S. Zein started his post-doc in September 2007 on the application of optimization algorithm for the method of random walk on rectangles using importance sampling techniques . The idea is then to look for optimal parameters through an automatic procedure in order to reduce the variance of the Monte Carlo estimator, or to simulate rare events.

A. Lejay and S. Maire have worked on Monte Carlo computations of the first eigenvalue of linear operators like the Laplace operator or the neutron transport operator in a bounded domain . This method relies on a branching mechanism and improves our previous algorithm , : The method is more robust, more accurate and provides the first eigenelement of the adjoint operator.

A. Lejay and S. Maire are developing a Monte Carlo method to simulate a diffusion process in discontinuous media. This method relies on a kinetic approximation of the diffusion operator.

All these methods are motivated by problems in physics and finance: estimation of the electrical conductivity in the brain (MEG problem), simulation of diffusions in heterogeneous media, computation of effective coefficients for PDEs at small scales, computation of option prices, ...

In numerous applications, one chooses to model a complex dynamical phenomenon by stochastic differential equations or, more generally, by semimartingales, either because random forces excite a mechanical system, or because time–dependent uncertainties disturb a deterministic trend, or because one aims to reduce the dimension of a large scale system by considering that some components contribute stochastically to the evolution of the system. Examples of applications respectively concern mechanical oscillators submitted to random loading, prices of financial assets, molecular dynamics.

Of course the calibration of the model is a crucial issue. A huge literature deals with the statistics of stochastic processes, particularly of diffusion processes: However, somewhat astonishingly, it seems to us that most of the papers consider that the dimension of the noise is known by the observer. This hypothesis is often questionable: there is no reason to a priorifix this dimension when one observes a basket of assets, or a complex mechanical structure in a random environment. Actually we were motivated to study this question by modelling and simulation issues related to the pricing of contracts based on baskets of energy prices within our past collaboration with Gaz de France.

Jointly with J. Jacod (Université Paris 6), A. Lejay and D. Talay have tackled the question of estimating the Brownian dimension of an Itô process from the observation of one trajectory during a finite time interval. More precisely, we aim to build estimators which provide an “explicative Brownian dimension” r_B: a model driven by a r_BBrownian motion satisfyingly fits the information conveyed by the observed path, whereas increasing the Brownian dimension does not allow to fit the data any better. Stated this way, the problem is obviously ill posed, hence our first step consisted in defining a reasonable framework to develop our study. We developed several estimators and obtained partial theoretical results on their convergence. We also studied these estimators numerically owing to simulated observations of various models.

The Feynman-Kac formula is a well-known tool to achieve stochastic representations of the pointwise solution of numerous partial differential equations like diffusion or transport equations. In , the authors introduced sequential Monte Carlo algorithms to compute the solution of the Poisson equation with a great accuracy using spectral methods. In , S. Maire and E. Tanré have proposed new variants of the Euler scheme and of the walk on spheres method for the Monte Carlo computation of Feynman-Kac representations. We optimize these variants using quantization for both source and boundary terms. Numerical tests are given on basic examples and on Monte Carlo versions of spectral methods for the Poisson equation.

Beskos et al. , proposed an exact simulation algorithm for one dimensional SDE with constant diffusion coefficient and restrictive assumptions on the drift coefficient. E. Tanré and V. Reutenauer (Calyon) have adapted the results. We extend this algorithm to more general SDEs and we apply it to SDEs with non Lipschitz coefficients (for instance, the CIR model for interest rates). During her internship supervised by E. Tanré, I. Henchiri has performed numerical tests on this subject.

The phenomenon of coagulation was first described by the Polish physicist Marian von Smoluchowski in 1916, in a paper on the precipitation in colloidal suspensions. It applies and is studied in many domains as: the formation of stars and planets, the behaviour of fuels in combustion engines, the polymerisation phenomenon, etc. Our expertise in this domain is now developed in a new direction.

