TALAY Denis

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In this paper we analyze a stochastic interpretation of the one-dimensional parabolic-parabolic Keller-Segel system without cut-off. It involves an original type of McKean-Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean-Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed for any values of the parameters of the model.

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In this paper we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi– Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterizes it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure. A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?.

This thesis studies the optimal control of McKean-Vlasov type dynamics and its applications in financial mathematics. The thesis contains two parts. In the first part, we develop the dynamic programming method for solving McKean-Vlasov type stochastic control problems. By using the appropriate admissible controls, we can reformulate the value function in terms of the law (resp. the conditional law) of the process as the only state variable and obtain the flow property of the law (resp. the conditional law) of the process, which allow us to obtain the principle of dynamic programming in all generality. Then we obtain the corresponding Bellman equation, based on the notion of differentiability with respect to probability measures introduced by P.L. Lions [Lio12] and the Itô formula for the probability stream. Finally we show the viscosity property and the uniqueness of the value function of the Bellman equation. In the first chapter, we summarize some useful results from differential calculus and stochastic analysis on the Wasserstein space. In the second chapter, we consider stochastic optimal control of nonlinear mean-field systems in discrete time. The third chapter studies the stochastic optimal control problem of McKean-Vlasov type EDS without common noise in continuous time where the coefficients can depend on the joint state and control law, and finally in the last chapter of this part we are interested in the optimal control of McKean-Vlasov type stochastic dynamics in the presence of common noise in continuous time. In the second part, we propose a robust portfolio allocation model allowing for uncertainty in the expected return and the correlation matrix of multiple assets, in a continuous time mean-variance framework. This problem is formulated as a mean-field differential game. We then show a separation principle for the associated problem. Our explicit results provide a quantitative justification for underdiversification, as shown in empirical studies.

In this work, we prove the well–posedness of a singularly interacting stochastic particle system and we establish propagation of chaos result towards the one-dimensional parabolic-parabolic Keller-Segel model.

In chemotaxis, the classical parabolic-parabolic Keller-Segel model in dimension d describes the time evolution of the density of a population of cells and the concentration of a chemical attractor. This thesis deals with the study of the parabolic Keller-Segel equations by probabilistic methods. To this end, we construct a nonlinear stochastic differential equation in the McKean-Vlasov sense whose drift coefficient depends, in a singular way, on the whole past of the marginal laws in time of the process. These marginal laws coupled with a judicious transformation allow to interpret the Keller-Segel equations in a probabilistic way. As far as the particle approximation is concerned, we have to overcome an interesting and, it seems to us, original and difficult difficulty : each particle interacts with the past of all the others through a strongly singular space-time kernel. In dimension 1, whatever the values of the model parameters, we prove that the Keller-Segel equations are well posed in all space and that the same is true for the corresponding McKean-Vlasov stochastic differential equation. Then, we prove the well-posedness of the associated system of non-Markovian and singular interacting particles. We also establish the propagation of the chaos to a unique mean field limit whose marginal laws in time solve the parabolic Keller-Segel system. In dimension 2, too large model parameters can lead to a finite time explosion of the solution to the Keller-Segel equations. Indeed, we show the well-posedness of the nonlinear process in the McKean-Vlasov sense by imposing constraints on the initial parameters and data. To obtain this result, we combine techniques of partial differential equation analysis and stochastic analysis. Finally, we propose a fully probabilistic numerical method to approximate the solutions of the two-dimensional Keller-Segel system and we present the main results of our numerical experiments.

This thesis deals with various control and optimization problems for which only approximate solutions exist to date. On the one hand, we are interested in techniques to reduce or eliminate approximations in order to obtain more precise or even exact solutions. On the other hand, we develop new approximation methods to deal more quickly with larger scale problems. We study numerical methods for simulating stochastic differential equations and for improving expectation calculations. We implement quantization-type techniques for the construction of control variables and the stochastic gradient method for solving stochastic control problems. We are also interested in clustering methods related to quantization, as well as in information compression by neural networks. The problems studied are not only motivated by financial issues, such as stochastic control for option hedging in incomplete markets, but also by the processing of large media databases commonly referred to as Big data in Chapter 5. Theoretically, we propose different majorizations of the convergence of numerical methods on the one hand for the search of an optimal hedging strategy in incomplete market in chapter 3, on the other hand for the extension of the Beskos-Roberts technique of differential equation simulation in chapter 4. We present an original use of the Karhunen-Loève decomposition for a variance reduction of the expectation estimator in chapter 2.

