality for parabolic differential equations[END_REF] that prove the continuity of the solution of the Cauchy problem and also the celebrated paper by D.G. Aronson [5], which gives upper and lower Gaussian estimate bounds for the fundamental solution of the operator L.

In

In the particular case where the matrix a is assumed to be discontinuous along the regular boundaries of some nice connected open sets in R d , a refined analysis of the parabolic equation may be found in the monograph [START_REF] Ladyženskaja | Linear and quasi linear equations of parabolic type, Izdat[END_REF], where the parabolic equation is interpreted as a diffraction problem with transmission conditions along the discontinuity boundaries (for a more modern perspective of parabolic transmission problems see also [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]).

Up to my knowledge, the first construction of a Markovian semi-group associated to L = ∇.(a∇) (in the general case where a is only supposed to be measurable) may be found in the seminal paper of D. Stroock [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]. To understand our difficulty, the conclusion of [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] is :

It should be obvious that the results obtained in this section can be used to construct a diffusion process on Ω corresponding to anyone of the semigroups discussed herein. In addition, the convergence results for the semigroups give rise to weak convergence of the corresponding measures on Ω.

It remains an open and challenging problem to provide a better probabilistic interpretation of these essentially analytic facts.

Since then, there have been many works that try to provide this 'better probabilistic interpretation' in this multidimensional framework.

In the general case where the symmetric matrix a is only supposed to be elliptic and measurable, the theory of Dirichlet forms as exposed in the monograph by Fukushima [START_REF] Fukushima | Dirichlet forms and Markov processes[END_REF][START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] gives surely the best possible answer to this question under these general hypothesis. The symmetric operator L is naturally attached to its corresponding symme-14 Put(0,-845.04684) Put(0,-845.04684) tric Dirichlet form, giving rise to a stochastic Dirichlet process X that is described as the addition of a continuous martingale and a continuous additive functional of zero energy.

Going further in the analysis, A. Roskosz [START_REF] Rozkosz | Stochastic representation of diffusions corresponding to divergence form operators[END_REF][START_REF]Weak convergence of diffusions corresponding to divergence form operators[END_REF] proves that X satisfies a Lyons-Zheng decomposition, namely X may be written as the solution of a complex stochastic equation that is the addition of three processes : a martingale, an increment of a reversed time martingale (whose quadratic variations depend on the unknown process X), and an additive functional involving the logarithm derivative of the fundamental solution of the parabolic operator evaluated at X. This description permits to retrieve a sort of Itô's formula for φ(X . ) where φ belongs to some 'good' Sobolev space.

However, the Lyons-Zheng decomposition provides a description of X that is so strongly nonlinear (time reversion and logarithm derivative of the density of the unknown process), that it seems quite impossible to analyze with full precision the behavior of the trajectories of X and even less to propose a direct stochastic numerical scheme for the approximation of X at this stage : here, by direct, we mean that we do not allow any kind of regularization of the coefficient a with the help of some regularization convolution kernel.

Still, one may hope to be able to describe in more detail the behavior of the trajectories of X when the coefficient matrix a -instead of being assumed to be only measurable -is now assumed to be smooth outside discontinuities that take place along nice and regular surfaces Γ.

In this direction some papers (see for example [START_REF] Trutnau | Multidimensional skew reflected diffusions, Stochastic analysis : classical and quantum[END_REF], [START_REF]A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE[END_REF]) go deeper and manage to apply the stochastic calculus tools attached to the theory of Dirichlet forms to investigate the solutions of sophisticated stochastic differential equations but none of these papers deals precisely with our case.

In Chapter 4 we provide a Skorokhod representation of the process X related to L when he coefficient matrix a -instead of being assumed to be only measurable -is now assumed to be smooth outside discontinuities that take place along nice and regular surfaces Γ. This representation describes the process as the solution of some stochastic differential equation whose right-hand side is the addition of the expected martingale term driven by some Brownian motion and the expected additive drift term (both capturing the regular part of the coefficients) and an additive functional A Γ that encapsulates Put(0,-845.04684) the behavior of the process at the discontinuity boundaries Γ. The additive functional is rigorously constructed through its Revuz correspondence with some transformation of the natural surface measure of Γ involving the discontinuity jumps of a along Γ. Formally, the conclusion is that when the coefficient matrix a is smooth everywhere except for a discontinuity along some regular boundary, it is possible to retrieve that X is a diffusion process writing as the solution of some kind of multidimensional extension of (3) : all objects of (3) are now turned in their multidimensional counterpart ; in particular the local time term is now replaced by the additive functional A Γ and X remains a semimartingale (see Chapter 4, Equation (4.13)).

When turning to the objective of constructing a direct stochastic numerical scheme, the description of A Γ via its Revuz correspondence measure does not provide a plain and unambiguous natural way for the discretization of X (see however the Phd thesis of L. Lenôtre [START_REF] Lenôtre | étude et simulation des processus de diffusion biaisés[END_REF] and the walk on spheres algorithm in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] in the case of a diagonal coefficient matrix a).

Together with P. Etoré, we propose in [START_REF]A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF] to tackle the problem of the construction of a direct numerical scheme for X from another perspective. Instead of taking X as our starting point, we begin to focus on the solution of the parabolic equation involving L. Inspired by the proof of the convergence of the Euler scheme constructed in the one dimensional case in [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF], we build a multidimensional Euler scheme purposely designed to capture the multidimensional transmission conditions of the parabolic PDE associated to L. Our scheme has been constructed without being concerned by the description of the limiting process X and the additive functional A Γ . All our concern was to guarantee that the error difference between the expectation of our process -visualized through some very smooth arbitrary test functions (belonging to the iterated domains of the operator L) -and the corresponding solution of the parabolic equation (with the test functions as initial condition) converges to zero.

The resulting numerical scheme may be viewed as an extension of the symmetrized Euler scheme of [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF] for reflected diffusion with smooth coefficients to partially reflected diffusions with discontinuous coefficients, when the Skorokhod representation of the local time at the boundary Γ traduces a partial reflection.

When turning to the proof of the convergence, we face the difficulty of getting precise Put(0,-845.04684) estimation bounds for the solution of the parabolic equation and its partial derivatives (up to order four in the space variable) outside the discontinuity boundary Γ for all strictly positive times (not just strictly positive times occurring after some ε > 0) and all the way up to the boundary (not only interior estimates). Unfortunately, the analysis performed in [START_REF] Ladyženskaja | Linear and quasi linear equations of parabolic type, Izdat[END_REF], which is somewhat difficult to read and to understand in full detail, does not provide the refined estimations we need for our purpose (see also the recent work for the estimation of the gradient in [START_REF] Fan | A gradient estimate for solutions to parabolic equations with discontinuous coefficients[END_REF] extending [START_REF] Li | Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients[END_REF] to the parabolic case).

Instead of trying to adapt and extend the results in [START_REF] Ladyženskaja | Linear and quasi linear equations of parabolic type, Izdat[END_REF], we preferred to look at the more modern results stated in McLean [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] that have been obtained for the solutions of elliptic divergence transmission problems. We extend the results of [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] to the parabolic case by performing the classical Hille-Yosida theorem. The price to pay is to strengthen severely our assumptions on the regularity of the coefficient matrix a outside its discontinuity boundary and to assume that the initial condition function (the test function) belongs to some iterated domain of the operator L. As expected, with this methodology the orders of smoothness and iteration that we require increase linearly with the dimension d. Using then the classical Sobolev injections, we obtain all the estimates on the solution we need to prove the convergence of our stochastic numerical scheme.

Note that the analysis of the weak error -overlooked through restricted test functions belonging to some large enough iteration of the domain of the underlying operator -is relatively natural. This is what is done for example in [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF] for the symmetrized Euler scheme corresponding to reflected diffusions, where the spatial derivatives of the test functions are assumed to verify some compatibility conditions at the reflection boundary ; (whereas in [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF] the compatibility conditions are fixed once and for all, here the compatibility conditions we require depend crucially on the dimension d).

Finally, we prove that the weak error of our scheme is of order √ ∆t (improving slightly the results of [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF] in the one dimensional case).

The last Chapter of these notes is devoted to a brief presentation of some of my other research interests outside the field of asymmetric diffusions.

In a first section I present a series of issues that may be labeled beneath the scope of filtering problems although the investigated questions are somewhat different. First, 17

Put(0,-845.04684) Put(0,-845.04684) I briefly expose the results of an old article [START_REF] Martinez | Approximations of a continuous time filter. Application to optimal allocation problems in finance[END_REF] written with S. Rubenthaler and E.

Tanré concerning the application of filtering techniques to an optimal portfolio allocation problem in the context of an unstable financial market. Second, I present a preprint together with C. Denis and C. Dion concerning the problem of multi-class statistical classification of diffusion paths. The common point between these papers is that both involve the approximation of a conditional expectation.

A subsequent section is devoted to the presentation of some results that may be referenced under the gigantic topic of stochastic control problems that I am discovering little by little with the help of my co-authors. I briefly present a joint work with D. Goreac and M. Kobylanski [START_REF] Goreac | A piecewise deterministic Markov toy model for traffic/maintenance and associated Hamilton-Jacobi integrodifferential systems on networks[END_REF] where we investigate optimal control problems in infinite horizon when the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks and corresponding to a specific toy traffic model. Put(0,-845.04684) Chapitre 1

Time inhomogeneous skew Brownian motion

In this chapter we present the paper [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] On the existence of a time inhomogeneous skew Brownian motion and some related laws, written with Pierre Étoré.

Introduction

In the seminal paper by S. Weinryb [START_REF] Weinryb | étude d'une équation différentielle stochastique avec temps local, Seminar on probability[END_REF], the following stochastic differential equation is considered

dB α t = B t + (2α(t) -1)dL 0 t (B α ), t ∈ [0, T ], B α 0 = x 0 . (1.1)
It is proved there that a pathwise uniqueness property holds true for the possible solutions of (1.1) under the general assumption that α is a deterministic function taking values in (0, 1). But, however, concerning the existence of weak solutions for (1.1), it is said that "partial existence results were obtained by Watanabe [START_REF] Watanabe | Construction of diffusion processes by means of Poisson point process of Brownian excursions[END_REF]149]". The second reference available by the time we were working on [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] comes from the fundamental book (posterior to [START_REF] Weinryb | étude d'une équation différentielle stochastique avec temps local, Seminar on probability[END_REF]) [ Put(0,-845.04684) of possible solutions to (1.1). Up to my knowledge, there has been only the paper [START_REF] Bouhadou | On the time inhomogeneous skew Brownian motion[END_REF] published on this subject (posterior to [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF]), where the authors provide another proof for the construction of a solution to (1.1). Note that our paper is also cited in [START_REF] Çetin | Filtered Azéma martingales[END_REF] (which studies the projection of the Brownian motion on the filtration generated by an observation process that provides information on the sign of the excursion of the Brownian motion) and [START_REF] Pahlajani | Stochastic averaging for a Hamiltonian system with skew random perturbations[END_REF] (which studies the orbits of a perturbed Hamiltonian system of ordinary differential equations).

In this chapter, we present one of the main results of [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF]. We provide an existence result for the solution of (1.1) from the computation of its explicit transition probability density. In [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] we also compute some joint laws : namely at some given time, a joint law for the state of the process, its current local time, and its last exit from zero.

Main ideas 1.2.1 Construction of the inhomogeneous Skew Brownian motion from its transition density

In order to understand the basic ideas of [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF], let us start with an illustrative computation.

Let {B t : s ≥ 0} a standard Brownian motion constructed on some filtered probability space (Ω, F, (F t ) t≥0 , P) satisfying the usual hypothesis. We introduce the last exit of 0 before time 1 for (B t ), namely

g 1 = sup (s ≤ 1 : B s = 0) . (1.2)
If B (R) denotes the Borel sigma-field, we may write for A, A ∈ B(R), s ∈ (0, 1)

{g 1 ≤ s} ∩ {|B s | ∈ A , |B 1 | ∈ A } = {∀u ∈ [s, 1], B u = 0} ∩ {|B s | ∈ A , |B 1 | ∈ A }.
In particular, if we introduce { Bu = B s+u -B s , u ≥ 0} and T0 (z) = inf(u ≥ 0 : z + Bu = 20 ut(0,-845.04684) Put(0,-845.04684) 0), we have from Markov's property applied at time s that

P (g 1 ≤ s; |B s | ∈ dz; |B 1 | ∈ dy) = P (|B s | ∈ dz) P z + B1-s ∈ dy; T0 (z) > 1 -s
(where we use the usual abuse of notations to denote probability laws on B(R)). Following up the computation, we see that

P (g 1 ≤ s; |B s | ∈ dz; |B 1 | ∈ dy) = 2 πs e -z 2 /2s q (1 -s, z, dy) dz = 2 πs 1 2π(1 -s) e -(y-z) 2 2(1-s) - 1 2π(1 -s) e -(y+z) 2
2(1-s)

I s∈(,) I y≥ I z≥ dz dy
where we made use of the well known explicit formulae for the density of the absolute value of a standard Brownian motion and its taboo probability law q(t, x, dy).

We may now integrate this expression w.r.t the variable z and we finally arrive at the following expression

P (g 1 ∈ ds; |B 1 | ∈ dy) = y e -y 2 /2(1-s) π s(1 -s) 3 I s∈(,) I y≥ ds dy, (1.3)
from which we deduce the independence relation

P g 1 ∈ ds; |B 1 | √ 1 -g 1 ∈ dy = 1
πs(1 -s) I s∈(,) ds ⊗ ye -y 2 /2 I y≥ dy.

(1.4)

Starting from (1.3) and using the symmetry of Brownian motion, we recover an interesting integral representation of the standard Gaussian law, namely

P (B 1 ∈ dy) = 1 0 1/2 π s(1 -s) 3 |y|e -y 2 /2(1-s) I y∈R dy. (1.5)
Suppose for a moment that -instead of using the symmetry of Brownian motion in the last step -we toss a rigged coin in order to choose the sign of B 1 . Surely, the resulting random variable is no longer the value of a Brownian motion at time 1. Let us call B α 1 the random variable that we thus obtain (where α denotes the load of the rigged coin). Note that by construction, the absolute value of this random variable is still that of a Brownian motion at time 1 : in fact, the random variable B α 1 is the value of a (homogeneous) skew 21 Put(0,-845.04684) Put(0,-845.04684) Brownian motion at time 1. Now, let us render this procedure more dynamic and instead of flipping the same fixed rigged coin through time, let us decide to make the load of the coin s → α(s) ∈ (0, 1) depend on the current time. Since the sign B α 1 is chosen at time g 1 , in view of the computations above and the independence relation (1.4), we should generalize (1.5) to

P (B α 1 ∈ dy) = 1 0 ds α(s) π s(1 -s) 3 |y|e -y 2 /2(1-s) I y≥ dy + 1 0 ds 1 -α(s) π s(1 -s) 3
|y|e -y 2 /2(1-s) I y< dy.

(1.6)

The identity (1.6) may be generalized to an arbitrary horizon time t. Also, we may start from an arbitrary point x at some time s ≤ t instead of starting from zero. Before hitting point zero the process solution of (1.1) should behave like a standard Brownian motion. Applying Markov's property at the first hitting time of zero shifts the function α. Performing these changes and using the explicit laws of the standard Brownian motion, after some tedious but easy computations, we naturally arrive at the following nice expression p α (s, t; x, y) = p(t -s, x, y)

+ t-s 0 β(s + u) 2 y π e -y 2 /2(t-(s+u)) e -x 2 /2u √ u(t -(s + u)) 3/2 du (1.7)
where p(t, x, y) = 1 √ 2πt e -(y-x) 2 /2t denotes the transition density of the Brownian motion and β : s → (2α(s) -1) stands for the skewness parameter function.