More precisely, M. Deaconu and E. Tanré are trying to adapt their methodology for a problem arising in the Chilean copper industry. Copper ore is ground in crushers by using steel balls, the aim being to fragment the ore by using the least amount of energy. A stochastic algorithm that evaluates the time spent in the crusher has been already constructed.

A new study is underway, aiming to construct a model which effectively illustrates the phenomenon of fragmentation and accounts for new parameters, such as the position of particles in the crusher, the crusher geometry, the shape, size and number of steel balls used as projectiles and the damage factor. This factor takes into account the effect of low-impact collisions, which may occur before particle fracture. A balance must be found between the model's complexity and its ability to approximate the copper industry crushers. A successful outcome will allow us to optimize crushers efficiency at a minimal cost.

E. Tanré participated to the supervision of the Master internship of Tomás Neumann (Pontificia Universidad Catòlica de Chile) on this subject, defended in June 2007.

In her Ph.D. thesis supervised by M. Bossy and D. Talay, A. Ganz studies stochastic particle numerical methods to approximate the limit, when tgoes to infinity, of the McKean Vlasov partial differential equation in dimension 1.

We suppose that bsatisfies a monotonicity condition, and that is a non-linear interaction kernel and a small perturbation of b.

Previously, we have used the method introduced in and written V( x, t)as a cumulative distribution function of a nonlinear stochastic process X_t. We have also established the existence and uniqueness of the equilibrium measure of X_t, and obtained an exponential convergence rate to the stationary measure in the 2-Wasserstein distance. We have estimated the convergence rate of to the distribution function of the equilibrium measure, where is the discretization by Euler scheme for the particle system associated to X_t. We also obtained a bound for the norm of the approximation error.

In 2007 we completed the study of the dimension 1: we established the convergence to the equilibrium measure in the norm and obtained a bound for the approximation error in the norm. In and norms, our approximation converges with a rate proportional to the discretization time step, inversely proportional to , and with an exponential decay in T.

In , under the assumptions of rare mutations and large population, N. Champagnat obtained a microscopic interpretation of a jump process of evolution where the evolution proceeds by a sequence of mutant invasions in the population, in the specific case where only one type can survive at any time. This process, called “trait substitution sequence” (TSS) of adaptive dynamics, is the basis of the theory of adaptive dynamics . This theory provides in particular an interpretation of the phenomenon of evolutionary branching, where the distribution of the phenotypic traits under consideration in the population, initially concentrated around a single value, is driven by selection to divide into two (or more) clusters corresponding to sub-populations with different phenotypes, that stably coexist and are still in competition.

In collaboration with M. Benaïm (Université de Neuchatel, Switzerland) and S. Méléard (École Polytechnique, Palaiseau), we continued the study of a similar scaling limit of an individual-based evolutionary system, in the case where coexistence is possible. We were able to justify the convergence to a generalized TSS under the assumption of symmetric competition, thanks to results from dynamical systems. From this generalized TSS, we were also able to mathematically justify the evolutionary branching criterion proposed by biologists in the limit of small mutation, using results on the asymptotic dynamics of 3-dimensional competitive Lotka-Volterra systems . We are preparing a publication.

In collaboration with A. Bovier (WIAS, Berlin), N. Champagnat is studying the different time scales of evolution as a function of the parameters of a similar model as the one of the previous paragraph. Three parameters are involved: the population size, the mutation probability and the size of mutations. When the first parameter goes to infinity and the two others go to zero, three distinct time scales appear. The first one corresponds to directional evolution, the second one to fast evolutionary branching and the last one to slow evolutionary branching. In the first phase, the number of coexisting types is (essentially) constant and in particular cannot increase. The types evolve by maximizing their relative “fitness” according to an ODE known as the “canonical equation of adaptive dynamics” . Once they reach an equilibrium, evolutionary branching occurs on the second time scale similarly as in the previous paragraph. In particular we are able to compute the asymptotic branching time. Finally, on the last time scale, non-local evolutionary branching occurs. Here again, we are able to compute the asymptotic behaviour of the branching using large deviations arguments. All these time scales have been mathematically justified when the initial population is monotype in a simplified model. Some technical difficulties, linked with stability properties of dynamical systems and with convergence of stochastic processes on long time scales, still remain for the generalization of our results to more realistic models and to multitype populations.