We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter $H$ of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ and of the Laplace transform of the first passage time of $X_H$ at a given threshold. We also present an improvement of already known Gaussian estimates on the density of $X^H_t$ to estimates with constants which are uniform w.r.t. $t$ in the whole half-line $\mathbb{R}_+ \setminus \{0\}$ and w.r.t. $H$ when $H$ tends to $\frac{1}{2}$.

The main objective of this thesis is to provide new theoretical results concerning the performance of investments based on stochastic models. To do so, we consider the optimal investment strategy in the framework of a risky asset model with constant volatility and a hidden Ornstein Uhlenbeck process. In the first chapter, we present the context and the objectives of this study. We present, also, the different methods used, as well as the main results obtained. In the second chapter, we focus on the feasibility of trend calibration. We answer this question with analytical results and numerical simulations. We close this chapter by also quantifing the impact of a calibration error on the trend estimate and exploit the results to detect its sign. In the third chapter, we assume that the agent is able to calibrate the trend well and we study the impact that the non-observability of the trend has on the performance of the optimal strategy. To do so, we consider the case of a logarithmic utility and an observed or unobserved trend. In each of the two cases, we explain the asymptotic limit of the expectation and the variance of the logarithmic return as a function of the signal-to-noise ratio and the speed of reversion to the mean of the trend. We conclude this study by showing that the asymptotic Sharpe ratio of the optimal strategy with partial observations cannot exceed 2/(3^1.5)∗100% of the asymptotic Sharpe ratio of the optimal strategy with complete information. The fourth chapter studies the robustness of the optimal strategy with calibration error and compares its performance to a technical analysis strategy. To do so, we characterize, analytically, the asymptotic expectation of the logarithmic return of each of these two strategies. We show, through our theoretical results and numerical simulations, that a technical analysis strategy is more robust than the poorly calibrated optimal strategy.

In this note, we propose two different approaches to rigorously justify a pseudo-Markov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis [9]. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.

In this paper we develop sensitivity analyses w.r.t. the long-range/memory noise parameter for solutions to stochastic differential equations and the probability distributions of their first passage times at given thresholds. Here we consider the case of stochastic differential equations driven by fractional Brownian motions and the sensitivity , when the Hurst parameter $H$ of the noise tends to the pure Brownian value, of probability distributions of certain functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$. We first get accurate sensitivity estimates w.r.t. $H$ around the critical Brownian parameter $H = \tfrac{1}{2}$ of time marginal probability distributions of $X^H$. We second develop a sensitivity analysis for the Laplace transform of first passage time of $X^H$ at a given threshold. Our technique requires accurate Gaussian estimates on the density of $X^H_t$. The Gaussian estimate we obtain in Section 5 may be of interest by itself.

We consider rate swaps which pay a fixed rate against a floating rate in presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit optimal strategy minimizing a risk measure of the hedging error. We here propose an efficient algorithm, based on the stochas-tic gradient method, to obtain an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numer-ical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost.

During the last decades, the development of technological means and particularly computer science has allowed the emergence of numerical methods for the approximation of Stochastic Differential Equations (SDE) as well as for the estimation of their parameters. This thesis deals with these two aspects and is more specifically interested in the efficiency of these methods. The first part will be devoted to the approximation of SDEs by numerical schemes while the second part deals with the estimation of parameters. In the first part, we study approximation schemes for EDSs. We assume that these schemes are defined on a time grid of size $n$. We will say that the scheme $X^n$ converges weakly to the diffusion $X$ with order $h in mathbb{N}$ if for all $T>0$, $vert mathbb{E}[f(X_T)-f(X_T^n)] vertleqslant C_f /n^h$. Until now, except in some particular cases (Euler and Ninomiya Victoir schemes), the research on the subject imposes that $C_f$ depends on the infinite norm of $f$ but also on its derivatives. In other words $C_f =C sum_{green alpha green leqslant q} Green partial_{alpha} f Green_{ infty}$. Our goal is to show that if the scheme converges weakly with order $h$ for such $C_f$, then, under assumptions of nondegeneracy and regularity of the coefficients, we can obtain the same result with $C_f=C Green f Green_{infty}$. Thus, we prove that it is possible to estimate $mathbb{E}[f(X_T)]$ for $f$ measurable and bounded. We then say that the scheme converges in total variation to the diffusion with order $h$. We also prove that it is possible to approximate the density of $X_T$ and its derivatives by that $X_T^n$. In order to obtain this result, we will use an adaptive Malliavin method based on the random variables used in the scheme. The interest of our approach lies in the fact that we do not treat the case of a particular scheme. Thus our result applies to both Euler ($h=1$) and Ninomiya Victoir ($h=2$) schemes but also to a generic set of schemes. Moreover the random variables used in the scheme do not have imposed probability laws but belong to a set of laws which leads to consider our result as a principle of invariance. We will also illustrate this result in the case of a third order scheme for one-dimensional EDSs. The second part of this thesis deals with the estimation of the parameters of a DHS. Here, we will consider the particular case of the Maximum Likelihood Estimator (MLE) of the parameters that appear in the Wishart matrix model. This process is the multi-dimensional version of the Cox Ingersoll Ross process (CIR) and has the particularity of the presence of the square root function in the diffusion coefficient. Thus this model allows to generalize the Heston model to the case of a local covariance. In this thesis we construct the MLE of the Wishart parameters. We also give the convergence speed and the limit law for the ergodic case as well as for some non-ergodic cases. In order to prove these convergences, we will use various methods, in this case: ergodic theorems, time change methods, or the study of the joint Laplace transform of the Wishart and its mean. Moreover, in this last study, we extend the domain of definition of this joint transform.