Once arrived at the expression (1.7) the program becomes quite clear. Prove first that the formula (1.7) defines a (inhomogeneous) family of transition density function satisfying the Chapman-Kolmogorov equations. As a consequence, there must exist a probability measure Wx 0 α on ) ) such that the coordinate process ω = (ω t ) t≥0 defined on this measurable space is a Markov process , starting from x 0 under Wx 0 α with t.f. p α (s, t.x, y). Second, prove using the explicit transition density that a Kolmogorov-Centsov criterion is satisfied and conclude that we can construct a modification of ω with Wx 0 α -a.s continuous paths. Thus, there exists a probability measure W x 0 α under which the coordinate process 22 ut(0,-845.04684) Put(0,-845.04684) ω = (ω t ) t≥0 is a continous Markov process, still with t.f. p α (s, t; x, y). It remains to prove that ω satisfies (1.1) for some W x 0 α Brownian motion B. From the explicit expression of p α (s, t; x, y), one can prove that, under W x 0 α , (|ω| t ) t≥0 is distributed as the absolute value of a Brownian motion started at x 0 .

R [0,∞) , B(R [0,∞
The Markov property ensures

E W x 0 α (ω t |C s ) = ∞ -∞ yp α (s, t; ω s , y)dy = w s + t-s 0 β(s + u) e -|ωs| 2 /2u √ 2πu du.
Since |ω| is a reflected Brownian motion it admits a symmetric local time which is a continuous additive functional of ω. So that for s ≤ t :

E W x 0 α t 0 β(u)dL 0 u (ω)|C s = s 0 β(u)dL 0 u (ω) + E W x 0 α t s β(u)dL 0 u (ω)|C s .
But,

E W x 0 α t 0 β(u)dL 0 u (ω)|C s = E W ωs α(s+.) t-s 0 β(s + u)dL 0 u (ω) = E ωs t-s 0 β(s + u)dL 0 u (|B|)
(Here B stands for some Brownian motion starting from ω s under the measure P ωs ).

Combining these facts ensures that

ω t - t 0 β(u)dL 0 u (ω) : t ≥ 0 is a (C t ) local martingale under W x 0 α . Since ω t = |ω| t = t,
we deduce that it is in fact a Brownian motion under W x 0 α . Performing all these steps permits to construct a weak solution of (1.1).

In [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF], we do not perform these steps in such a linear way. The difficulty arises right from the first step. Instead of proving by hand that the Chapman-Kolmogorov identity is satisfied for p α (s, t; x, y) defined in (1.7), we circumvent these tedious computations using a convergence argument. We assume first that α is a smooth function taking values 23 Put(0,-845.04684) Put(0,-845.04684) in [κ, k] ⊂ (0, 1). In this case, applying a powerful result of [START_REF] Peskir | A change-of-variable formula with local time on curves[END_REF], it is possible to apply a space-time transformation that removes the local time in the equation (1.1). Using general results for Stochastic Differential Equations without local time, we know that the resulting process necessarily satisfies the (strong) Markov property. As a consequence, for smooth α there must exist a solution to (1.1) with p α (s, t; x, y) as its transition density, and therefore p α (s, t; x, y) must satisfy the Chapman-Kolmogorov identity in this case.

A convergence argument allows then to retrieve the Chapman-Kolmogorov identity for non-smooth functions α. Only then can we perform all the steps of the above announced program and prove an existence result for solutions of (1.1) in the general case of nonsmooth measurable functions α (see [START_REF] Étoré | On the existence of a time inhomogeneous skew Brownian motion and some related laws[END_REF] for all the details).

The existence of a weak solution, together with the pathwise uniqueness results of [START_REF] Weinryb | étude d'une équation différentielle stochastique avec temps local, Seminar on probability[END_REF] (see also [130, Chap. VI Exercise 2.24]) ensures the existence of a unique strong solution to (1.1) (see [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF]) and we get the following result Theorem 1.2.1. Let α : R + → [0, 1] a Borel function, and B a standard Brownian motion. For any fixed x 0 ∈ R, there exists a unique (strong) solution to (1.1). It is a (strong) Markov process with transition function p α (s, t; x, y) given by (1.7).

Further investigations 1.3.1 Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process

Inspired by our investigation of the ISBM and aiming at characterizing the possible generators of related diffusions, we also studied with P. Etoré some class of timeinhomogeneous skewed stochastic differential equations and their associated parabolic operators in [START_REF]Time inhomogeneous stochastic differential equations involving the local time of the unknown process, and associated parabolic operators[END_REF].

A natural generalization of the inhomogeneous skew Brownian motion is to investigate time inhomogeneous versions of SDEs with local time, namely equations of the form

dX t = σ(t, X t )dW t + b(t, X t )dt + I i=1 β i (t)dL x i t (X), t ∈ [0, T ], X 0 = x 0 , (1.8) 24
Put(0,-845.04684) Put(0,-845.04684) where the coefficients σ and b are now allowed to depend on time, the functions x i : t → x i (t) are now time-curves, so that (L x i t (X)) t∈[0,T ] stands for the symmetric local time of the unknown process (X t ) t∈[0,T ] along the time-curve x i .

In the article Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process and associated parabolic operators [START_REF]Time inhomogeneous stochastic differential equations involving the local time of the unknown process, and associated parabolic operators[END_REF] written with Pierre Étoré, we investigate these equations and establish the link between the process X and solution of some parabolic PDE with time inhomogeneous transmission conditions. This PDE implies operators of the form

ρ 2 ∂ x (a∂ x .) + b - ρa x,± 2 ∂ x (1.9)
with ρa = σ 2 and a(t, x i (t)+) -a(t, x i (t)-) proportional to β i (t) (for any i ≤ i ≤ I and any t ∈ [0, T ]).

Inhomogeneous Brownian spider ?

Another natural extension of our construction of the time inhomogeneous skew Brownian motion would be to adapt the ideas developed in the previous section to construct inhomogeneous Brownian spiders.

In 1965, Itô and McKean [81, Section 4.2, Problem 1] introduced the simple but intriguing diffusion process that they called skew Brownian motion. This process was revisited by Walsh [START_REF] Walsh | A diffusion with discontinuous local time[END_REF] in 1978. Walsh introduced it as a Brownian motion with excursions around zero in random directions on the plane. The random directions are values of a random variable in [0, 2π) that are independent for different excursions with a constant value during each excursion. This "definition" can be made precise as, e.g., in Barlow, Pitman and Yor [START_REF] Barlow | On Walsh's Brownian motions[END_REF]. This motion is now called Walsh's Brownian motion. Following [START_REF] Barlow | On Walsh's Brownian motions[END_REF] (see also

Example 1 in Evans and Sowers [START_REF] Evans | Pinching and twisting Markov processes[END_REF]), we may consider a version of Walsh's Brownian motion which lives on N semi-axes on the plane, called the "legs of the spider", that are joined at the origin, the so-called Brownian spider, or Walsh's spider. Loosely speaking, this motion performs a regular Brownian motion on each one of the legs and, when it arrives to the origin, it continues its motion on any of the N legs with a given probability.

Thus, one can construct the Brownian spider by independently putting the excursions Put(0,-845.04684) from zero of a standard Brownian motion on the j-th leg of the spider with probability p j , j = 1, 2, . . . , N with N j=1 p j = 1.

One could try to generalize this framework as to include the inhomogeneous case where the probability of choosing each leg varies through time. We would call such a process, if it exists, an inhomogeneous Brownian spider. Is it possible to construct such a process ? At least intuitively, inspired by the results of the previous section, we could try to compute its transition probability function.

Let us fix N . We set

V N = v(r, j) = r exp 2iπ j N , r ≥ 0, j = 1, . . . , N , v(0) = (0, 0). (1.10)
We are given a family p = {(p j (s)) j∈{1,...,N } , s ≥ 0} of probabilities on 1, . . . , N indexed by the time satisfying N j=1 p j (s) = 1 for all s ≥ 0.

Reproducing (1.6), we would define then

p p (t, v(0), v(y, )) = t 0 ds p (s) π s(1 -s) 3 ye -y 2 /2(t-s) (1.11)
From (1.11) we could define a family of probability transition densities function satisfying the Chapman-Kolmogorov equations. In order to do so, one could rely on the computations presented in the above section for the case of the inhomogeneous skew Browian motion. The main difficulty is that we cannot hope to have a nice stochastic differential equation at our disposal (see however the results of [START_REF] Freidlin | Diffusion processes on graphs : stochastic differential equations, large deviation principle[END_REF] and [START_REF]Diffusion processes on an open book and the averaging principle[END_REF]). This comes from the fact that the filtration of a Brownian spider is no longer a Brownian filtration when N ≥ 3 (see e.g. the results in [START_REF] Barlow | Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes[END_REF], see also [START_REF] Meyer | Une simplification de l'argument de Tsirelson sur le caractère nonbrownien des processus de Walsh[END_REF]). In particular, it does not seem easy to prove with all the required rigor a last exit decomposition.

More fundamentally, an interesting extension of this research, would be to seek for theoretical results that describe Markov processes with inhomogeneous partial reflections when the underlying state space possesses a topology that is not that of R or R d (e.g. star shaped networks, graphs). We feel that the theory of Ray and Right processes, as exposed for example in [START_REF] Getoor | Markov processes : Ray processes and right processes[END_REF], provides the fundamental tools that should be used to construct such processes. In the homogeneous context, this is done abstractly in [START_REF] Salisbury | Construction of right processes from excursions[END_REF], where as an

Introduction

Consider (B t ) t≥0 a standard Brownian motion on some filtered probability space (Ω, F, (F t ) t≥0 , P) where the filtration satisfies the usual right continuity and completeness conditions. The skew Brownian motion X x,β is defined as the solution of the stochastic differential equation with singular drift coefficient

X x,β t = x + B t + βL 0 t (X x,β ), (2.1)
where β ∈ (-1, 1) is the skewness parameter, x ∈ R and L 0 t (X x,β ) is the symmetric local time at 0.

In this chapter, we are interested in the structure of the flow for solutions of (2.1).

A quite intriguing fact about these solutions is that they do not satisfy the usual flow 29 Put(0,-845.04684) Put(0,-845.04684) property of differential equations, which prevents two solutions with different initial positions to meet in finite time. Indeed, it is shown in [START_REF]Coalescence of skew Brownian motions[END_REF], that almost surely, the two paths t → X 0,β t and t → X x,β t , driven by the same Brownian motion (B t ) t≥0 , meet at a finite random time. Moreover, the law of the valued of the local times of these processes at their coalescence time is computed in [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF].

In the papers [START_REF] Gloter | Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time[END_REF][START_REF]Bouncing skew brownian motions[END_REF], we study the time dynamic of the distance between the two processes X 0,β 1 and X x,β 2 where the skewness parameters β 1 and β 2 are possibly different.

Assume x > 0, 0 < β 1 , β 2 < 1 and β 1 > β 2
1+2β 2 . We show that, looking at the distance between the two processes through a random time change and before the coalescence time, the distance between X 0,β 1 and X x,β 2 is solution to an explicit stochastic differential equation with jumps, driven by the Poisson process attached to the excursions of X 0,β 1 .

The dynamic of this stochastic differential equation enables us to compute the law of the hitting time of zero for the distance between the two skew Brownian motions in this time-scale. The other cases are handled using a similar method, only the equations change.

The stochastic differential equation satisfied by the distance process degenerates when the distance gets near 0 and the process is studied only up to its first hitting time at 0. A natural question raised by the previous study becomes the understanding of the behavior of the distance process after its first hitting time at 0. We show that the distance process is in fact a self-similar process, so that its study may be investigated using the general theory of positive self-similar processes. In the language of this theory, one of our main results asserts that the difference process after its first hit at 0 behaves as the unique self-similar Markovian extension of the difference process killed when it hits 0 that leaves 0 continuously. Although the jump measure degenerates near 0, the description of the difference process before the first hit of zero remains valid after this time. Thus, the distance process appears as a solution of a jump S.D.E. driven by the excursion process of X 0,β 1 at all times.

Once this study is achieved and as an application of our results, we manage to characterize the inhomegeneous Markovian dependence of the distance process with respect to β. In particular, we give an answer to an open question raised in [31, Open problem 1.9]. Put(0,-845.04684)

Hitting of two skew Brownian motions driven by the same underlying Brownian motion

Consider two skew Brownian motions,

X x,β 2 t = x + B t + β 2 L 0 t (X x,β 2 ) (2.2) X 0,β 1 t = B t + β 1 L 0 t (X 0,β 1 ) (2.3)
We introduce the rcll process defined as

Z x,β 1 ,β 2 u = X x,β 2 τu(X 0,β 1 ) , (2.4)
where τ u (X 0,β 1 ) is the inverse of the local time, given as,

τ u (X 0,β 1 ) = inf{t ≥ 0 : L 0 t (X 0,β 1 ) > u}.
Note that, since X 0,β 1 τu(X 0,β 1 ) = 0, we have

Z x,β 1 ,β 2 u = X x,β 2 τu(X 0,β 1 ) -X 0,β 1 τu(X 0,β 1 )
. This explains why we choose below to call Z x,β 1 ,β 2 the "distance process". Our first result shows that the "distance process" is solution to a stochastic differential equation with jumps, driven by the excursion Poisson process of X 0,β 1 0 .

Description of the distance process before its first hit at zero

Let us introduce (e u ) u>0 the excursion process associated to X 0,β 1 namely,

e u (r) = X 0,β 1 τ u-(X 0,β 1 )+r , for r ≤ τ u (X 0,β 1 ) -τ u-(X 0,β 1 ).
The point process (e u ) u>0 takes values in the space of C 0→0 of rcll excursions with finite lifetime, endowed with the usual uniform topology. We denote by n β 1 the Itô excursion measure attached to (e u ) u>0 .

Let us define

T = inf{t ≥ 0 | X 0,β 1 t = X 0,β 2 t } ∈ [0, ∞] and U = L 0 T (X 0,β 1 t
). Since 31 Put(0,-845.04684) Put(0,-845.04684) X x,β 2 and X 0,β 1 are driven by the same Brownian motion, it is easy to see that they can only meet when X 0,β 1 = 0. As a consequence, we have

U = inf{u ≥ 0 | Z x,β 1 ,β 2 u = 0} ∈ [0, ∞], and Z x,β 1 ,β 2 > 0 on [0, U ).
Our first result about Z x,β 1 ,β 2 is the following. Almost surely, we have for all t < U ,

Z x,β 1 ,β 2 t = x -β 1 t + 0<u≤t β 2 (Z x,β 1 ,β 2 u- , e u ), (2.5)
where

: (0, ∞) × C 0→0 → [0, ∞) is a measurable map.
In particular, almost surely, for

all t < U L 0 τt (X x ) = 0<u≤t (X x τ u-, e u ). (2.6)
For h > 0, we can describe the law of e → (h, e) under n β 1 by

n β 1 ( (h, e) ≥ a/β 2 ) = 1 -β 1 2h 1 + a h - 1+β 2 2β 2 , ∀a > 0. (2.7)
Remark 2.2.2. Theorem 2.2.1 fully details the dynamic of the "distance process" before it (possibly) reaches 0. The "distance process" decreases with a constant negative drift, and has positive jumps. Moreover, the value of a jump at time u is a function of the level

Z x,β 1 ,β 2 u-
and of the excursion e u .

The image of the excursion measure under this function, with a fixed level h > 0, is given by the explicit expression (2.7).

In [START_REF]Coalescence of skew Brownian motions[END_REF][START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] it is shown that the processes X 0,β 1 and X x,β 2 meet in finite time under some appropriate conditions for the skewness coefficients.

Theorem 2.2.3 ( [START_REF]Coalescence of skew Brownian motions[END_REF][START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF]). Assume x > 0 and 0

< β 1 , β 2 < 1 with β 1 > β 2 1+2β 2 . Then the hitting time T = inf{t > 0 | X 0,β 1 t = X x,β 2 t } is almost surely finite.
Remark 2.2.4. Actually in [START_REF]Coalescence of skew Brownian motions[END_REF] the case β 1 = β 2 is considered with x > 0. In [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] the situation β 1 = β 2 is treated in the case x = 0 and with the condition

β 2 1+2β 2 < β 1 < β 2 32
Put(0,-845.04684) Put(0,-845.04684) (see [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF]Theorem 1.4 iii]). Nevertheless, it is rather clear that the additional condition β 1 < β 2 is mainly related to the choice x = 0 and could be removed if x > 0.