N. Champagnat and Amaury Lambert (Laboratoire d'écologie, ENS Paris) have recently obtained  the counterpart of the TSS described above when the population size is finite. A diffusion process has also been obtained when the size of the jumps in the TSS vanishes, together with analytical expressions of its coefficients as series. From a biological viewpoint, this diffusion process quantifies the interplay between genetic drift and directional selection, and allows us to give an interpretation of the phenomenon of punctualism, where the evolution of the population is an alternance of long phases of stasis (constant state) and short phases of quick evolution.

We are now working on the numerical computation of the coefficients of the diffusion. They involve the first-order derivatives of the probability of invasion of a mutant with respect to the mutant birth, death and competition rates. We were able to decompose these derivatives into five fundamental components that allow one to determine which mutant type is more likely to invade the population (a problem related to notion of robustness of the population). This numerical study was initiated in  and allowed to obtain the convergence of the canonical diffusion to the canonical equation of adaptive dynamics mentioned above when the population size goes to infinity. We also plan to apply our numerical results to a bistable ecological model with two habitats.

In collaboration with P. Del Moral (INRIA Bordeaux) and L. Miclo (CNRS, Marseille), N. Champagnat and P.-E. Jabin are working on nonlinear Feynman-Kac formulæ giving a probabilistic interpretation of PDEs of the form

The Laplacian corresponds to mutations in the phenotype space, and the nonlinear term corresponds to birth and death with interaction. The parameter scales the mutation size. Barles and Perthame showed that, in very specific cases, when 0, converges to a single Dirac mass which support is characterized by a Hamilton-Jacobi PDE with constraint on its maximum. Using large deviations results, our probabilistic interpretation gives a variational characterization of the solution of this Hamilton-Jacobi equation, that could allow one to extend the results of Barles and Perthame to cases where several phenotypes can coexist in the population, i.e. where evolutionary branching can occur.

In collaboration with P. Diaconis (Stanford University and Université de Nice – Sophia Antipolis) and L. Miclo (CNRS, Marseille), N. Champagnat is studying two-types birth and death processes conditioned on non-extinction. The motivation is to study the competition between two populations (for example a resident one and a mutant one) and to study which population survives when the whole population does not go extinct for a long time. We obtained criterions on the eigenvalues of sub-matrices of the transition matrix that characterizes the cases where one population goes extinct or when both populations survive. In the symmetric cases, where both populations are identical, we obtained a full description of the eigenvalues of the transition matrix in terms of the monotype transition matrix. We next aim at studying small deviations from the symmetric case to identify the largest eigenvalue and the corresponding asymptotic behaviour in the neighbourhood of symmetry.

In  , N. Champagnat and S. Méléard (École Polytechnique, Palaiseau) have studied some large population scalings of individual-based ecological birth-death-mutation-competition models where each particle moves in the (physical) space according to a diffusion reflected at the boundary of a fixed domain. A particular care is put on the spatial interaction range, in order to recover integro-PDE models with local spatial interactions proposed by Desvillettes et al. . Several examples are given, illustrating spatial and phenotypic clustering and the importance of such models, that combine space and evolution, for the study of invasion phenomena.

P.-E. Jabin is mainly working on systems of differential equations with singular force fields. He was recently able to obtain quantitative estimates of compactness for solutions to ODE's with a BV field and a sort of compressibility condition (weaker than bounded divergence). More precisely it is possible to show that solutions to

_tX( t, x) = b( X( t, x)), X(0, x) = x,

satisfy a uniform bound on a quantity like

provided that the field is BV and the Jacobian of the flow is bounded. The above quantity was already introduced in a paper by Crippa and DeLellis , but with a W^{1, p}condition on the field. This bound also implies well posedness for the system and it improves the well-known result of Ambrosio which required a bounded divergence.