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In this note, we clarify the well-posedness of the limit equations to the mean-field N-neuron models proposed in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) and we prove the associated propagation of chaos property. We also complete the modeling issue in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.

In this article we study the convergence of a stochastic particle system that interacts through threshold hitting times towards a novel equation of McKean-Vlasov type. The particle system is motivated by an original model for the behavior of a network of neurons, in which a classical noisy integrate-and-fire model is coupled with a cable equation to describe the dendritic structure of each neuron.

In this thesis, we focus on the combination of variance reduction and complexity reduction methods of the Monte Carlo method. In a first part of this thesis, we consider a continuous diffusion model for which we build an adaptive algorithm by applying importance sampling to the Romberg Statistical method. We prove a Lindeberg Feller type central limit theorem for this algorithm. In this same framework and in the same spirit, we apply importance sampling to the Multilevel Monte Carlo method and we also prove a central theorem for the obtained adaptive algorithm. In the second part of this thesis, we develop the same type of algorithm for a non-continuous model, namely the Lévy processes. Similarly, we prove a central limit theorem of the Lindeberg Feller type. Numerical illustrations have been carried out for the different algorithms obtained in the two frameworks with jumps and without jumps.

In this paper we review some recent results on stochastic analytical and numericalapproaches to parabolic and elliptic partial differential equationsinvolving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. In the one-dimensional case existence and uniqueness results for such PDEscan be obtained by stochastic methods. The probabilistic interpretation of the solutionsallows one to develop and analyze a low complexity Monte Carlo numerical resolutionmethod. In addition, it allows one to get accurate pointwise estimates for thederivatives of the solutions from which sharp convergence rate estimates are deducedfor the stochastic numerical method.A stochastic approach is also developed for the linearized Poisson-Boltzmannequation in Molecular Dynamics.As in the one-dimensional case, the probabilistic interpretation of the solution involves the solution of a SDE including a non standard localtime term related to the discontinuity interface. We presentan extended Feynman-Kac formula for the Poisson-Boltzmann equation. Thisformula justifies various probabilistic numerical methods to approximate thefree energy of a molecule and bases error analyzes.We finally present probabilistic interpretations of the non-linearizedPoisson-Boltzmann equation in terms of backward stochastic differentialequations.

Stochastic computational modelsare used to simulate complex physical or biological phenomena and to approximate (deterministic) macroscopic physical quantities by means of probabilistic numerical methods.By nature, they often involve singularities and are subject to the curseof dimensionality.Their efficient and accurate simulation is still an open question in many aspects.The aim of this lecture is to review some recent developments concerningthe numerical approximation of singular stochastic dynamics,and to illustrate novel issues in stochastic analysis and PDE analysis thatthey lead to.

The objective of this thesis is to develop the theory of stochastic control and its applications in population dynamics. From a theoretical point of view, we present the study of stochastic control problems with finite horizon on diffusion, nonlinear branching and branch-diffusion processes. In each case, we reason by the dynamic programming method, taking care to carefully prove a conditioning argument analogous to the strong Markov property for controlled processes. The principle of dynamic programming then allows us to prove that the value function is a solution (regular or viscosity) of the corresponding Hamilton-Jacobi-Bellman equation. In the regular case, we also identify a Markovian optimal control by a verification theorem. From an application point of view, we are interested in the mathematical modeling of cancer and its therapeutic strategies. More precisely, we build a hybrid model of tumor growth that accounts for the fundamental role of acidity in the evolution of the disease. The targets of therapy are explicitly included as parameters of the model in order to use it as a support for the evaluation of therapeutic strategies.