Let us give some insight about the representation (2.6). Assume τ u -τ u-> 0, then using (2.3), we have for r ≤ R(e u ),

e u (r) = X 0 τ u-+r = X 0 τ u-+ B τ u-+r -B τ u-+ β 1 [L 0 τ u-+r (X 0 ) -L 0 τ u-(X 0 )] = B τ u-+r -B τ u-.
Recalling (2.2), we deduce,

X x τ u-+r = X x τ u-+ B τ u-+r -B τ u-+ β 2 [L 0 τ u-+r (X x ) -L 0 τ u-(X x )] = X x τ u-+ e u (r) + β 2 [L 0 τ u-+r (X x ) -L 0 τ u-(X x )] = X x τ u-+ e u (r) + β 2 L 0 r (X x τ u-+• ). (2.8)
The relation (2.8) shows that (X x τ u-+r ) r<R(eu) satisfies a skew Brownian motion type of equation, but governed by the excursion path e u , and starting from the value X x τ u-. By solving this equation, we will show that the process (X x τ u-+r ) r<R(eu) can be obtained as a functional of the excursion e u and of the initial value X x τ u-. When one can solve this equation, the process (X x τ u-+r ) r<R(eu) potentially can be obtained as a functional of the excursion e u and of the initial value X x τ u-. This suggests that (X x τ u-+r ) r<R(eu) could be obtained as a functional of the excursion e u and of the initial value X x τ u-. As a consequence the local time L 0 τu (X x ) -L 0 τ u-(X x ) will be written too as a functional (X τ x u-, e u ). Remark that in a general, it is not true that the equation X r = h + e(r) + β 2 L 0 r ( X) admits a unique solution for all h ∈ R and all e ∈ C 0→0 . This makes the rigorous construction in [START_REF] Gloter | Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time[END_REF] a bit delicate.

Hitting time at zero of the distance process

In [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] the law of U = L 0 T (X 0,β 1 ) is computed in the particular situation β 1 = β 2 . In the following theorem we give the law without this restriction.

Theorem 2.2.5. Assume x > 0 and 0

< β 1 , β 2 < 1 with β 1 > β 2 1+2β 2 . Denote U = 33
Put(0,-845.04684) Put(0,-845.04684) L 0 T (X 0,β 1 ) then the law of U has the density

p U (x, du) = 1 b(1 -ξ , 1-β 1 2β 1 ) β 1 x β 1 u x ξ -2 1 - x β 1 u 1-3β 1 2β 1 1 [ x β 1 ,∞) (u)du (2.9) where b(a, b) = 1 0 u a-1 (1 -u) b-1 du = Γ(a)Γ(b) Γ(a + b) and ξ = 1 2β 1 -1 2β 2 . Hence, x β 1 U is distributed as a Beta random variable B(1 -ξ , 1-β 1 2β 1 ).
Remark 2.2.6. For β 1 = β 2 we retrieve the result of [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF]. However, in [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] the cumulative distribution function of U was explicitly derived using a max-stability argument for the law of U . By (2.9) we see that for β 1 = β 2 the cumulative distribution function cannot be computed explicitly. Actually, arguments similar to [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] do not seem to apply directly here.

In [START_REF] Gloter | Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time[END_REF] we also give corresponding results for the hitting time of X 0,β 1 and X x,β 2 when one of the skewness parameters is negative or when both are negative. These results may be derived using the same method of proof as for Theorem 2.2.5.

The proof of Theorem 2.2.5 relies on the description of the dynamics of the distance process Z x,β 1 ,β 2 . Let us define A the generator of the process Z x,β 1 ,β 2 by

Af (h) = -β 1 f (h) + ∞ 0 [f (h + a) -f (h)]ν(h, da), (2.10) = -β 1 f (h) + ∞ 0 [f (h + a) -f (h)] κ h 2 1 + a h -γ da, (2.11)
for h > 0 and f an element of

C 1 (0, ∞) bounded on [0, ∞).
The idea is to compute the Laplace Transform of U using Dynkin's formula. We show a "Dynkin's formula" that relates the generator of the process with U and the exit time from (0, ∞). For λ > 0 we denote u λ (x) = E x [e -λU ] where the subscript x emphasizes the dependence upon the starting point of the process Z x,β 1 ,β 2 . Put(0,-845.04684) 2. The function u λ is solution to the integro-differential equation :

Au λ (x) = λu λ (x),
for all x > 0.

(2.12)

A scaling argument takes care of the first point. For the second point, we first prove using Markov's property that {M t = u λ (Z x t∧U ) e -λ(t∧U ) : t ≥ 0} is a martingale. Once this is proved we may apply Itô's formula. Using the continuity of s → Z x,β 1 ,β 1 at zero permits to retrieve the announced result.

Remark 2.2.8. The proof of Lemma 2.2.7 uses the fact that the laws of the processes

t → Z x,β 1 ,β 2 t and t → xZ 1,β 1 ,β 2 t/x
coincide until they reach zero. This is not surprising, since one can show that the compensators of the point processes

t → Z x,β 1 ,β 2 t + β 1 t and t → x(Z 1,β 1 ,β 2 t/x
+ β 1 t/x) are the same. For point processes with finite intensities, it is known that the compensator characterizes the law of the process (see [START_REF] Jacod | Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 1.26 in Chapter III]).

This will be the key point to carry on our investigations on process Z x,β 1 ,β 2 after its first hit of zero.

The integro-differential equation (2.12) can be transformed into an ordinary differential equation. Related techniques were used in [START_REF] Chen | An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model[END_REF] for computing the ruin time of Levy processes. In [START_REF] Chen | An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model[END_REF], a crucial fact is that the generator of a Levy process acts as a multiplier in the Fourier domain. Such simplifications in the Fourier domain do not occur for the generator of the process Z x,β 1 ,β 2 , however the self-similar invariance of the process (see Remark 2.2.8) suggests the use of the Mellin's transform. In fact, from (2.12) and using Mellin's transform we prove that the function x → u λ (x) is solution to

β 1 xu λ (x) + u λ (x)(λx + β 1 ξ ) -λ(γ -2)u λ (x) = 0, for all x ∈ (0, ∞), where ξ = 1 2β 1 -1 2β 2 and γ = 1+3β 2 2β 2 . Then, if we define ω λ (x) = λ x u λ -xβ 1 λ (for x < 0),
we recover a Kummer's equation for x → ω λ (x). We know from the formulae of [1] that Kummer's equation admits two independent explicit solutions given in terms of confluent geometric functions. Using the asymptotic behavior of u λ permits to select the good solution and the expression (2.9) is obtained using simple algebra. In his pioneering study [START_REF] Lamperti | Semi-stable Markov processes. I[END_REF] of the structure of self-similar Markov processes with state space [0, ∞) Lamperti posed the problem of determining those self-similar Markov processes that agree with a given self-similar Markov process up to the time the latter process first hits 0. A few years ago, Vuolle-Apiala [START_REF] Vuolle-Apiala | Itô excursion theory for self-similar Markov processes[END_REF], Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF], [132], and Fitzsimmons [START_REF] Fitzsimmons | On the existence of recurrent extensions of self-similar Markov processes[END_REF], pushed forward the studies of this problem by giving necessary and sufficient condition for the existence of such a â ȂIJrecurrent extensionâ Ȃİ that, in addition, leaves 0 continuously. To state the main results of this beautiful theory precisely, we introduce some notation and recall some of the basic theory of self-similar Markov processes. A Borel right process Z † = ((Z † t ) t≥0 , (P x ) x≥0 ) with values in [0, ∞) is self-similar provided there exists α > 0 such that, for each c > 0 and x ≥ 0, the law of the rescaled process (c -α Z ct ) t≥0 , is P x/c when Z † has law P x . The number α is the order of Z † . Thanks to [START_REF] Lamperti | Semi-stable Markov processes. I[END_REF],

we can assume that Z is a Hunt process ; thus in addition to being a right-continuous strong Markov process, the sample paths of Z are quasi-left-continuous. One of several zero-one laws developed by Lamperti states that if T 0 := inf{t > 0 : Z † t = 0} then either

P x [T 0 < ∞] = 0 for all x > 0 or P x [T 0 < ∞] = 1 for all x > 0.
Let us assume the ladder in this section.

We assume that 0 is a trap for Z † , so that each of the laws P x governing Z † is carried by

{ω ∈ Ω : ω(t) = 0, ∀t ≥ T 0 (ω)} .
The natural filtration on Ω is (G t ) t≥0 , and G ∞ := t≥0 G t . We write

P t f (x) = P t (x, f ) := P x [f (Z † t )]
Put(0,-845.04684) for the transition semigroup of Z † , and U q := ∞ 0 e -qt P t dt, (q > 0), for the associated resolvent operators.

Define, for c > 0, φ c : Ω → Ω by φ c ω(t) := c -α ω(ct). The α-self-similarity of Z † means that

φ c P x (B) := P x [φ -1 c B] = P x/c α (B) ∀B ∈ G ∞ , x ≥ 0, c > 0. Definition 2.3.1. A Borel right Markov process Z = (Z t , P x ) with state space [0, ∞) is
a recurrent extension of Z † provided :

(i) the stopped process ((Z t∧T 0 ) t≥0 , P x ) has the same law as ((Z † t ) t≥0 , P x ), for each x ≥ 0 (ii) 0 is not a trap for Z.

Note that if a recurrent extension Z is self-similar, then its order must be the same as that of Z † .

Excursions associated to markovian extensions

   Z t • θ s = Z t+s , 0 ≤ t < T 0 • θ s , 0, t ≥ T 0 • θ s ,
where θ s is the shift operator on D. Let ( 0 t (Z)) t≥0 denote the Z-local time at 0, normalized so that P 0 ∞ 0 e -t d 0 t (Z) = 1.

Then, there is a σ-finite measure n on (Ω, G ∞ ) such that for any predictable V ≥ 0 37 Put(0,-845.04684) Put(0,-845.04684) and G ∞ -measurable F ≥ 0,

E 0   s∈G V s .F (e s )   = E 0 ∞ 0 V s d 0 s (Z) .n(F ). (2.13)
This formula determines n uniquely, and under n, the process (Z t ) t≥0 is a strong Markov process.

We will use a slight extension of (2.13), namely

E 0   s∈G V s .F (s, e s )   = E 0 ∞ 0 V s n (F (s, .)) d 0 s (Z) (2.14)
holding for any measurable

F : (R × C 0→0 , B(R) ⊗ U) → R (where U stands for the Borel σ-algebra of C 0→0 ).
It is shown in [START_REF] Vuolle-Apiala | Itô excursion theory for self-similar Markov processes[END_REF] (see Theorem 1.2) that if Z is a self-similar recurrent extension of

Z † then ∞ 0 I {} (Z s )ds = 0 and either n (Z 0 > 0) = 0 or n (Z 0 = 0) = 0.
Definition 2.3.2. We say that the extension (Z t ) t≥0 leaves 0 continuously if

n (Z 0 > 0) = 0. (2.15)
2.4 Description of the distance process after its first hit at zero in the reflecting case

Throughout this section, we will make the follwing assumption

h : 0 < β 2 1 + 2β 2 < β 1 < β 2 < 1.
38 Put(0,-845.04684) Put(0,-845.04684) 2.4.1 The distance process as the self-similar Markovian extension that leaves 0 continuously In Remark 2.2.8, we saw that the distance process Z x,β 1 ,β 2 is a positive self-similar process. One of our main results asserts that the difference process after its first hitting time at zero U behaves as the unique positive self-similar Markovian extension of the process killed when it hits zero that leaves zero continuously. In particular, we show that the description of the distance process given at Theorem 2.2.1 remains valid after U . So that Z x,β 1 ,β 2 appears as a solution of a jump S.D.E. driven by the excursion process of X β 1 .

Using this description of the difference process Z x,β 1 ,β 2 on R + permits to retrieve an Itô-Dynkin formula. Using again the Mellin's transform, we manage to compute explicitely the law of Z 0,β 1 ,β 2 .

Remember that

U := inf{t > 0 : X x,β 2 τt(X 0,β 1 ) = 0}. Let Z †,x t t≤U * = X x,β 2 τt(X 0,β 1 ) t≤U *
the process killed when it first reaches 0. We may extend Z †,x t t≤U * on the whole time line [0, ∞) to a process -that we still note abusively

Z †,x t t≥0
-such that 0 is a trap for Z †,x .

We have the following result.

Theorem 2.4.1. Assume h and x > 0.

The process

Z x,β 1 ,β 2 t t≥0
is the positive Markov 1-self-similar recurrent extension of

Z †,x t t≥0
that leaves 0 continuously.

The proof of this Theorem relies on the following crucial and quite technical lemma.

Lemma 2.4.2. Assume -1 < β < 1. Under P

(3) 0 , there is strong existence and uniqueness for X 0,β (ρ) solution of the skew Brownian motion equation driven by a 3-dimensional Bessel process.

Moreover if -1 < β < 1, L 0 t X 0,β (ρ) = 0 for any t > 0, P

0 -a.s., and X 0,β t (ρ) = ρ t for any t > 0, P

0 -a.s.

Intuitively, this lemma implies that whenever X 0,β 1 and X x,β 2 meet at some starting point of an excursion with positive length, the two processes are solutions of the same Put(0,-845.04684) skew equation driven by the excursion, and the result of the lemma asserts that these solutions they remain pasted together. So that the two processes can only separate on the set of points where the local time of X 0,β 1 increases, and the two processes must separate continuously in time. For all t > 0, the law of Z 0,β 1 ,β 2 t has density

p Z (t, 0, y) = c 1 1 y tβ 1 y γ-1 1 + tβ 1 y 1-γ-ξ I y> ,
where c 1 is defined by c -1

1 := ∞ 0 z γ-2 (1 + z) 1-γ-ξ * dz = Γ(γ -1)Γ(ξ )/Γ (γ + ξ -1).
Once again, the main ingredient in the proof is that because of the self-similarity of Z, the generator A acts as a multiplier for Mellin's transform. Hence, we may apply the same method of proof as that of Theorem 2.2.5.

An explicit formula for the entrance law

Let us denote by n the σ-finite excursion measure attached to Z x,β 1 ,β 2 . Let us introduce the killing kernel operator

Q u f (x) = E x f (Z †,x u )I u<U (x) . (2.16)
We have

Q u (x, A) = Q au (ax, aA).
Let (n t (dy)) t>0 := n I et∈dy I t≤R(e) t>0 denote the family of entrance laws satisfying n t Q s = n t+s (t, s > 0) and related to n. The family of entrance measures (n t (dy)) t>0 may then be described in terms of the underlying Levy process (H t ) t≥0 thanks to the result 40 ut(0,-845.04684) Put(0,-845.04684) of [132] (See Theorem 2. formula (3)), which in our case and with our notations reads

n t (f ) = 1 t 1-ξ Γ(1 -ξ )E (J -ξ ) E f t J J -ξ , with J := ∞ 0 e -H 0 s ds (2.17)
and where E denotes the Lamperti transform of the canonical Lévy process under the h-transform probability measure (with respect to h : y → e (1-ξ )y ) of the law of (H t ) t≥0 .

In [START_REF]Bouncing skew brownian motions[END_REF], we prove the following result, which gives an explicit formula of the family of entrance laws (n t (dy)) t>0 .

Theorem 2.4.4. Assume h.

The family of entrance laws (n t (dy)) t>0 related to the description of n(de) are given by n t (dy) ∝ t γ+ξ -2 (y + β 1 t) -γ dy.

(2.18)

A study of the Markovian dependence on parameter β

For β = 0 let L 0 t := 1 × L 0 t (X 0,0 ) = L 0 t (B) and for β ∈ (0, 1) let L β t := βL 0 t X 0,β , where L 0 t X 0,β is the local time process corresponding to X β t with parameter β and starting from x = 0. Unique strong solutions exist for all rational β ∈ [0, 1) simultaneously.

Fix some rational β 2 ∈ (0, 1) ∈ Q and let β 1 < β 2 < β 3 . The process X . Here G β 2 + is the set of all s ≥ 0 such that for some 0 < g s < d s , we have s

= L 0 gs X 0,β 2 , X 0,β 2 gs = 0 = X 0,β 2 ds , X 0,β 2 v > 0 for v ∈ (g s , d s ) and e + s (u) = X 0,β 2 gs+u for u ∈ [0, d s -g s ).
We define in analogous way the Poisson point process

E -= {(s, e s ) -} s∈G β 2 - of negative excursions of X β 2 t .
The processes E + and E -are independent.