Moreover, in collaboration with Julien Barré (Université de Nice – Sophia Antipolis) and Maxime Hauray (Université Paris 6), we used this idea of quantitative estimates for the study of interacting particles. Considering a number of particles interacting with each other through a potential (electrostatic for example), the question is to obtain the limit when the number of particles goes to infinity. This limit is classically a kinetic equation in the phase space but a rigorous proof was not known whenever the potential was singular. We prove that in a suitable sense, for almost all initial conditions, the trajectories of almost all particles are close to the trajectories predicted by the mean field limit.

M. Bossy, N. Champagnat, P.-E. Jabin, S. Maire and D. Talay are working in collaboration with T. Malliavin (Institut Pasteur, Paris) on a probabilistic interpretation of the Poisson-Boltzmann equation of molecular dynamics that allows one to compute the electrostatic potential around a bio-molecular assembly (for example a protein). We also aim at developing specific Monte Carlo methods linked to the singularities of the model (operator with a divergence form with discontinuous coefficients, singular source terms, and discontinuous non-linear terms).

Motivated by applications to risk analysis in Finance and reliability problems in Random Mechanics, P.-L. Lions and D. Talay are getting precise estimates on lower bounds of certain hypo-elliptic diffusion processes. Lower bounds of Gaussian type are well known for strongly elliptic diffusions. The techniques are very different in the hypo-elliptic case.

In his Ph.D. thesis supervised by M. Bossy and D. Talay, J.-F. Jabir studies theoretical problems involved by ( ) in the case where the pressure gradient is removed while kand are supposed to be constants.

M. Bossy and J.-F. Jabir study the well-posedness of a simplified version of equation ( ) (in particular, without the pressure term) where is replaced by the kernel . The conditional expectation involved in ( ) implies that this equation is a McKean-Vlasov equation with singular kernels. When bis a bounded continuous function, we proved the well-posedness of the Lagrangian system. In particular, the existence result has been obtained by studying related smoothened interacting particle systems and by proving a propagation of chaos result.

Motivated by the downscaling application (see Section  ), we construct and investigate the well-posedness of Langevin system confined within a regular domain of with the imposed boundary condition

for denoting the outward normal unit vector related to . The basic idea of our construction is to add a “confinement” term, , to the velocity dynamic at each time the particle hits the boundary. Assuming that the law of the Lagrangian system admits a trace component along the boundary with suitable integrability properties, the law of satisfies the specular boundary condition (see )

and show that ( ) implies ( ).

When , (strong) existence and uniqueness for the “confined” Langevin system have been established in the simplified case of classical SDE with a constant diffusion thanks to the results of Lachal . Moreover, under suitable conditions on ₀and the drift coefficient we also show that the law of satisfies ( ) and ( ). These results have been also extended for the class of McKean-Vlasov equations with smooth and bounded kernels.

When is a bounded regular domain of , in collaboration with P.-L. Lions, we study the Fokker-Planck equation associated to

When bis bounded, we establish the well-posedness of this equation using Maxwellian super-solutions and sub-solutions which give upper and lower bounds for the solution.

We continue our collaboration with A. Rousseau (MOISE Inria Grenoble – Rhône-Alpes) and F. Bernardin (CETE Clermont-Ferrand) on this topic (see section ). We combine a “fractional step” method and a “particles in cell” method to simulate the time evolution of the Lagrangian variables of the flow described by equation ( ).

The most noticeable problem arises from the mass-density conservation between two time steps within a given cell. The aim is to move the particles in a box such that:

This problem is not classical at all and is referred in the literature as a problem of discrete optimal transportation, which is known to be nonlinear and numerically very hard to solve in dimension 3.

During summer 2007, A. Rousseau managed a research program at CEMRACS, and the group adapted an algorithm from D. Bertsekas , in order to compute the solution of an optimal transport problem that concerns the particles involved in SDM: see . This research program was funded by CEMRACS and INRIA (Service de Formation par la Recherche). Next January, C. Chauvin, from the CEMRACS group, will start a 12-months post-doc position in the TOSCA and MOISE teams.