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The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.

In this note, we rigorously justify a conditioning argument which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. To this end, we set up controlled martingale problems in an unusual way.

A rather detailed study of Markov processes with discrete state space is provided. It focuses on sample path techniques in a perspective inspired by simulation needs. The relationship of these processes with Poisson processes and with discrete-time Markov chains is shown. Rigorous constructions and results are provided for Markov process with uniformly bounded jump rates. To this end, elements of the theory of bounded operators are introduced, which explain the relation between generator and semigroup, and provide a useful framework for the forward and backward Kolmogorov equations and the Feynman–Kac formula.

In order to effectively implement Monte Carlo methods, the random approximation errors must be controlled. For this purpose, theoretical results are provided for the estimation of the number of simulations necessary to obtain a desired accuracy with a prescribed confidence interval. Therefore absolute, i.e., non-asymptotic, versions of the Central Limit Theorem (CLT) are developed: Berry–Esseen’s and Bikelis’ theorems, as well as concentration inequalities obtained from logarithmic Sobolev inequalities. The difficult subject of variance reduction techniques for Monte Carlo methods arises naturally in this context, and is discussed at the end of this chapter.

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In the first part of this thesis, we obtain the existence of a density and Gaussian estimates for the solution of a backward stochastic differential equation. This is an application of Malliavin's calculus and more particularly of a formula of I. Nourdin and F. Viens. The second part of this thesis is devoted to the simulation of a stochastic partial differential equation by a probabilistic method based on the representation of the stochastic partial differential equation in terms of a backward stochastic differential equation, introduced by E. Pardoux and S. Peng. We extend in this framework the ideas of F. Zhang and E. Gobet et al. on the simulation of a backward stochastic differential equation. In the last part, we study the weak error of the implicit Euler scheme for diffusion processes and the stochastic heat equation. In the first case, we extend the results of D. Talay and L. Tubaro. In the second case, we extend the work of A. Debussche.

This chapter develops discretization schemes for stochastic differential equations and their applications to the probabilistic numerical resolution of deterministic parabolic partial differential equations. It starts with some important properties of Ito’s Brownian stochastic calculus, and the existence and uniqueness theorem for stochastic differential equations with Lipschitz coefficients. Then, using probabilistic techniques only, existence, uniqueness, and smoothness properties are proved for solutions of parabolic partial differential equations. To this end, we show that stochastic differential equations with smooth coefficients define stochastic flows, and we prove some properties of such flows. We are then in a position to prove an optimal convergence rate result for the discretization schemes.

This thesis presents two independent research topics concerning the application of stochastic methods to problems arising from molecular dynamics. In the first part, we present work related to the probabilistic interpretation of the Poisson-Boltzmann equation which is used to describe the electrostatic potential of a molecular system. After introducing the Poisson-Boltzmann equation and the main mathematical tools used, we focus on the linear parabolic Poisson-Boltzmann equation. Before stating the main result of the thesis, we extend the existence and uniqueness results of stochastic backward differential equations. We then give a probabilistic interpretation of the nonlinear Poisson-Boltzmann equation in the form of the solution of a backward stochastic differential equation. Finally, in a second prospective part, we start the study of a method proposed by Paul Malliavin for the detection of slow and fast variables of a molecular dynamics.

From now on, Markov processes with continuous state space (\(\mathbb{R}^{d}\) for some or one of its closed subsets) are considered. Their rigorous study requires advanced measure-theoretic tools, but we limit ourselves to developing the reader’s intuition, notably by pathwise constructions leading to simulations. We first emphasize the strong similarity between such Markov processes with constant trajectories between isolated jumps and discrete space ones. We then introduce Markov processes with sample paths following an ordinary differential equation between isolated jumps. In both cases, the Kolmogorov equations and Feynman–Kac formula are established. This is applied to kinetic equations coming from statistical Mechanics. These describe the time evolution of the instantaneous distribution of particles in phase space (position-velocity), when the particle velocity jumps at random instants in function of the particle position and velocity.