Let L β 3 s = L β 3 ,β 2 s := inf{L β 3 t : L β 2 t > s} = L β 3 τ β 2 s/β 2 -
. Define L β 1 s and L 0 s in a similar way.

We have the inequalities L β 2 t ≥ L β 1 t for all t ≥ 0 a.s. On the intervals where X β 2 t is strictly negative the value of the processes L β 2 t , L β 1 t and L 0 t do not change. In turn this implies that L β 1 s , L 0 s are measurable with respect to the filtration F + = σ{(s, e s ) + } s∈G β 2 + ,s≤u . The same reasoning ensures that L β 3 s is adapted to σ{(s, e s ) -} s∈G β 2 + ,s≤u . The random time T = inf (s :

L 0 s ≥ 1) = L β 2 τ 0 1
is a stopping time relative to F + . By independence of E + and Put(0,-845.04684) E -, the random elements L β 1

T and e + T are independent of L β 3 T given the value of

T = L β 2 τ 0 1 .
But this can be restated as the independence of

L β 1 τ 0 1 and L β 3 τ 0 1 given L β 2 τ 0 1 , since L β 3 T = L β 3 τ 0 1 and L β 1 τ 0 1 is a function of L β 1
T and e + T . This proves the Markov property for the process

β ∈ [0, 1) ∩ Q → L β τ 0 1 . Conditionally on L β τ 0 1 = a, we have that X β+ε τ 0 1 -X β τ 0 1 = X β+ε τ β a β -X β τ β a β .
In particular, conditionally on

L β τ 0 1 = a X β+ε τ 0 1 = Z β+ε a β + a. In particular for β + ε > β > β+ε 1+2(β+ε) > 0 i.e. for 2β 2 1-β > ε > 0 X β+ε τ 0 1 = Z β+ε a β + a.
Hence, we find that for any ε ∈ Q such that

2β 2 1-β > ε > 0, L X β+ε τ 0 1 |X β τ 0 1 = a = c 1 (β, β + ε) 1 y -a a y -a 1+(β+ε) 2(β+ε) 1 + a y -a -1+β 2β I y>a dy = c 1 (β, β + ε) 1 y -a a y -a 1+(β+ε) 2(β+ε) y y -a -1+β 2β
I y>a dy := q β,β+ε (a, y)dy.

A bit of algebra permits to check the validity of the Chapman-Kolmogorov equation. For

any y > a, ε ∈ Q such that 2β 2 1-β > ε > 0, and θ ∈ Q such that 2(β+ε) 2 1-(β+ε) > θ > 0 :
q β,β+ε (a, z)q β+ε,β+ε+θ (z, y)dz = q β,β+ε+θ (a, y)dy.

(2.19)

The infinitesimal generators (G β ) β∈(0,1)∩Q of the inhomogeneous Markov process β ∈ 42 Put(0,-845.04684) Put(0,-845.04684) (0, 1) ∩ Q → X β τ 0 1 may be computed. Using 1 Γ(z) ∼ 0 ze ρz (ρ is the Euler constant) :

G β g(a) = lim ε 0+ 1 ε ∞ 0 (g(a + z) -g(a)) 1 Γ 1 2β -1 2(β+ε) 1 z a z 1+(β+ε) 2(β+ε) z + a z -1+β 2β dz = ∞ 0 (g(a + z) -g(a)) 1 2β 2 1 z a a + z 1+β 2β dz.
Note that the ladder integral is not well definite in general if g is only assumed to be bounded in some neighborhood of a, so it is not clear to answer what is the domain of G β . If we assume that g is bounded and is uniformly a Hölder function in the sense that there exists η > 0 and δ > 0 such that sup

a∈R + sup z∈[0,δ[ g(a + z) -g(a) z η < ∞,
then G β g is well-defined of all β ∈ (0, 1) ∩ Q.

Further investigations ?

Once again, it appears challenging to try to prove similar results for the Brownian spider. The first enormous difficulty turns out to determine what is the driving process of a Brownian Spider ?

I have a strong feeling that 'a trajectorial construction' of the Brownian Spider should be possible by gathering iteratively together the branches of the spider with the help of N -1 independent Skew Brownian motions with respective parameters

(β 1 , β 2 , . . . , β N -1 ) := (2α 1 -1) , 2α 2 N i=2 α i -1 , . . . , 2α n-1 N i=N -1 α i - 1
These Skew Brownian motions are themselves driven by N -1 independent Brownian motions B (1) , . . . , B (N -1) .

The advantage of such a construction -if it is valid -is that the filtration of the resulting Brownian Spider would then be set to the natural filtration generated by B (1) , . . . , B (N -1) so that the measurability ambiguities due to the choice of the branch at each excursions 43 Put(0,-845.04684) Put(0,-845.04684) away from the origin are erased. If this construction is valid, then we could give some sense to the study of the distance of two spiders driven by the 'same (N -1) dimensional standard Brownian motion' and hopefully give some sense to the meaning of a possible extension of the results of this chapter.

As an example of a potential application in Biology, one could decide to model rivers and affluents by graphs ; the results of such a study might then give some information concerning the contacts at the vertices of two populations of fishes driven by the same river flows on each branch of the graph map of rivers (see [START_REF] Vasilyeva | Population dynamics in river networks : analysis of steady states[END_REF] for a PDE model along these lines). 44 ut(0,-845.04684) Put(0,-845.04684) Chapitre 3

Exact simulation for skew diffusions

This chapter is based on the papers Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process [START_REF]Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process[END_REF] and

Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift [START_REF]Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift[END_REF] written with Pierre Étoré.

Introduction

In this chapter, we present our ideas concerning the objective of producing an exact sampling method for the solution of the following skew stochastic differential equation

dX β t = dW t + b(X β t )dt + βdL 0 t (X β ), t ∈ [0, T ] (3.1)
where |β| < 1 and b is a smooth bounded function, possibly discontinuous at point zero.

We are inspired by the methodology developped in Beskos et al. [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF][START_REF] Beskos | Exact simulation of diffusions[END_REF].

Exact simulation methods for trajectories of one-dimensional SDEs have been a subject of much investigation in the last years : see for example [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF][START_REF]A factorisation of diffusion measure and finite sample path constructions[END_REF][START_REF] Beskos | Exact simulation of diffusions[END_REF][START_REF] Reutenauer | Exact simulation of prices and greeks : application to cir[END_REF][START_REF] Sermaidis | Markov chain Monte Carlo for exact inference for diffusions[END_REF]. The interest in such methods relies on the fact that unlike the classical simulation methods which all involve some kind of discretization error, the exact simulation methods do not present any discretization error, under the strong assumption that the diffusion coefficient is constant (this is usually claimed to be without loss of generality, by the use of a Lamperti transformation).

45

Put(0,-845.04684) Put(0,-845.04684) An important problem that arises when one tries to adapt the methodology of Beskos et al. to the solutions of (3.1) comes from the fact that the laws of theses solutions are no longer absolutely continuous with respect to the Wiener measure. Thus, we cannot use the Wiener measure as reference measure in the rejection sampling procedure as in [START_REF] Beskos | Exact simulation of diffusions[END_REF].

It appears that in order to adapt the method of [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF][START_REF] Beskos | Exact simulation of diffusions[END_REF] for an exact sampling algorithm for the solutions of (3.1), we have to be able to simulate bridges of the SBM with drift. This issue is raised in [START_REF]Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process[END_REF], where a solution is presented using the explicit computation of the transition density of the SBM with a constant drift component.

Unfortunately, this idea applies only in the case β = 0 because the convenient constant drift for the SBM with drift is no longer defined in the case β = 0. However, we know that X β tends strongly to X 0 as β tends to zero. This leads naturally to investigate what happens to the convergence at the level of the rejection sampling algorithms as β tends to zero. We prove that although the drift explodes as β tends to zero, the rejection bounds of the algorithm converge. So that there exists a limiting rejection algorithm as β tends to zero. The main issue becomes then to prove that this algorithm is indeed an exact rejection algorithm for the simulation of X 0 and to give an interpretation of its corresponding reference measure. We prove that the reference measure may be described in terms of a standard Brownian motion conditionned on prescribed laws for the couple of its final position and local time.

In a recent paper [START_REF] Papaspiliopoulos | Exact sampling of diffusions with a discontinuity in the drift[END_REF], propose an algorithm for the simulation of X 0 using conditioned paths of the couple (Brownian motion, its local time). However their method seems more complicated to implement and less suitable for generalizations to the inhomogeneous case or in the presence of several discontinuities.

The methods presented in this Chapter have been generalized in [START_REF] Dereudre | Exact simulation of Brownian diffusions with drift admitting jumps[END_REF] for the treatment of several (say n) local times and discontinuities of the drift (in [START_REF] Dereudre | Exact simulation of Brownian diffusions with drift admitting jumps[END_REF], the necessary explicit density formulas are computed for n = 2 and the numerical illustrations are shown in this case). Note also that in the very recent paper [START_REF] Frikha | On the weak approximation of a skew diffusion by an Euler-type scheme[END_REF], the presented Euler scheme is constructed using the fact that the solution of (3.1) may be exactly simulated.

Put(0,-845.04684)

Notations and assumptions

A time horizon 0 < T < ∞ is fixed.

We make the following assumptions -The function b : R → R is bounded, with bounded first derivatives on R +, * and R -, * with a possible discontinuity at point {0}. We suppose that the limits

for some finite constant M .

-We make the natural assumption |β| < 1 that guarantees strong existence and uniqueness for solutions of (3.1).

We set

φ(x) := b 2 (x) + b (x) 2 , φ(x) = φ(x) -m with m = inf x∈R φ(x). (3.3)
We define

θ := b(0+) -b(0-) 2 . ( 3.4)
Let µ ∈ R. We will denote by B β,µ the SBM of parameter β with drift µ. That is to say where

B β,
N c (y) = 1 √ 2π
∞ y e -z 2 /2 dz. We have the following proposition. Proposition 3.2.1. We have for all t > 0, for all x, y i nR, p β,µ (t, x, y) = p 0,µ (t, x, y)v β,µ (t, x, y).

(3.7)

We recall that C is the set of continuous mappings from [0, T ] to R, and C the Borel σ-field on C induced by the supreme norm. We denote by (C t ) t∈[0,T ] the usual canonical filtration. We denote by w = (w t ) t∈[0,T ] the canonical process.

The Wiener measure on (C, C) is denoted P and W = w (W t (w) = w t for any w ∈ C and any t ∈ [0, T ]). We will also consider Brownian motions starting from x ∈ R and will simply denote

P x = P(. | W 0 = x).
Let us define Px β (resp. Ŵx β,µ ), the probability measure induced on (C, C) by the law of X β (resp. B β,µ ) under P x . Our aim is to provide exact samples of w T under the measure Px β . Those are distributed as X β T solution of (3.1) under P x .

Main ideas 3.3.1 Case β = 0

We set Put(0,-845.04684) We have by application of Girsanov's theorem, see Theorem 3.5.1 in [START_REF]Brownian motion and stochastic calculus[END_REF] (note that Novikov's condition is verified),

µ β = 1 + β 2β b(0+) - 1 -β 2β b(0-) (3.
dX β t = dW t + b β (X β t )dt + µ β dt + βdL 0 t (X β ) = dW SD t + µ β dt + βdL 0 t (X β )
where

W SD t = W t + t 0 b β (X β s
)ds is a Brownian motion starting from x under W x β,µ β defined by

dP x dW x β,µ β = exp T 0 b β (X β t ))dW t + 1 2 T 0 b 2 β (X β t )dt = exp T 0 b β (X β t ))dW SD t - 1 2 T 0 b 2 β (X β t )dt
We may apply the symmetric Itô-Tanaka formula [130, Exercise VI.I.25], we have

B β (X β T ) -B β (X β 0 ) = T 0 b β (X β s -) + b β (X β s +) 2 dX β s + 1 2 R B β (x)L x T (X β ) = T 0 b β (X β s )I X β s = dW SD s + T 0 µ β b β (X β s )ds + β b β (0-) + b β (0+) 2 L 0 T (X β ) + R b β (x) 2 I x = L 0 T (X β )dx + b β (0-) + b β (0+) 2 L 0 T (X β ).
By the occupation time formula, we then get

B β (X β T ) -B β (X β 0 ) = T 0 b β (X β s )I X β s = dW SD s + T 0 µ β b β (X β s )ds + T 0 b β (X β s ) 2 I X β s = ds + β b β (0-) + b β (0+) 2 -βµ β + b β (0-) -b β (0+) 2 ( * )=0 L 0 T (X β ). Finally, T 0 b β (X β s )dW SD s - 1 2 T 0 b 2 β (X β s )ds = B β (X β T ) -B β (X β 0 ) - T 0 φ β (X s )ds with φ β (z) = b 2 β (z)+b β (z)+2µ β b β (z) 2 = b 2 (x)+b (x) 2 - µ 2 β 2 = φ(z) - µ 2 β 2 .
49 Put(0,-845.04684) Put(0,-845.04684) Notice that denoting m β = inf x∈R φ β (x) we have

φ β (x) -m β = φ(x) - µ 2 β 2 -inf x∈R φ(x) - µ 2 β 2 = φβ (x)
On the canonical space, these computation lead to

d Px β d Ŵx β,µ β (w) = e -m β T exp B β (w T ) -B β (x) - T 0 φβ (w s )ds .
We are now in position to apply the methodology of [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF]. Denote now by Ẑx β the probability measure induced on (C, C) by the law of B β,µ (under P x ) conditioned to B β,µ T hx β (z)dz, where h x β is a probability density. Proposition 1 in [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF] asserts that The idea is now to sample along Px using rejection sampling with Ẑx β as a reference measure.

d Ẑx β d Ŵx β,µ β (w) = h β p β,µ (T, x, .) (w T ). (3.9) Choosing then h x β (y) = C x β exp (B β (y) -B β (x)) p β,µ (T, x, y) (C x β is the normalizing constant that makes h x β a
Applying the procedure proposed in Beskos et al. [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF], we get the following algorithm.

Let K denote an upper bound for φβ (x).

EXACT SIMULATION ALGORITHM FOR

SIMULATE A SKELETON (w

t 1 , . . . , w t N , w T ) WHERE w ∼ Ẑx β . 3. IF ∀i ∈ {1, . . . , N } φ(w t i ) ≤ z i ACCEPT THE SKELETON. ELSE RE- TURN TO STEP 1.
This algorithm produces an exact sampling of X β T under P x : it is the final instance w T of an accepted skeleton.

The main issue in the above algorithm is to sample a skeleton of the canonical process under Ẑx β . Recall that under Ẑx β the canonical process follows the law of a SBM with drift whose terminal position must be sampled from a prescribed density. This makes it tractable for a numerical perspective. However, note that in the case β = 0, the constant µ β is no more defined. In fact in the case β = 0 and b(0+) = b(0-), there is no constant µ such that proceeding as in the above computations with b 0 (x) = b(x) -µ we can cancel the local time term appearing in the exponential weight. We need a specific approach to handle the case β = 0.

Case β = 0

We assume that β = 0 and that b(0+) = b(0-).