In collaboration with R. Gibson (Zürich University), C. Blanchet and S. Rubenthaler (Université de Nice – Sophia-Antipolis), B. de Saporta (Université Bordeaux 4), D. Talay and E. Tanré elaborate an appropriate mathematical framework to develop the analysis of the financial performances of financial techniques which are often used by the traders. This research is funded by NCCR FINRISK (Switzerland) and is a part of its project “Conceptual Issues in Financial Risk Management”.

In the financial industry, there are three main approaches to investment: the fundamental approach, where strategies are based on fundamental economic principles, the technical analysis approach, where strategies are based on past prices behavior, and the mathematical approach where strategies are based on mathematical models and studies. The main advantage of technical analysis is that it avoids model specification, and thus calibration problems, misspecification risks, etc. On the other hand, technical analysis methods have limited theoretical justifications, and therefore none can assert that they are risk-less, or even efficient.

Consider an unstable financial economy. It is impossible to specify and calibrate models which can capture all the sources of instability during a long time interval. Thus it is natural to compare the performances obtained by using erroneously calibrated mathematical models and the performances obtained by technical analysis techniques. To our knowledge, this question has not been investigated in the literature.

We deal with the following model for a financial market, in which two assets are traded continuously. The first one (the bond) is an asset without systematic risk. The second one (the stock) is subject to systematic risk and its instantaneous rate of return changes at independent exponentially distributed random times. The trader does not observe the times of change.

This year we extended our earlier results , by taking transaction costs into account. At any time, the trader is allowed to invest all his wealth in the bond or in the stock. This leads to a non-classical stochastic control problem. We have proved that the value function of this control problem is continuous, satisfies a dynamic programming principle, and is the unique viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equation. We developed an algorithm to compute the value function and a sub-optimal trading strategy, and compared its performance to the performance of the moving average strategy. When transactions costs are high, it is difficult for the technical analyst to outperform miscalibrated mathematical strategies. We are now working on the numerical analysis of the HJB equation. We aim to precise the rate of discretization methods.

Company strategists in the energy sector find themselves faced with new constraints, such as emission quotas, that can be exchanged through a financial market.

In collaboration with Nadia Maïzi (CMA ENSMP Sophia Antipolis) and Odile Pourtallier (CORPIN INRIA Sophia Antipolis – Méditerranée), M. Bossy and A. Floryszczak propose an approach in order to assess the impact of CO2 tax and quota levels on CO2 market prices. We model the expectation of the yield of an industrial with and without CO2 market, in order to derive the indifference price. This price is established through an optimal dynamic approach leading to the resolution of an Hamilton-Jacobi-Bellman equation. First results, based on simplified tax functions, have been presented in .

Under M. Bossy and D. Talay's supervision, M. Cissé studied the problem of artificial boundary conditions for nonlinear PDEs. The motivation of this research concerns American option pricing and the numerical resolution of the variational inequality characterizing prices of American options.

First, we have extended the theorem on existence and uniqueness of solutions of the reflected backward stochastic differential equations (RBSDE) with fixed terminal time to the case of bounded random terminal time. We obtained a probabilistic interpretation of the previous solution as a viscosity solution of variational inequalities with Dirichlet boundary conditions. We obtained a general expression of the localization error in terms of the boundary conditions.

Secondly, we were interested in variational inequalities with Neumann boundary conditions. They can be interpreted as a generalized RBSDEs coupled with a reflected forward stochastic differential equations (RSDE). We used the derivative in the sense of distributions with respect to the initial data of the RSDE and the representation of the gradient of Ma and Zhang to establish a representation theorem for the space derivative of the viscosity solution of the variational inequalities. We apply this result to get a priori estimates on the error induced in numerical simulations by artificial boundary conditions for the PDEs related to American options pricing. The differentiability of the reflected forward SDE in the sense of distributions with respect to the initial data is deduced from results by Bouleau and Hirsch on the absolute continuity of probability measures.