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This chapter deepens the variance reduction subject, and focuses on the Monte Carlo methods for deterministic parabolic partial differential equations. This topic requires advanced notions in stochastic calculus, particularly the Girsanov theorem, which we state and discuss first. We strongly emphasize that universal techniques do not exist: most often, effective variance reduction methods depend on the numerical analyst’s knowledge and experience. We will see that it is rather easy to construct perfect variance reduction methods which are irrelevant from a numerical point of view. a contrario, the construction of an effective method often lies on the approximation of a perfect method, the approximation method needing to be adapted to each particular case. Interesting examples can be found in Duffie and Glynn (Ann. Appl. Probab. 5(4), 897–905, 1995).

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The principles of Monte Carlo methods based on the Strong Law of Large Numbers (SLLN) are detailed. A number of examples are described, some of which correspond to concrete problems in important application fields. This is followed by the discussion and description of various algorithms of simulation, first for uniform random variables, then using these for general random variables. Eventually, the more advanced topic of martingale theory is introduced, and the SLLN is proved using a backward martingale technique and the Kolmogorov zero-one law.

We first introduce some practical and theoretical issues of modeling by means of Markov processes. Point processes are introduced in order to model jump instants. The Poisson process is then characterized as a point process without memory. The rest of the chapter consists in its rather detailed study, including various results concerning its simulation and approximation. This study is essential to understand the abstract constructions and the simulation methods for jump Markov processes developed in the following chapters.

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In this thesis, we are interested in hedging derivatives in incomplete markets. The chosen approach can be seen as an extension of M. Schweizer's work on local minimization of quadratic risk. Indeed, while remaining within the framework of asset modeling by semimartingales, our method consists in replacing the quadratic risk criterion by a more general risk criterion, in the form of a convex functional of the local cost. We first obtain existence, uniqueness and characterization results for optimal strategies in a frictionless market, in discrete and continuous time. Then we explain these strategies in the framework of diffusion models with and without jumps. We also extend our method to the case where liquidity is no longer infinite. Finally, we show through numerical simulations the effects of the choice of the risk functional on the constitution of the optimal portfolio.

This thesis studies variance reduction techniques for the problem of approximating functionals of diffusion processes, motivated by applications in computational finance to derivatives pricing and hedging. The main tool is Malliavin's stochastic calculus of variations, which yields simulatable representations of both sensitivities and the optimal strategy for variance reduction. In the first part we present a unified view of the control variates and importance sampling methodologies, and give a practical factorization of the optimal strategies. We introduce a parametric importance sampling algorithm and carry out its study in detail. To solve the corresponding optimization problem, we validate two procedures based respectively on stochastic approximation and minimizing an empirical counterpart. Several numerical examples are given which highlight the method's potential. In a second part we combine integration by parts with a Girsanov transform to obtain several stochastic representations of sensitivities. Going beyond a strictly elliptic framework, we show on a class of HJM models with stochastic volatility how to efficiently construct a covering vector field in the sense of Malliavin-Thalmaier. The last chapter, of a more applied nature, deals with a practical case of pricing and hedging exotic rates options.

This work presents some results concerning deterministic and probabilistic numerical methods for partial differential equations with random coefficients, with applications to hydrogeology. We first focus on the flow equation in a porous medium in steady state with a homogeneous lognormal permeability coefficient, including the case of a weakly regular covariance function. We establish estimates in the strong and weak sense of the error committed on the solution by truncating the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates from which we derive an extension of the existing error estimate for the stochastic collocation method, as well as an error estimate for a multilevel Monte-Carlo method. Finally, we focus on the coupling of the flow equation considered above with an advection-diffusion equation, in the case of large uncertainties and a small correlation length. A numerical analysis of a numerical method for calculating the average velocity at which the area contaminated by a pollutant expands is proposed. It is a Monte-Carlo method combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.

This thesis develops a new methodology to establish analytical approximations for European option prices. Our approach cleverly combines stochastic developments and Malliavin calculus to obtain explicit formulas and accurate error estimates. The interest of these formulas lies in their computation time which is as fast as that of the Black-Scholes formula. Our motivation comes from the growing need for real-time calculations and calibration procedures, while controlling the numerical errors related to the model parameters. We treat four categories of models, performing specific parameterizations for each model in order to better target the right proxy model and thus obtain easy to evaluate correction terms. The four parts treated are: diffusions with jumps, local volatilities or Dupire models, stochastic volatilities and finally hybrid models (rate-share). It should also be noted that our approximation error is expressed as a function of all the parameters of the model in question and is also analyzed in terms of the regularity of the payoff.