Let us apply again Girsanov's theorem. We see that X 0 is a Brownian motion under

Q x defined by dP x dQ x = exp T 0 b(X 0 t )dX 0 t - 1 2 T 0 b 2 (X 0 t )dt . (3.11)
We set B(x) =

x 0 b(y)dy. Applying the Itô-Tanaka formula

B(X 0 T ) -B(X 0 0 ) = T 0 b(X 0 t )dX 0 t + 1 2 R B (dx)L x T (X 0 ) = T 0 b(X 0 t )dX 0 t + 1 2 T 0 b (X 0 t )dt + b(0-) + b(0+) 2 L 0 T (X 0 ).
DINATES (t 1 , z 1 ), . . . (t N , z N ) (with t 1 < • • • < t N ). 2. SIMULATE A SKELETON (w t 1 , . . . , w t N , w T ) WHERE w ∼ Ẑx 0 . 3. IF ∀i ∈ {1, . . . , N } φ(w t i ) ≤ z i ACCEPT THE SKELETON. ELSE RE- TURN TO STEP 1.
Now the issue is to sample a skeleton along Ẑx 0 . The interpretation of Ẑx 0 is the following. Under Ẑx 0 , w is a Brownian motion conditioned on (w T , L 0 T (w)) ∼ h(y, )dyd with h(y, )dyd ∝ exp (B(y) -B(x) -θ ) P x w T ∈ dy, L 0 T (w) ∈ d . [START_REF] Papaspiliopoulos | Exact sampling of diffusions with a discontinuity in the drift[END_REF]. However, their method seems computationally complicated. In our paper [START_REF]Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift[END_REF], taking advantage of our algorithm to sample paths under Ẑx β (β = 0), we use convergence arguments at the level of algorithms in order to produce a limit algorithm that returns a sample along Ẑx 0 . We use the following proposition that ensures an abtract convergence at the level algorithms, when there is convergence of the underlying probability measures. Proposition 3.3.2. Assume that we have a sequence (xi n ) of probability measures on a measurable space (S, S), and ξ dom a probability measure on (S, S), satisfying for any

n ∈ N dξ n dξ dom = 1 ε n f n , with ε n > 0 and 0 ≤ f n ≤ 1. Assume that f n n→+∞ ----→ f point-wise on S.
Then, (ξ n ) converges towards a probability measure ξ satisfying

dξ dξ dom = 1 ε f, with ε = lim n→+∞ ε n .
Our basic idea is the following. If we denote by q β,µ (t, T, a, b, y) the density defined (for t < T ) by

P (a) B β,µ t ∈ dy | B β,µ 0 = a, B β,µ T = b = q β,µ (t, T, a, b, y)dy.
The function (t, y) → q β,µ (t, T, a, b, y) is the transition density function of a bridge of a SBM with drift relating points a and b in T unit time. Since B β,µ is a homogeneous Markovian process we have that

q β,µ (t, T, a, b, y) = p β,µ (t, a, y)p β,µ (T -t, y, b) p β,µ (T, a, b) .
Besides, q 0,µ (t, T, a, b, y) = q 0,0 (t, T, a, b, y), thus by Proposition 3.2.1, we have from (3.7)

53

Put(0,-845.04684) Put(0,-845.04684) that q β,µ (t, T, a, b, y) = q 0,0 (t, T, a, b, y) v β,µ (t, a, y)v β,µ (T -t, y, b) v β,µ (T, a, b) .

(3.12)

Observe that the quantity q 0,0 does not depend on β, neither on µ. Now having a close look at (3.6), we see that the quantities v β,µ appearing in (3.12) all depend on β and µ only through the product βµ. In particular, recalling our definition of µ β , we see that the quantities v β,µ β admit a limit as β tends to 0 because of the fact that the product βµ β tends to θ as β tends to 0 (even though the drift µ β alone slips to infinity). This argument is, in a nutshell, the key fact that gives the following proposition.

Numerical Experiments (case β = 0)

Example 1. We first deal with a toy example. We consider the following SDE

dX t = dW t - π 2 cos( π 5 X t )dt + βdL 0 t (X), X 0 = x 0 , (3.14)
with β = 0.6, and x 0 = 0.2. Note that, here, the drift b(x) = -π 2 cos( π 5 x) is bounded and of class C ∞ on the whole real line.

Our goal is to sample values of X T with T = 1 and (X t ) 0≤t≤T following (3. (Step 1 of the Algorithm). This can be done by rejecting standard normal random variables with mean x 0 .

We plot on Figure 3.1 (up and down figures) the histogram obtained with 10 6 simulations of X T , sampled with our exact procedure (the histogram has been normalized in order to represent an approximated density function). On the up figure we plot the approximated densities obtained with 10 6 simulations of the Euler Scheme introduced in Chapter 4, for decreasing time steps. We can observe the convergence of Euler type simulations to exact ones. Note that to have the Euler scheme fitting the exact procedure we have to take a fine time step (namely ∆t = 10 -4 ). This is because, as shown in 4, the rate of weak convergence of the Euler scheme in this situation is of order (∆t) 1/2-, for smooth initial condition.

On the down figure the histogram is compared with the approximated densities obtained with 10 6 simulations of the random walk based method studied in [START_REF] Étoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF], for decreasing space steps. Again we can observe the convergence of the process with discretization error.

One may wish to compare the confidence intervals provided by the three methods.

On Figure 3.2 we used 5.10 4 simulations for each method. Let {C i } i be the cells used to plot the normalized histograms, that is to say, to plot (normalized) estimations of the probabilities P(X T ∈ C i ) (bold lines on the graphs). Together with these histograms we plot confidence intervals at level 90% for the estimated P(X T ∈ C i )'s (shadow lines).

On these graphs the wideness of confidence intervals is nearly the same for the three Put(0,-845.04684) Exact

Euler

Random Walk (∆t = 10 -n , n = 4, 6) (h = We now have a closer look at what happens near zero. Using a fine cell C 0 = [-0.02, 0.02] and 10 6 simulations we compute confidence intervals for P(X T ∈ C 0 ), using successively the three methods (again this estimated probability has been normalized by |C 0 |). We observe that confidence intervals for the exact method and the Euler scheme with ∆t = 10 -4 have a null intersection, although ∆t = 10 -4 is a rather fine step (see Figure 3.1 ; we have checked that for cells away from zero confidence intervals match very well).

In Table 3.2 we report the CPU times needed to get the 10 6 simulations, with the three different methods (and with the different discretization steps we have used). Programs were written in C-language and executed on a personal computer equipped with an Intel Core 2 duo processor, running at 2.23 Ghz. On this example exact simulations is competitive, compared to schemes with very fine grids.

Example 2. We want now to sample along the law of the continuous Markov process with

X generated by L = 1 2 d dx a d dx • (3.
a(x) =          x 2 +x+1 (2x+1) 2 if x ≥ 0 3x 2 -x+2 (6x-1) 2 if x < 0. Note that a(0+) = 1 = 2 = a(0-). The coefficient a(x) is of class C 1 on R * ,-and R * ,+ ,
and uniformly strictly positive and bounded, which ensures the existence of X ; from the results of Chapter 4, we know that X solves

dX t = a(X t )dW t + a (X t ) 2 dt + a(0+) -a(0-) a(0+) + a(0-) dL 0 t (X), (3.16)
We plot on is always continuous. We refer to Stroock [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] for a proof based on the self-adjoint properties of these semi-groups and Nash's inequality.

Numerical Experiments (case β = 0)

Note that the simplest case of an equation of type 3.1 with β = 0 and b discontinuous at zero, is surely the so called 'Brownian motion with two valued drift solution of

dX 0 t = W t θ 0 I X  t > + θ 1 I Xt< dt, t ∈ [0, T ], (3.17)
where (θ 0 , θ 1 ) ∈ R 2 . For a general reference concerning these types of motions, we refer to [START_REF]Brownian motion and stochastic calculus[END_REF] p.440-441 or [START_REF] Karatzas | Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control[END_REF]. These motions appear in stochastic control problems (see for example [START_REF] Beneˇs | Some solvable stochastic control problems, Analysis and optimisation of stochastic systems[END_REF], [START_REF] Karatzas | Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control[END_REF]). Even though there exist explicit representation formulae for the densities of such Brownian motions with two valued drift in terms of linear combinations of convolution integrals (see [START_REF]Brownian motion and stochastic calculus[END_REF] p. 440-441), up to our knowledge there were no exact numerical simulation algorithm for such motions available in the literature by the time the article [START_REF]Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift[END_REF] was published.

Exact simulation of a Brownian motion with two-valued (or alternate) drift

In this paragraph, we choose to exhibit numerical results obtained with the exact limit algorithm for the simplest non-trivial cases (3.17). Indeed, in this symmetric case a benchmark is provided by the explicit and computable density of X T given in [START_REF]Brownian motion and stochastic calculus[END_REF] p. 440-441.

dX t = dW t ± sgn(X t )dt, X 0 = 0, corresponding to θ 0 = -θ 1 = ±1 in
Put(0,-845.04684) Exact Euler

CPU times 2111s 9521s

Table 3.3 -CPU times for 10 6 simulations of a Brownian motion with two-valued drift with θ 0 = 2 and θ 1 = -1 (x = 0.0 and T = 1).

Exact Algorithm Bridges

Acceptance Ratio 20.4% 58,6%

Table 3.4 -Acceptance ratios for the case of a Brownian motion with two-valued drift with θ 0 = 2 and θ 1 = -1 and (x = 0 and T = 1).

We draw the renormalized histogram of 10 6 samples of X T and compare it to the explicit density of X T (Figure 3.1 for the outgoing case θ 0 = 1 and Figure 3.4 for the incoming case θ 0 = -1).

In the non-symmetric case we can still use our limit algorithm but the density of X T becomes less explicit (see formula (6.5.12) in [START_REF]Brownian motion and stochastic calculus[END_REF]). Thus we will use as a benchmark the renormalized histogram of 10 6 samples of X ∆ T , where (X ∆ ) denotes an Euler Scheme with time step ∆ = T.10 -5 . We chose θ 0 = 2, θ 1 = -1, T = 1 and x = 0.0. We plot the corresponding renormalized histograms on Figure 3.5.

In Table 3.3 we report the CPU times needed to get the 10 6 samples, with the exact limit algorithm and the Euler scheme. Programs were written in C-language and executed on a personal computer equipped with an Intel Core 2 duo processor, running at 2.23 Ghz. We report in Table 3.4 the acceptance ratios.

On this example the acceptance ratios are good and the exact method is nearly four times faster than the Euler scheme with time step ∆ = T.10 -5 . Put(0,-845.04684) Figure 3.4 -Brownian motion with two-valued drift cases θ 0 = -θ 1 = 1 and θ 0 = -θ 1 = -1 (x = 0, T = 1).

CPU times 11813s 20s 12952s

Table 3.5 -CPU times for 10 6 simulations of X T for the case where b is given by (3.18) (x = 0.0 and T = 1).

Exact simulation of an SDE with a discontinuous drift coefficient

We consider now the SDE (3.1) with

b(x) =          -π 2 cos π 5 x if x ≥ 0 3π 2 -π 2 cos π 5 x if x < 0. (3.18)
Let 0 < T < ∞. We wish to sample along X T .

We have θ = -3π/4 and

φ(x) = b2 (x) + b (x) 2 + π 2 20 .
We take K = 2π 2 + π 2 10 as an upper bound for φ. This allows to use the limit Algorithm. Figure 3.5 shows a comparison between a renormalized histogram of 10 6 samples of X T obtained with the exact limit algorithm, and a renormalized histogram of 10 6 samples of X ∆ T , where (X ∆ ) denotes an Euler Scheme with time step ∆. We chose x = 0.0, T = 1 and time-steps ∆ = T.10 -2 and ∆ = T.10 -5 .

In Table 3.5 we report the CPU times needed to get the 10 6 samples, with the exact limit algorithm and the Euler scheme (and, for the later one, with the different time steps we have used). We report in Table 3.6 the acceptance ratios. Though the acceptance ratios are very low, we found on this example that the exact simulation is competitive compared to schemes with very fine grids.

62

Put(0,-845.04684) Put(0,-845.04684) Exact Algorithm Bridges Acceptance Ratio 3.6% 50,7% Table 3.6 -Acceptance ratios for the case where b is given by (3.18) (x = 0.0 and T = 1).

63

Put(0,-845.04684) Put(0,-845.04684) 64

Put(0,-845.04684) Put(0,-845.04684) Chapitre 4

Euler scheme for skew diffusions related to divergence form operators

In this chapter we briefly present the results of the preprint A transformed stochastic Euler-scheme for multidimensional parabolic transmission PDE [START_REF]A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF], which may be viewed as an extension of [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF] to the multidimensional case.

Preliminaries

In this section, we present the results of the preprint [START_REF]A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF] written with P. Étoré. We construct a probabilistic numerical method for solutions of multidimensional parabolic diffraction problems (in divergence form) and provide an analysis of the error corresponding to this Euler type scheme.

The divergence form operator now writes

A = ∇. [a(x)∇] ,
where the matrix valued function a : R d -→ R d ⊗ R d is discontinuous along hypersurfaces. In the case of discontinuity hypersurfaces with smooth boundary, a suitable paramatrization allows one to locally map the boundary into a hyperplane with a nice smooth diffeomorphism. The difficulty arises in the fact that the problem cannot be reduced to a one-dimensional framework since the matrix a(x) need not be diagonal. In the case of 65 Put(0,-845.04684) Put(0,-845.04684) a diagonal matrix, this method was followed by Bossy et al. [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] to construct stochastic representation of the Poisson-Boltzmann equation with piecewise constant function a(x) and prove the convergence of the related Walk on Spheres algorithm.

Here, we present an Euler type scheme that overcomes the problem of having a diagonal matrix. We prove a convergence rate of √ ∆t for this numerical method when the initial condition of the parabolic PDE associated to A belongs to some iterate space of the domain of the operator.

∀x ∈ R d , ∀ξ ∈ R d , λ|ξ| 2 ≤ ξ * a(x)ξ. (4.1)
To the coefficient matrix a, we may associate a closed symmetric Dirichlet form

(E, D(E)) defined on L 2 (R d ) by    D [E] = H 1 (R d ), E(u, v) = (a∇u, ∇v) u, v ∈ D [E] .
(see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]).

On the underlying Hilbert space L 2 (R d ), we denote by (A, D(A)) the self-adjoint operator associated to (E, D [E]) and characterized by

   D(A) ⊂ D [E] , E(u, v) = -(Au, v) u ∈ D(A), v ∈ D [E] .
Note that by the one-to-one correspondence between Thus, it is proved that

D(A) ⊆ {u ∈ H 1 (R d ) | ∇.(a∇u) ∈ L 2 (R d )}.
Let us now give some hints for the reverse inclusion.

Let f ∈ {u ∈ H 1 (R d ) | ∇.(a∇u) ∈ L 2 (R d )}
. By the symmetry of the coefficient matrix a and successive integration by parts, it is not hard to prove that for any v ∈ [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). So that we get the reverse inclusion

D(A), (Av, f ) = (∇.(a∇v), f ) = (v, ∇.(a∇f )) and in particular f ∈ D(A * ) def = {g ∈ L 2 (R d ) | ∃h g ∈ L 2 (R d ) s.t. (Av, g) = (v, h g ), ∀ v ∈ D(A)} (see
{u ∈ H 1 (R d ) | ∇.(a∇u) ∈ L 2 (R d )} ⊆ D(A * ) = D(A)
where the equality comes from the fact that (A, D(A)) is known to be self-adjoint.

Finally,

D(A) = {u ∈ H 1 (R d ) | ∇.(a∇u) ∈ L 2 (R d )}. (4.2)
Note that since a is only assumed to be measurable, C ∞ c -which is a core for the Dirichlet form (E, D(E)) -is not even a subset of D(A).

Since (-A, D(A)) is a self-adjoint operator on the Hilbert space L 2 (R d ) that is nonnegative definite, it admits a spectral resolution of the identity

{E γ : γ ∈ [0, ∞)} such that (-Af, g) = [0,∞) γd(E γ f, g) ∀f ∈ D(A), g ∈ L 2 (R d ) 67
Put(0,-845.04684) Put(0,-845.04684) and [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]

D(A) = f ∈ L 2 (R d ) : [0,∞) γ 2 d(E γ f, f ) < ∞ . (see
(T t f, g) = [0,∞) e -γt d(E γ f, g) ∀f ∈ L 2 (R d ), g ∈ L 2 (R d ).
Note that for any γ ≥ 0, t > 0, and any functions f ∈ L 2 (R d ) and g ∈ L 2 (R d ), we have the commutation property

(T t E γ f, g) = (E γ f, T t g) = (T t f, E γ g) = (E γ T t f, g).
Using the properties of the spectral resolution of the identity, we prove that for any

f ∈ L 2 (R d ) and any t > 0, [0,∞) γ 2 d(E γ T t f, T t f ) ≤ 4 t 2 e -2 (T t f, f ) ≤ 4 t 2 e -2 ||f || 2 2 < +∞,
which ensures that T t f ∈ D(A) for any t > 0.