In his thesis under D. Talay's supervision, J. Huang continues the work on the numerical analysis of HJB equations.

As Karatzas have mentioned in , some portfolio optimization problems corresponding to the given HJB equations can also be solved by the martingale method. Then we can expect the martingale method may give a good approximation of boundary conditions for the aforementioned HJB equations. The martingale method is, however, (roughly speaking) a special application of the approach of Lagrange multipliers, which is the basic tool in nonlinear constrained optimization. The Lagrange multiplier often acts as the root of some constraint equation. This implies that in order to approximate the boundary conditions of the HJB equation by using the martingale method one has to solve the constraint equation and find the corresponding root. To this end, we invoke a particular stochastic algorithm firstly proposed by Robbins and Monro and then developed by several other authors.

Generally speaking, for a given equation h( ) = 0, the Robbins-Monro algorithm is like

where _nis the step size, y_nis stochastic and so is _n. Under some assumptions, people have proved that is asymptotically normal, where ^*is the sought-for root. We, however, are interested in the Berry-Esseen bound for the aforementioned term because this kind of bound usually provides a more accurate and neat estimate of the error term in the normal approximation. Thorough analyzes show that is a sum of two terms: one is a sum of martingale increments and the other is a random variable converging to zero almost surely. Thus our work resides in solving two problems: one is to give the Berry-Esseen bound for a sum of martingale increments and the other is to specify the convergence rate to zero of the second variable mentioned above.

P. Protter (Cornell University) and D. Talay are addressing the following question. We have the possibility of trading in a risky asset (which we refer to as a stock) with both liquidity and transaction costs. We further assume that the stock price follows a diffusion, and that the stock is highly liquid. We limit our trading strategies to those which change our holdings only by jumps (i.e., discrete trading strategies), and we begin with $0 and 0 shares, and we end with a liquidated portfolio (that is, we no longer hold any shares of the stock), on or before a predetermined ending time T. The question then is, given the structure of the liquidity and transaction costs, what is the optimal trading strategy which will maximize our gains? This amounts to maximize the value of our risk free savings account. This problem can be solved in this context if it is formulated as a non classical problem in stochastic optimal control. We prove existence and uniqueness results for the related Hamilton–Jacobi–Bellman equation.

N. Champagnat, T.-H. Mai, S. Maroso, D. Talay and E. Tanré are working on various aspects of portfolio management in incomplete markets within a contract with Natixis (see Section  ).

In collaboration with P. Patie (University of Bern, Switzerland), M. Cissé and E. Tanré solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. This generalizes a result by Beibel and Lerche , . The approach relies on a combination of Doob's h-transform, time-changes and martingales techniques. Our results combined with Patie's results allows us to treat a few examples, one of them being the evaluation of a perpetual American type option with payoff defined by

where the process Xis the instantaneous interest rate modeled by the Vasicek or Ornstein-Uhlenbeck process.

In , P. Vallois and C. Tapiero (ESSEC) have considered a memory-based persistent counting random walk, based on a Markov memory of the last event. The usefulness to some problems in insurance, finance and risk analysis are discussed.

The articles , present some method to estimate parameters in vivo uptake kinetics of photo-sensitizing drugs into cancer cells in presence of random timing errors.

The article presents a statistical analysis of sequence variations of 17 abundantly expressed genes in a large set of human EST's originating from either normal or cancer samples, which allows to show that cancer EST's have greater variations than normal for 70%of the tested genes.

Jointly with M. Yor (Université Paris 6), P. Vallois and B. Roynette have continued their study of the penalization of diffusion processes , , , . The penalization procedure allows one to modify a given process (for example a Brownian motion or a Bessel processes) in order to get a new path property (for instance boundedness or a different behavior in a neighborhood of 0). This procedure can be very interesting in modelling to go further than the approach based on diffusions.

In , , P. Vallois and B. Bérard-Bergery (Université Nancy 1) have proposed new approximation schemes related to the local time process of the standard Brownian motion. Some rates of convergence have been exhibited explicitly.