From the fact that | d dt e -γt | ≤ γ is integrable w.r.t. d(E γ h, g) whenever h ∈ D(A), we deduce from the commutation property that for any f, g ∈ L 2 (R d ) and for any s > 0

- d dt (T t f, T s g) = [0,∞) γe -γt d(E γ f, T s g) ---→ t 0+ [0,∞)
γd(E γ T s f, g) = (-AT s f, g)

where the limit exists and is well defined (since we have shown that T s f ∈ D(A)).

If moreover g ∈ D [E] then - d ds (T s f, g) = - d dt (T s+t f, g)| t=0+ = - d dt (T t f, T s g)| t=0+
= (-AT s f, g) = E(T s f, g).

||∇T s f || 2 ≤ ||f || 2 √ λ s , ∀s > 0. (4.4)
In turn this estimate implies that for any f ∈ L 2 (R d ), g ∈ D [E], the function

s → E(T s f, g) is integrable on (0, t],
and from (4.3) and the right continuity of s → T s f at time s = 0+ (one may extend

T 0 f = f as long as no differentiation of s → T s f is implied at s = 0+ when f / ∈ D(A)),
we deduce the integrated version of (4.3) namely

∀f ∈ L 2 (R d ), ∀g ∈ D [E], (T t f, g) -(f, g) = - t 0 (a∇T s f, g) ds = - t 0 E(T s f, g)ds, t ∈ (0, ∞). ( 4
S 0 := µ ∈ S : sup v∈D[E]∩Cc(R d ) R d |v(x)| ||v|| E 1 µ(dx) < ∞ . 69
Put(0,-845.04684) Put(0,-845.04684) Finally, introduce

S 00 := {µ ∈ S 0 : µ(R d ) < ∞, ||R 1 µ(.)|| ∞ < ∞}.
Let us denote respectively by A + c and A + c,1 the families of all Positive Continuous Additive Functionals (PCAF in short) (resp. the family of all PCAF in the strict sense associated) of M (for the distinction between A + c and A + c,1 , see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] the introduction of Section 5.1).

The Revuz correspondence (see for e.g. Theorem 5.1.4 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]) asserts that there is a one-to-one correspondence (up to equivalence of processes) between A + c and S through the so-called Revuz correspondence. This correspondence permits to construct for any µ ∈ S 00 a unique PCAF in the strict

sense A ∈ A + c,1 such that ∀x ∈ R d , E x ∞ 0 e -t dA t = R 1 µ(x). (4.6)
In order to get a bijective map, introduce a new subset S For u ∈ D [E] b , we may associate a unique positive Radon measure µ u ∈ S, satisfying

R d f (x)µ u (dx) = 2E(uf, u) -E(u 2 , f ), ∀f ∈ D [E] ∩ C c (R d ).
If Put(0,-845.04684) on the above assertions, please refer to [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] Section 3.2).

Note that obviously (E, D [E]

) is strong local, so we may apply Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF].

Suppose that a function u satisfies the following conditions :

1. u ∈ D [E] b,loc , u is finely continuous on R d .
2. I G .µ u ∈ S 00 for any relatively compact open set G.

3. ∃ = (1) - (2) with I G . (1) , I G . (2) ∈ S 00 for any relatively compact open set G and

E(u, v) = ( , v), ∀v ∈ C ∞ c (R d ).
(Note that even though u is not formally in D [E], the quantity E(u, v) is welldefined because v has compact support and u ∈ D [E] b,loc ).

Let A (1) , A (2) , and B be PCAF's in the strict sense with Revuz measures (1) , (2) , and

µ u respectively. Then, u(X t ) -u(X 0 ) = M [u] t + N [u] t , P x -a.s, ∀x ∈ R d . (4.7)
Here,

N [u] = -A (1) + A (2) , P x -a.s, ∀x ∈ R d (4.8)
and M [u] is a local Additive Functional in the strict sense such that for any relatively compact set G,

E x M [u] t∧τ G = 0, ∀x ∈ G and E x (M [u] t∧τ G ) 2 = E x B t∧τ G , ∀x ∈ G,
where τ G = inf(s > 0 : X s / ∈ G) stands for the first leaving time from G (with the convention inf ∅ = ∞). We denote

D = D + ∪ D -= R d \ Γ ⊂ R d . For any multi-index α = (i 1 , . . . , i d ) ∈ N d and x = (x 1 , . . . , x d ) ∈ R d , we note x α the product x i 1 1 . . . x i d d and |α| = i 1 + • • • + i d .
In this subsection we adopt the notations from [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF] and do the following assumptions.

∈ Γ, γ + (x), ν(x) ≥ λ > 0 ∀x ∈ Γ, γ -(x), ν(x) ≤ -λ < 0. Proposition 4.1.4. Assume (D) and (Γ). There is constant R > 0 such that : 1. (a) for any x ∈ V - Γ (R), there are unique s = π γ + Γ (x) ∈ Γ and F γ + (x) ≤ 0 such that : x = π γ + Γ (x) + F γ + (x)γ + (π γ + Γ (x)) ; (4.9) (b) for any x ∈ V + Γ (R), there are unique s = π γ - Γ (x) ∈ Γ and F γ -(x) ≤ 0 such that : x = π γ - Γ (x) + F γ -(x)γ -(π γ - Γ (x)) ; ( 4
γ + Γ (x) is called the projection of x on Γ parallel to γ + : this is a C 4 function on V - Γ (R) ; (b) the function x → π γ - Γ (x) is called the projection of x on Γ parallel to γ -: this is a C 4 function on V + Γ (R) ; 3. Let us set F γ ± (x) = F γ ± (x)||γ ± (π γ ± Γ (x))
|| the normalized version of F γ ± corresponding to the unit vector field γ± : x → γ ± (x) ||γ ± (x)|| . (a) the functions x → F γ ± (x) are called the algebraic distance of x to Γ parallel to γ ± (to γ± ) : these are C 4 functions on V ∓ Γ (R). One has

F γ + , F γ + ≤ 0 on V - Γ (R) and F γ -, F γ -≤ 0 on V + Γ (R). (b) It is possible to extend F γ + , F γ + and F γ -, F γ -to C 4 b (R d , R) functions, with the conditions F γ ± , F γ ± > 0 on D ± and F γ ± , F γ ± < 0 on D ∓ .
4. The above extensions for F γ ± and F ν can be performed in a way such that the functions F γ ± and F ν are equivalent in the sense that

1 c 1 d(x, Γ) = 1 c 1 |F ν (x)| ≤ F γ ± (x) ≤ c 1 |F ν (x)| = c 1 d(x, Γ) for all x ∈ R d (4.11)
for some constant c 1 > 1.

For

x ∈ Γ, ∇ F γ ± (x) = ν * ν, γ± (x)
. (4.12)

We sometimes use the notation ν(x) or γ

± (x) even if x / ∈ Γ. For x ∈ V ± Γ (R), we set ν(x) = ν(π γ ± Γ (x)) and γ ± (x) = γ ± (π γ ± Γ (x)) and for x / ∈ V ± Γ (R)
, arbitrary values are given. Let us now assume that Assumption (D) is fulfilled and that a

ij ∈ C 1 b (D + ; R) ∪ C 1 b (D -; R)
for all 1 ≤ i, j ≤ d and a ij possesses a possible discontinuity on Γ. Our objective is to apply the results of Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] in this context for the coordinate functions p k (x 1 , . . . , x d ) := x k (k ∈ {1, . . . , d}) and to show that the Hunt process M associated to (E, D [E]) is a diffusion and possesses the following Skorokhod decomposition :

X k t = x k + t 0 d j=1 √ 2σ kj (X s )dW j s + t 0 d j=1 ∂ j a kj (X s )I Xs∈D ds - 1 2 t 0 (γ +,k (X s ) + γ -,k (X s )) dK s , t ≥ 0, P -a.s., ∀x ∈ R d . ( 4
∈ D [E] b such that p k = f k on G. Let M [f k ] the square bracket of M [f k ] .
Then, an easy computation shows that the energy measure of

M [f k ] is µ f k (dy) = µ M [f k ] (dy) =< a(y)∇f k (y), ∇f k (y) > R d (dy)
and we know that µ p k = µ p k on G. It is easy to show that I G .µ p k is a finite Radon measure belonging to S 00 so that µ p k is a smooth measure.

Moreover, using the First Green Identities (extended version), we have for any v ∈

C ∞ c (R d ) : E(p k , v) = E + (p k,+ , v) + E -(p k,-, v) = D + (-A + p k,+ )v -B + ν p k,+ , γ(v) Γ + D - (-A -p k,-)v + B - ν p k,-, γ(v) Γ = ( k , v) with k (dy) := d j=1 ∂ j a kj (y)I y∈D (dy) -[(γ + ) k + (γ -) k ] (y)σ(dy).
It remains to show that I G . k belongs to S 00 .

Since ||∂ j a kj I D || ∞ < ∞, it is not difficult to prove that the measure ∂ j a kj (y)I y∈D (dy) is a smooth measure. Let us now turn to the surface measures [(ã ± ν) k ] dσ. Using Aronson's gaussian estimates (see [5] or [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]), we may proceed in the same manner as in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] (Example 5.2.2 p.255) to prove that these measures belong to S 00 .

Put(0,-845.04684) 4.2 Smoothness properties for solutions a parabolic transmission problem Throughout this paper we make the following assumption. We denote γ : H 1 (D ± ) → H 1/2 (Γ) the usual trace operator on Γ.

For a point x ∈ Γ we denote by ν(x) ∈ R d the unit normal to Γ at point x, pointing to D + .

We denote

D = D + ∪ D -= R d \ Γ ⊂ R d . For u ∈ C(R d ; R) we denote for a point y ∈ Γ u(y±) = lim z→y , z∈D ± u(z). For u ∈ C(R d ; R) ∩ C 1 (D; R) we denote ∇ x u = ( ∂u ∂x 1 , . . . , ∂u ∂x d ) * and, for a point y ∈ Γ ∇ x u ± (y) = lim z→y , z∈D ± ∇ x u(z), (4.14)
if this limit exists.

For a vector field

G ∈ C(R d ; R d ) ∩ C 1 (D; R d ) we denote by ∇ • G its divergence i.e. ∇ • G = d i=1 ∂G i ∂x i .
Let a(x) be a matrix valued and time homogeneous diffusion coefficient. In the following we will always make the below defined ellipticity assumption.

∈ R d , ∀ξ ∈ R d , λ|ξ| 2 ≤ ξ * a(x)ξ. (4.15)
In the sequel we will frequently note a ± the restrictions of a to D ± .

We define the unbounded operator A :

D(A) ⊂ L 2 (R d ) → L 2 (R d ) by D(A) = u ∈ H 1 (R d ) with d i,j=1 D i (a ij D j u) ∈ L 2 (R d ) and ∀u ∈ D(A), Au = d i,j=1 D i (a ij D j u).
Let us introduce now the iterated domains defined recursively by

D(A k ) = {v ∈ D(A k-1 ) : Av ∈ D(A k-1 )}, k ≥ 2. For u ∈ C(R d ; R) ∩ C 2 (D; R) we denote Lu(x) = ∇ • (a(x)∇ x u(x)) , ∀x ∈ D. ( 4
(P T )                                  ∂ t u(t, x) -Lu(t, x) = 0 ∀(t, x) ∈ (0, T ] × D a + ∇ x u + (t, y) -a -∇ x u -(t, y), ν(y) = 0 ∀(t, y) ∈ (0, T ] × Γ ( ) u(t, y+) = u(t, y-) ∀(t, y) ∈ [0, T ] × Γ u(0, x) = u 0 (x) ∀x ∈ R d .
We will say that (t, x) → u(t, x) is classical solution to (P T ) if it satisfies

u ∈ C [0, T ]; C 2 ( D+ ) ∩ C 2 ( D-) ∩ C 1 [0, T ]; C( D+ ) ∩ C( D-) ∩ C [0, T ]; C(R d )
and satisfies the following requisites. First, u satisfies the first line of (P T ), where the derivatives are understood in the classical sense. Second, for all 0 < t ≤ T the function

x → u(t, x) is such that the limits lim z→y , z∈D ± ∇ x u(t, z) exist for all y ∈ Γ and satisfy the transmission condition ( ). Third, u is continuous accross Γ (third line). Fourth, it satisfies the initial condition at the fourth line of (P T ).

We use the Hille-Yosida theorem in L 2 (R d ), together with some Sobolev embedding arguments. This requires to study in a first time the associated resolvent equation, of (elliptic) type u -Au = f (4.17)

(in particular for a source term f ∈ L 2 (R d ) we will seek for a solution u in D(A) to (4.17).

Following closely [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] and then applying the Hille-Yosida theorem in L 2 (R d ), we prove the following result. -Denote

k 0 =    d 4 + 2 if d is even ; 3 2 + d/2 2 + 2 if d is odd. (4.18)
Assume that the coefficients a ij satisfy (a

± ) ij ∈ C 2k 0 -3 b (D ± ). For u 0 ∈ D(A k 0 ) the 77
Put(0,-845.04684) Put(0,-845.04684) parabolic transmission problem (P T ) admits a classical solution.

-Furthermore, if u 0 ∈ D(A k ) for k ≥ k 0 and the coefficients a ij satisfy (a

± ) ij ∈ C 2k-1 b (D ± ), this classical solution u is such that u ∈ C k-j [0, T ] ; C n(j) ( D+ ) ∩ C n(j) ( D-) , d/4 ≤ j ≤ k with n(j) = 2j -d 2 .
Remark Here we can notice that we have asked additional smoothness on the (a ± ) ij 's inside the D ± 's. This is because our technique of proof is very different : the use of the Hille-Yosida theorem and Sobolev embeddings requires this smoothness, unlike the use of the parametrix method in the classical case. But by doing so, we fully treat the transmission condition aspects in a modern way.

Going a bit further in the analysis, and using additional Sobolev embedding arguments, we can state the following results. for any j ≤ p and any multi-index α, with |α| ≤ q. random times between each sub-period of the considered unstable financial economy are very difficult to know. Thus, it might be useful to search for optimal strategies when the model leading the financial economy is perfectly known and afterwards to compare them with the sub-optimal allocation policies obtained when one uses erroneously calibrated parameters. Other questions of interest appear when we consider the following fact : even if the underlying unstable financial economy market evolves in continuous time, the data concerning its evolution can only be observed and collected at discrete times in practice.

In particular, a trader can only reasonably make his decisions and give his orders to buy or sell at discrete times. This is a difficulty that mathematicians have to face since the optimal strategies concerning mathematical financial models are given theoretically in continuous time in general. Thus, it might be also useful to introduce mathematical tools in order to evaluate the manner in which the time step affects the continuous theoretical optimal strategy. In particular, a natural question is to seek for numerical schemes that approximate the continuous theoretical optimal strategy in an appropriate way, so that it minimizes the effect of time discretization in both ways (the time discretization due to observations/decisions and the time discretization due to the numerical procedure used when we implement the theoretical optimal strategy in a computer). These questions have been investigated in [START_REF] Blanchet-Scalliet | Technical analysis techniques versus mathematical models : boundaries of their validity domains, Monte Carlo and quasi-Monte Carlo methods[END_REF][START_REF] Blanchet-Scalliet | Technical analysis compared to mathematical models based methods under misspecification[END_REF] in a particular case : the authors assume that the prices of the risky asset evolve according to a log-normal model whose return rate performs a rupture occurring at a random time that cannot be directly observed.

In the paper Misspecified Filtering Theory applied to Optimal Allocation Problems in Finance written with Sylvain Rubenthaler and Etienne Tanré [START_REF] Martinez | Approximations of a continuous time filter. Application to optimal allocation problems in finance[END_REF], we consider the case of an asset whose instantaneous expected rate of return changes at unknown random times. We give results concerning :

-A strategy or allocation procedure which is optimal when the parameters of our mathematical model are perfectly specified and calibrated : the result is given in continuous time. We will see that this optimal allocation procedure can be described in terms of a mathematical object arising in filtering theory : a continuous time filter that satisfies a stochastic differential equation.

-Mathematical strategies in misspecified situations :

-the underlying market is observed only at multiples of some time step ;

Put(0,-845.04684) -the decisions of the trader can only be taken at discrete times ; -the parameters of the model are not exactly known, we make some errors of calibration.

In this case we have to deal with a mathematical object that corresponds intuitively to some kind of misspecified filter. Our problem, in which an asset has an instantaneous rate of return changing at unknown random times, is in relation with the rupture detection.

One can quote the reference book [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems : some statistical aspects, Analysis and optimization of systems[END_REF] on the subject. There are some differences between our setting and the framework of [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems : some statistical aspects, Analysis and optimization of systems[END_REF]. First, we work in continous time and [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems : some statistical aspects, Analysis and optimization of systems[END_REF] is entirely written for discrete time models. Second, we want to detect the changes in the return rate with the objective to maximize our wealth whereas [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems : some statistical aspects, Analysis and optimization of systems[END_REF] has the objective to optimize other quantities. Finally, we suppose the dynamic of the return rate is completly known which is not the case in [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems : some statistical aspects, Analysis and optimization of systems[END_REF].

To be more specific, in [START_REF] Martinez | Approximations of a continuous time filter. Application to optimal allocation problems in finance[END_REF] we analyse the performances of various approximations of the continuous time filter F t = P(µ t = µ 1 | F X T ) where (µ(t)) t≥0 is a Markov process which takes only two real values µ 1 and µ 2 , and the observation process describing the unstable market is given by

X t = x + T 0 µ(s)ds + σB t .
We study numerical approximations of (F t ) t≥0 in misspecified situations where the process X is observed at discrete times and all the parameters of the model (µ 1 , µ 2 and the jump rates λ 1 , λ 2 ) are unknown with only access to estimation of these parameters. By using the fact that the filter is a solution to a stochastic differential equation driven by the observation process, an Euler scheme is used for determining an approximation, and convergence rates are given. As a second approach a discrete time approximation is given by using an updating/prediction procedure. Our main result states that the second approach provides a better convergence rate than the Euler scheme. Finally, we give applications in allocation problems in finance.

Put(0,-845.04684) 5.2 Stochastic control problems 5.2.1 Hamilton-Jacobi systems on networks for a toy traffic/maintenance model

In the paper A piecewise deterministic Markov toy model for traffic/maintenance and associated Hamilton-Jacobi integrodifferential systems on networks [START_REF] Goreac | A piecewise deterministic Markov toy model for traffic/maintenance and associated Hamilton-Jacobi integrodifferential systems on networks[END_REF] written with D.

Goreac and M. Kobylanski, we study an optimal control problem in infinite horizon when the dynamics belong to a certain class of piecewise deterministic Markov processes (PDMP) constrained to a star-shaped traffic network.

Imagine a regulator who observes the traffic process X that has to remain on some star-shaped network containing several edges that meet at a common intersection. The regulator has the possibility to intervene in the regulation by imposing speed limits via some control. At the same time, the functionality of the networks evolves stochastically and damages a random set of specific edges. The dynamics evolve with a pure jump component Γ. and the damage occurs exponentially distributed with a parameter λ(X, Γ, α) depending on the traffic, on the previous state of the network, and on the regulator's control policy α. In this context of controlled switched PDMP, the regulator seeks to minimize the following discounted operating cost v δ (x, γ) := inf α,X x,γ,α ∈network E e -δt Γ x,γ,α t (X x,γ,α t , α t ) dt where γ denotes the state of the network in at the initial time.

Several recent results are available on this subject when dealing with deterministic systems (see for e.g. [2][3][4] and [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]). These works rely either on Bellman's approach or on Perron's method for the existence of solutions of the associated Hamilton-Jacobi equation and propose several methods for the uniqueness part.

Let us emphasize that random switch processes are often used to model various problems arising in biology [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF], finance [START_REF] Rolski | Stochastic processes for insurance and finance[END_REF], and also communication networks [9,[START_REF] Graham | Interacting multi-class transmissions in large stochastic networks[END_REF] (see also our paper [START_REF] Goreac | A piecewise deterministic Markov toy model for traffic/maintenance and associated Hamilton-Jacobi integrodifferential systems on networks[END_REF] and the references therein).

Put(0,-845.04684) We introduce the following Hamilton-Jacobi integro-differential system δv δ (x, γ) + sup where (λ, Q) are the characteristics of the jump process that govern the damages and reparations of the road, f γ governs the dynamics between the jumps and A γ,x is the set of admissible controls depending on the state of the network : for available (active) roads, the controllability assumptions are standard ; but entering inactive roads from the intersection is prohibited and vehicles on inactive roads should turn back to the intersection.

Under standard assumptions on the functions f γ , the rates λ, the transition kernel Q and the cost functions γ , we prove that the value function v δ is a bounded continuous generalized viscosity solution of (5.1).

The difficult part in proving this result is to check that the assumptions yield a dynamic programming principle and the uniform continuity of the value function v δ .

In a subsequent part of the paper, we focus on different other notions of uniqueness.

We prove that the value function is the pointwise supremum over regular subsolutions.

Then, we adapt Kylov's "shaking the coefficients" method (see [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF], [START_REF] Krylov | On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients[END_REF]) to exhibit a sequence of regular subsolutions of our Hamilton-Jacobi system converging to the initial control problem. The uniqueness is retrieved by looking at the dual value which allows to state that the initial value function is indeed the pointwise supremum over regular subsolutions.

Further investigations ?

Further lines of possible investigations would be to study two types of stochastic controlled systems -Find a stochastic formulation of a second order control problem on a junction, in finite time horizon, with controls at the junction point (see [START_REF] Ernst | Exercising control when confronted by a (Brownian) spider[END_REF] for the problem of controlling a Brownian spider). A first work has been proposed for the control of a diffusion on a junction by Isaac Wahbi in a recent preprint, but several technical Put(0,-845.04684) difficulties arise in the definition of the underlying stochastic control process (the least of all to make sure that the set of controls is not empty). We have now started to work together with Isaac Wahbi in order to dig further these issues.

-Stochastic control problems with partial reflection and controllability at the permeable boundary. With the help of Dan Goreac and Magdalena Kobylanski, a mid-term objective would be to extend the work [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under constraints[END_REF] in the context of solutions of partially reflected multidimensional stochastic differential equations of type (4.13).

In my opinion and up to my knowledge, a lot of work has still got to be done in this direction. 92 ut(0,-845.04684) Put(0,-845.04684)

  the multidimensional case the divergence operator now writes L = ∇.(a∇) and the symmetric elliptic matrix valued function a : R d -→ R d ⊗ R d is now discontinuous along the regular boundary of some nice connected open set in R d .

Theorem 2 . 2 . 1 .

 221 [70, Theorem 1, Proposition 2] Assume x > 0 and 0 < β 1 , β 2 < 1.

Lemma 2 . 2 . 7 (

 227 Dynkin's formula). 1. The function x → u λ (x) belongs to C ∞ (0, ∞). It satisfies lim x→0 u λ (x) = 1, and |u λ (x)| ≤ e -λx/β 1 . Moreover the derivatives of u λ decay exponentially near ∞ and satisfy x k u (k) λ (x) = O(1) near 0 (for any k ≥ 0). 34 Put(0,-845.04684)

  35ut(0,-845.04684) Put(0,-845.04684) Before presenting the results of[START_REF]Bouncing skew brownian motions[END_REF], we need to introduce the basic concepts of the theory regarding positive self-similar Markovian processes.2.3 About positive self-similar Markov processes 2.3.1 Positive self-similar extensions of self-similar Markov processes killed at zero

  Let M denote the closure of the zero set {t ≥ 0 : Z t = 0} and let G denote the set of strictly positive left endpoints of the maximal open sets in the complement of M . The excursions of Z from 0 are indexed by elements of G. The excursion e s associated with s ∈ G is the D valued path defined by e s (t) :=

2. 4 . 2

 42 The law of the distance process started at 0 Once Theorem 2.4.1 is proved, we can derive an Itô-Dynkin formula, which is the key to recover the following result. Theorem 2.4.3. Assume h.

β 2 t 2 +

 22 is a Skew Brownian Motion. Let E + = {(s, e s ) + } s∈G β be the Poisson point process of positive excursions of X β 2 t

  lim x→0+ b(z) = b(0+) and lim x→0-b(z) = b(0-) exist and are finite. The value b(0) of the function b at 0 is of no importance and can be fixed arbitrarily to some constant (possibly different from either b(0+) and b(0-)). We denote by b the derivative of b. We have sup z∈R |b(z)|, sup z∈R +, * ∪R -, * |b (z)| ≤ M

8 )

 8 and then b β (x) = b(x) -µ β and B β (x) = x 0 b β (z)dz. 48 Put(0,-845.04684)

  density), and settingc x β = (C x β ) -1 e -m β T ,

  A SKEW DIFFUSION SOLUTION OF (3.1) starting from x in the case β = 0 1. SIMULATE A POISSON POINT PROCESS WITH UNITY DENSITY ON [0, T ] × [0, K]. THE RESULT IS A RANDOM NUMBER N OF POINTS OF COOR-50 Put(0,-845.04684) Put(0,-845.04684) DINATES (t 1 , z 1 ), . . . (t N , z N ).

Remark 3 . 3 . 1 .

 331 Note that in the case b(0±) = b(0), we simply have µ β = b(0). Note that the definition of b β allows to get rid of the local time term involved in the exponential martingale of Girsanov's theorem, after the application of the symmetric Itô-Tanaka formula.

51 Put

 51 x (w) = e -mT exp B(w T ) -B(x) -θL 0T (w) expw) = C x exp B(w T ) -B(x) -θL 0 T (w) (C x is a normalizing constant)t )dt ≤ 1 with c x 0 = (C x ) -1 e -mT .Therefore the following exact algorithm in order to sample X 0 T under P x .Let K denote an upper bound for φ(x).EXACT SIMULATION ALGORITHM FORA DIFFUSION SOLUTION with DISCONTINUOUS DRIFT (3.1) starting from x (case β = 0) 1. SIMULATE A POISSON POINT PROCESS WITH UNITY DENSITY ON [0, T ] × [0, K]. THE RESULT IS A RANDOM NUMBER N OF POINTS OF COOR-

Proposition 3 . 3 . 3 .

 333 We have, Combining the result of Proposition 3.3.3 with the abstract convergence result of Proposition 3.3.2 paths the way to perform Step 2 of the Exact Simulation Algorithm in the case β = 0.

2 .Figure 3 . 1 -

 231 Figure 3.1 -Normalized histogram of the positions at time T = 1.0 of 10 6 paths of the solution of (3.14) starting from x 0 = 0.2 : exact versus Euler with time step ∆t = 10 -n , for n = 2, 4 (up) and exact versus random walk with space steps h = 1 10 , 1 200 (down).

Figure 3 . 2 -Figure 3 . 3 -

 3233 Figure 3.3 -DIVERGENCE FORM OPERATOR : normalized histogram of the positions at time T = 1 of 10 7 paths of approximated solutions of (3.16) starting from x 0 = 0.0. Exact versus random walk with space step h = 3.10 -3 and Euler scheme with ∆t = 10 -4 .

Figure 3 . 58 Put 1 .

 3581 3 the histogram of 10 7 simulations of X T for x 0 = 0.0 and T = 1, obtained from the exact procedure. We plot on the same figure the histograms obtained with the Euler scheme and the random walk approximation mentioned in Example 1. Note that, at least graphically and contrary to what we can see on Figures 3.1 and 3.4, the transition density plotted on Figure3.3 seems to be continuous at 0 : this matches the well-known theoretical result, which asserts that the transition density of diffusion semigroups corresponding to elliptic divergence form operator of the form(3.15)

Figure 3 . 5 -

 35 Figure 3.5 -Limit algorithm v.s. Euler Scheme for Brownian motion with two-valued drift with θ 0 = 2 and θ 1 = -1 and where b is given by (3.18) (x = 0 and T = 1).

4. 1 . 1

 11 Dirichlet form and Markovian semigroup associated to elliptic divergence form operators Let a : R d → R d ⊗ R d a uniformly bounded measurable symmetric coefficient matrix satisfying the ellipticity condition : Assumption 4.1.1. (E) : There exist λ > 0 such that

  (A, D(A)) and (E, D [E]), we know that D(A) ⊆ D [E] = H 1 (R d ) (see [68] Corollary 1.3.10 p.21).Put(0,-845.04684) By the very definition of (A, D(A)), for any f ∈ D(A) and any g ∈ H1 (R d ) -(Af, g) = E(f, g) = (a∇f, ∇g), = -(∇.(a∇f ), g),where ∇.(a∇f ) may be understood first in the distributional sense as an element ofH -1 (R d ).In turn (by the density of H 1 (R d ) in L 2 (R d )) this equality permits to identify for any f ∈ D(A),Af = ∇.(a∇f ) = d i=1 d k=1 ∂ i (a ik ∂ k f ) ,and consequently the distribution ∇.(a∇f ) may be regarded as a function in L 2 (R d ).

(4. 3 )

 3 And since -d ds (T s f, g) = -d ds (T s g, f ) by the symmetry property of T s , we deduceE(T s f, g) = E(f, T s g) for any f, g ∈ D [E].Consequently, for any f ∈ D [E] and using the ellipticity of the coefficient matrix a, 68 Put(0,-845.04684) Put(0,-845.04684) we deduce the fundamental estimate

. 5 ) 4 . 1 . 2

 5412 Representation of the Hunt process associated to elliptic divergence form operators using the Revuz correspondence for additive functionals Since (E, D [E]) is a regular Dirichlet form (with the space D [E] ∩ C c (R d ) or C ∞ c (R d ) as a special standard core, see e.g. Exercice 1.4.1 in [68]), we are in position to apply Theorem 7.2.1 p. 380 of[START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]. We may associate to (E, D [E]) a Hunt process, symmetric w.r.t the Lebesgue measure (dx) on R d . We shall denote this Hunt processM = (Ω, (F t ) t≥0 , F, (P x ) x∈R d ).Let us also denote by {R α : α > 0} the Markovian resolvent kernel of the Markovian transition function {p(t, x, dy) := p(t, x, y)dy : t > 0}. Then, for anyα > 0, f ∈ B b (R d ) and x ∈ R d , R α f (x) = R d r α (x,y)f (y)dy with r α (x, y) = ∞ 0 e -αt p(t, x, y)dt. Denote by S the set of positive Radon measures on (R d , B(R d )). For µ ∈ S define R 1 µ(x) = R d r 1 (x, y)µ(dy) (x ∈ R d ) and introduce the subset of finite energy measures

1 (

 1 1 of S defined by µ ∈ S 1 if there exists a sequence (E n ) n≥0 of Borel finely open sets increasing to R d satisfying that I En .µ ∈ S 00 for each n. Then, there is a one-to-one correspondence between S 1 and A + c,up to equivalence) which is given by relation (4.6) whenever µ ∈ S 00 . S 1 is called the set of smooth measures in the strict sense. Let us introduce D [E] b,loc the space of essentially bounded functions belonging locally to D[E] (u ∈ D [E] b,loc if for any compact set G, there exists a bounded function ω such that u = ω, (dx)-a.e. on G).

  D [E] b,loc , we may construct µ u ∈ S with the help of a sequence (G n ) n≥0 of relatively compact open sets such that G n ⊂ G n+1 and n≥0 G n = R d . Let (u n ) n≥0 a sequence of functions in D [E] b satisfying u n = u on G n . There is no ambiguïty in defining µ u = µ un on G n because the construction is consistent (since µ un = µ u n+1 on G n . (For an account 70 Put(0,-845.04684)

3

 3 Skorokhod representation of the Hunt process associated to transmission operators in divergence form Consider R d = D+ ∪ D -with D + and D -two open connected subdomains separated by a transmission boundary Γ that is to say Γ = D+ ∩ D-.

Assumption 4 . 1 . 2 .

 412 (D) : The transmission boundary Γ is bounded and of class C 5 . Assumption 4.1.3. (Γ) : The co-normal vector fields x → γ + (x) := a + (x)ν(x) and x → γ -(x) := -a -(x)ν(x) are of class C 5 and satisfy ∀x

  a) the function x → π

4. 2 . 1

 21 Framework and assumptionsConsider R d = D+ ∪ Dwith D + and D -two open connected subdomains satisfying ∅ = D + ∩ D -separated by a transmission boundary Γ that is to say Γ = D+ ∩ D-.

Assumption 4 .

 4 2.1. (D) : The transmission boundary Γ is of class C 5 .

Theorem 4 . 2 . 3 .

 423 Let a = (a ij ) 1≤i,j≤d satisfy (E).

Proposition 4 . 2 . 5 .

 425 Let p, q ∈ N with p + q/2 ≥ 2. Let m = q 2 + d 4 , m = m + 1 and k = m + p. Assume that the coefficients a ij satisfy (a ± ) ij ∈ C 2m -1 b (D ± ), and that u 0 ∈ D(A k ).Then the classical solution u(t, x) of (P T ) constructed in Theorem 4.2.3 satisfiesu ∈ C p ([0, T ] ; C q b ( D+ ) ∩ C q b ( D-)).

Corollary 4 . 2 . 6 .

 426 In the context of Proposition 4.2.5 we havesup t∈[0,T ] sup x∈ D± |∂ j t ∂ α u ± (t, x)| < ∞

-

  f γ (x, a), Dv δ (x, γ)γ (x, a) -λ(x, γ, a) γ ∈E Q(x, γ, γ , a) v δ (x, γ ) -v δ (x, γ)

Table 3 .

 3 1 -Confidence intervals for P(X T ∈ C 0 ), with 10 6 simulations.

	1 200 )

Table 3 .

 3 2 -CPU times for 10 6 simulations of X T .

methods. Intervals are the wider for cells C i near zero.

  paragraph 1.3.4 p.18). Consequently, the family of operators {T t def = e tA : t > 0} is a strongly continuous semigroup of self-adjoint contractions acting on L 2 (R d ) ( [68] Lemma 1.3.2 p.19) and

  In the above equality W = (W 1 , . . . , W d ) stands for a d-dimensional standard Brownian motion starting from zero, and (K t ) t≥0 denotes the unique PCAF associated to the surface measure σ(dξ) ∈ S on Γ through the Revuz correspondence. (K t ) increases only at times where X lies on Γ, Xs∈Γ dK s = K t , t ≥ 0. Indeed, of course p k ∈ D [E] b,loc and p k is finely continuous on R d . Let G a relatively compact open set and a function f k

	ut(0,-845.04684) Put(0,-845.04684)
	.13)
	73

t 0 I

  4.2.4. The result of Theorem 4.2.3 has to be compared with the classical results on parabolic PDE for smooth coefficients (and with no transmission condition). In this more classical situation, a unique classical solution to the parabolic PDE exists as soon as the a ij 's are bounded and Hölder continuous and satisfy (E), and u 0 is continuous and satisfies some growth condition (see for example[START_REF] Friedman | Partial differential equations of parabolic type[END_REF] Chap. 1 or[START_REF] Lieberman | Second order parabolic differential equations[END_REF], Theorem 5.14).

Remerciements

Je suis profondément reconnaissant à Mireille BOSSY, Krzysztof BURDZY et Arturo KOHATSU-HIGA d'avoir accepté la tâche de rapporter sur mes travaux. Tâche rendue d'autant plus lourde eu égard aux délais impartis et à la situation compliquée conséquente à l'émergence du virus. Je tiens également à remercier Valentine GENON-CATALOT,

Put(0,-845.04684) 4.3 Euler scheme

Our transformed Euler scheme

From now on t = h n = T n denotes the time step of our Euler scheme. The time grid is given by (t n k ) n k=0 with t n k = T k n for 0 ≤ k ≤ n. We denote by (∆W k+1 ) n k=0 the i.i.d. sequence of Brownian increments defined by

Finally let σ : R d → R d×d a matrix valued coefficient satisfying, σσ * (x) = a(x), ∀x ∈ D.

Note that this coefficient exists because a(x) is non-negative definite for all x ∈ D.

Set (∂a(x)) j = div(x → (a 1j (x), . . . , a nj (x))).

Our stochastic numerical scheme X t k n k=0 is defined as follows X 0 = x 0 and for t ∈ (t k , t k+1 ] we set

Put(0,-845.04684) Put(0,-845.04684) We describe the behavior of our scheme near the boundary in figure 4.3.2.

Convergence rate

Following the same ideas as [START_REF]One-dimensional parabolic diffraction equations : pointwise estimates and discretization of related stochastic differential equations with weighted local times[END_REF] we manage to prove the following convergence result Theorem 4.3.1. Let 0 < T < ∞. Let f : R d → R be in the space D(A k ). We have that there is a constant K(T ) such that for all n large enough and for all x 0 in R d ,

Apart from the control bounds on the partial derivatives of the solution u(t, x) of the transmission parabolic problem, the main ingredients of the proof are the following (where we use the same notations as in the previous sections of this chapter) :

-Purposely designed Euler scheme to capture the transmission condition ( )

where we have used the vector problem solved by (F γ + , π γ + Γ ), the transmission condition ( ) and the definition of γ ± (x).

-Control of a discounted error of the distance to the boundary For all c > 0, there exists a constant K(T ) such that Together with Pierre Étoré and Pierre-André Zitt, we plan to study other possible numerical schemes for multi-dimensional divergence form operators using kinetic operators by trying to adapt and generalize the methodology presented in [START_REF] Lejay | Simulating diffusions with piecewise constant coefficients using a kinetic approximation[END_REF][START_REF]New Monte Carlo schemes for simulating diffusions in discontinuous media[END_REF] and [START_REF] Rousset | A weak overdamped limit theorem for Langevin processes[END_REF].

Another line of investigation suggested by Arturo Kohatsu-Higa would be to construct the semi-group of the process associated to a divergence form operator by extending the parametrix method to this case (see [START_REF] Kohatsu-Higa | The parametrix method for skew diffusions[END_REF] for the construction in the case of Skew diffusions and the reference paper [8]). Though difficult, one could be inspired by the computations performed in the pioneering works of Portenko [START_REF] Portenko | Diffusion processes with a generalized drift coefficient[END_REF][START_REF]Stochastic differential equations with a generalized drift vector[END_REF]. In my opinionin order to overcome the inherent difficulty due to the singularity of the drift measurethe symmetry of the operator should be better exploited, these methods should maybe be combined with some type of time reversal technique. It is impossible to specify and calibrate models which can capture all the sources of instability during a long time interval. In other words, one can only pretend to divide a long investment period into sub-periods such that, in each one of these sub-periods, the market can reasonably be supposed to follow some particular model (e.g., a stochastic differential equation with a fixed volatility function). Because of the market instability, each sub-period is short. Therefore, one can only use small amounts of data during each sub-period to calibrate the model and the calibration errors can be substantial. However, hedging strategies, portfolio management, strategies, etc. highly depend on the underlying model for the market evolution, and also on the values of the parameters involved in the model. One can conclude that, in non-stationary unstable economies, it is possible to use strategies which have been optimally designed under the assumption that the market is perfectly described by a prescribed model, but these strategies may be extremely misleading in practice because the parameters of the model in each sub-period and the 83 Put(0,-845.04684) Put(0,-845.04684)

Supervised classification of misspecified diffusion paths

In the multiclass classification framework, it is assumed that we have at our disposal a learning sample of observations that consists of N independent realizations of (X, Y ) with the feature X ∈ X and the label Y ∈ {1, . . . , K} constructed on some probability space (Ω, F, P). For a new observation X the goal is to predict the associated unobserved label Y . This is done through a classifier g : X → {1, . . . , K}. The misclassification risk of g is P(g(X) = Y ). The accuracy of the classifier is then evaluated by comparison with the Bayes classifier g * . For x ∈ X , g * (x) is defined as the maximizer over {1, . . . , K} of the conditional probabilities P(Y = k|X = x). Moreover, the Bayes classifier minimizes the misclassification error over the set of all classifiers (see for e.g. [START_REF] Devroye | A probabilistic theory of pattern recognition[END_REF][START_REF] Vapnik | Statistical learning theory[END_REF]). Therefore, the performance of an arbitrary classifier g is measured by considering the excess risk P(g(X) = Y ) -P(g * (X) = Y ). In statistical learning, the joint distribution of (X, Y ) is unknown. Consequently, based on the learning sample, the objective is to build an empirical classifier g such that the expectation of its excess risk tends to zero as N tends to infinity (consistency).

Within this context, the present work focuses on the case where the feature X = (X t ) t∈[0,T ] is a diffusion process solution of some stochastic differential equation (s.d.e.) with an unknown drift function depending on the label Y . This kind of functional random data is widely used to model the behavior of an agent that produces real valued stochastic data features along time. Such type of random data are used in many domains such as medical sciences [START_REF] Donnet | A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models[END_REF], physics [START_REF] Parisi | Supersymmetric field theories and stochastic differential equations[END_REF], financial mathematics [START_REF] Karoui | Backward stochastic differential equations in finance[END_REF].

We propose statistical classification strategies based on the learning sample and relying on the diffusion model assumption. Naturally, our classification procedures involve drift coefficient estimators. One specificity of the paper is that diffusions are sampled at high frequency (time step ∆) over a fixed time interval [0, T ].

Motivation and state of the art

The classification problem for diffusion sample paths may be regarded as a particular case of functional data analysis problems. Many methods have been developed to solve such problems in general [START_REF] Ramsay | Applied functional data analysis : methods and case studies[END_REF][START_REF] Wang | Review of functional data analysis[END_REF]. Among all these methodologies, we may mention Put(0,-845.04684) k-nearest neighbors in Hilbert spaces [START_REF] Biau | Functional classification in hilbert spaces[END_REF][START_REF] Biau | Rates of convergence of the functional k-nearest neighbor estimate[END_REF] that could be applied to our classification problem. There is also some recent developments for related problems such as functional random forests [START_REF] Gregorutti | Grouped variable importance with random forests and application to multiple functional data analysis[END_REF], functional principal component analysis, kernel estimators, just to mention a few of them. Recent works on depth classification for functional data [START_REF] Cuevas | Robust estimation and classification for functional data via projection-based depth notions[END_REF][START_REF] Kuelbs | Limit theorems for quantile and depth regions for stochastic processes[END_REF][START_REF] Lange | Fast nonparametric classification based on data depth[END_REF][START_REF] López-Pintado | Depth-based classification for functional data[END_REF] propose various elegant computational solutions. These methods have the strong robustness of not specifying any model on the data, which makes them very interesting for practitioners. However, the counterpart is that the convergence may be difficult to obtain.

Let us now talk about specific methods for our classification problem of diffusion sample paths. Not too far from our problem, we mention [7] that studies supervised classification for a family of Gaussian processes and [START_REF] Delattre | Mixtures of stochastic differential equations with random effects : application to data clustering[END_REF] that uses mixed stochastic differential equations in order to solve a data clustering issue. Closer to our problem [START_REF] Denis | Classification in postural style based on stochastic process modeling[END_REF] investigates multiclass classification for Cox-Ingersoll-Ross processes.

To the best of our knowledge, the main theoretical contribution for the classification problem of general diffusion sample paths discriminated by the drift function is [START_REF] Cadre | Supervised classification of diffusion paths[END_REF].

The obtained results focus on binary classification for continuous observations and rely on the empirical risk minimization strategy. The author provides a consistent empirical classification rule. However, the resulting procedure cannot be implemented in practical situations.

Note that, by itself, the interesting question of providing an estimation of the drift coefficient from the observation of a single trajectory has been thoroughly studied. In the frequentist framework, one can cite for example [START_REF] Bibby | Martingale estimation functions for discretely observed diffusion processes[END_REF] and [START_REF] Kessler | Estimating equations based on eigenfunctions for a discretely observed diffusion process[END_REF] for martingale estimation functions, [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] in the low frequency context. In the case of continuous ergodic diffusions, the LAN properties are treated in [START_REF] Gobet | Lan property for ergodic diffusions with discrete observations[END_REF][START_REF] Yoshida | Estimation for diffusion processes from discrete observation[END_REF]. Nonparametric estimators are proposed for example in [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF][START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF]. This work takes place in the high-frequency observation framework. The sampling interval ∆ between successive observations is assumed to tend to zero as the number of observations n tends to infinity. The length of the observation time interval T = n∆ is supposed to be fixed and we do not assume any stationary property for the underlying process. In this context it is well-known that a consistent estimation of the drift from a single trajectory is impossible. However, in our framework we take advantage of the Put(0,-845.04684) repeated observations of the learning sample to derive consistent estimators of the drift.

Main contribution

We provide a closed formula for the optimal Bayes classifier which yields an explicit representation for the excess risk of a general classifier. Thus, the relation between the conditional probabilities P(Y = k|X) and the vector b of unknown drift functions is fully explicit. Our strategy relies on the plug-in principle (see for e.g. [6]). Based on an estimator b of b, we consider an estimator of the conditional probabilities. Then, for each estimator b, we consider the empirical classifier g := g b defined as the maximizer of the estimated conditional probabilities. The major part of the paper is then devoted to show that plug-in classification procedures derived from drift coefficient estimators are indeed consistent. In particular we first exhibit a sufficient condition on the estimator b which ensures the consistency of the resulting procedure. Secondly, we construct an estimator based on the minimization of the empirical risk over the learning sample. We show the consistency of this new procedure. Under mild assumptions, we show that the rate of convergence is comparable to the one obtained in [START_REF] Cadre | Supervised classification of diffusion paths[END_REF] but in the multiclass context with discrete observations. A substantial part of the paper is devoted to the study of the parametric case. We study minimum contrast estimators of the parameters that rule the drift and show their consistency and asymptotic normality. The resulting plug-in classification procedure is then shown to be consistent. Furthermore, we propose to use a convex version of the empirical risk minimizer which involves convex surrogates of the misclassification risks [START_REF] Bartlett | Convexity, classification, and risk bounds[END_REF][START_REF] Zhang | Statistical analysis of some multi-category large margin classification methods[END_REF]. We present here two new easily implementable classifiers and prove their consistency.

In comparison to [START_REF] Cadre | Supervised classification of diffusion paths[END_REF], the present work brings three main extensions. The first one is the generalization of the binary missclassification problem for diffusion paths to the corresponding multiclass classification problem. The second one is the discrete setting of our framework. Closer to reality, we assume that the data collected are recorded at discrete times. This introduces an additional error term due to the time step and we give the order of this additional error in the rates of convergence. Thirdly, in the parametric setting, we exhibit procedures that are easily implementable. We present convincing numerical results on some classical examples.

Put(0,-845.04684) Supervised classification deals with predicting the unknown nature Y of an observation X taking values in some metric space H. Generally, the label Y is assumed to take values in {1, . . . , K} called classes or labels. The objective of the statistician is to create a classification rule g : H → {1, . . . , K} that represents his guess on the label Y for the observation X. Since X does not fully characterize Y in practice, an error occurs. This error can be measured by the probability of error associated to g namely L(g) = P(g(X) = Y ).

In fact, one can show that the best possible rule is given by Bayes' rule defined by g (x) = argmax i∈{1,...,K} P(Y = i | X = x).

This means that Bayes' risk L(g ) satisfies L(g ) ≤ L(g) for any other possible rule g.

The supervised classification problem is to construct a rule ĝn based on a learning sample D n of independent random variables (X 1 , Y 1 ), . . . , (X n , Y n ) with same distribution as (X, Y ), whose performance is close to that of Bayes rule.

Further investigations

Since the redaction of these notes, our paper has been accepted for publication in the Scandinavian Journal of Statistics. Together with Christophe Denis and Charlotte Dion, we now aim to use the learning sample to adapt known results concerning the statistical inference of coefficients for diffusion processes in a non-parameter setting (see e.g. [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF]) when the diffusion process X is only observed at discrete times. Once the coefficients are properly estimated, we plan to use a data-driven procedure based on filtering to construct our decision rule. A pre-print of our work is now available at https ://hal.archives-ouvertes.fr/hal-02528092/.