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In this thesis, we give a localization error control on the system of parabolic partial differential inequalities with Dirichlet edge conditions. This error control is done via the probabilistic interpretation of the variational inequalities in the form of stochastic backward differential equations (SDFEs). Thus, the viscosity solutions of localized variational inequations with Dirichlet conditions at the edge are interpreted as solutions of the reflected EDSRs with bounded random final time. We establish an existence and uniqueness theorem for this type and give a definition to the notion of viscosity solution for our problem. In the last part of this chapter, we apply this control to the American option pricing problem. Next, we establish the almost everywhere derivability of the reflected diffusion with respect to its initial value and give the derivative in the one-dimensional case. We give the representation of the almost everywhere derivatives of the solutions of variational inequalities with Neumann edge condition. From these representations, we give the localization error over an entire portfolio of American options. In the second part, we explicitly solve the optimal stopping problem with random discounting and an additive functional as the cost of observations for a regular linear diffusion. This result generalizes the work of Beibel and Lerche who solved (1997 and 1998) this type of problem without any additional additive functional. We use in our approach the h-transformed method, the martingale technique, the time change.

In this thesis we propose a numerical approximation for the equilibrium measure of a McKean Vlasov stochastic differential equation (SDE), when the drift coefficient is given by a function with ergodic properties, which is perturbed by a Lipschitzian nonlinear interaction function. We establish a theorem of existence and uniqueness of the equilibrium measure, as well the exponential convergence rate to this equilibrium. We apply the method based on the obtention of Wasserstein contractions using the random coupling variables, as suggested by Cattiaux-Gullin-Malrieu (2006) for the convex potential drift case. After, using the particle system, the chaos propagation property and Euler’s scheme to approximate the SDE, we estimate numerically the integral of every Lipschit function w. R. T. The measure at fixed time, with a time-uniform estimation error. Then, using this numerical estimation we approximate the integral w. R. T. The equilibrium measure. Finally, in the one-dimensional case, we provide numerical estimations for the density and the cumulative distribution function of the equilibrium measure. We use the algorithm proposed by Bossy-Talay (1996) and obtain the optimal rate convergence of the approximation in different norms.

In this thesis, we are interested in the theoretical aspects of a new class of stochastic differential equations called Lagrangian stochastic models. These models have been introduced to model the properties of particles associated with turbulent flows. Motivated by a recent application of these models in the development of scale refinement methods for weather forecasting, we also consider the introduction of edge conditions in the dynamics. In the framework of McKean-type nonlinear equations, Lagrangian stochastic models designate a particular class of nonlinear dynamics due to the presence in the coefficients of conditional distribution. In simplified cases, we establish the well-posedness of these dynamics and their particle approximation. Concerning the introduction of edge conditions, we construct a stochastic confined model for the prototype condition of "no permeability on average". In the case where the confining domain is the hyperplane, we obtain an existence and uniqueness result for the considered dynamics, and show that the edge condition is satisfied. For general domains, we study the conditional McKean-Vlasov-Fokker-Planck equation satisfied by the law of systems. We develop the notions of Maxwellian over- and under-solutions, giving the existence of Gaussian bounds on the solution of the equation.

The asymptotic properties of Monte-Carlo type algorithms and of the usual functionals of ergodic diffusion processes are characterized using central limit theorems. The purpose of this thesis is to present results refining these theorems in four different settings. The first part of this work concerns the simulation of diffusion processes. The first chapter is devoted to the presentation of a method for adaptively controlling the variance during a Monte-Carlo simulation. Applications are given in finance. The second chapter proposes an estimator of the asymptotic variance of ergodic simulations. Its construction is based on results of the almost sure central limit theorem. Variance reduction techniques are proposed in this framework. The second part concerns the statistics of processes. The first chapter deals with ergodic diffusion processes. For different functionals of these processes, we prove Edgeworth developments specifying the speed of convergence of the central limit theorem. Applications in statistics are proposed, and in particular an opening towards the bootstrap. The second chapter proposes a theoretical framework for the parametric estimation of diffusion processes generalizing the asymmetric Brownian motion.

The analysis and approximation of solutions of stochastic differential equations (SDEs) with discontinuous coefficients is a subject that has not been treated in a fully satisfactory way. This problem becomes particularly motivating when one tries to approximate, by Monte-Carlo methods, the solutions of some partial differential equations (PDE) which also involve discontinuous coefficients. This is for example the case, well known in physics, of PDEs with divergence operator (DIO) whose coefficients are discontinuous and which we study in this thesis: the discontinuities then translate the irregularities of the medium in which the system studied evolves. This thesis proposes new results for the analysis and approximation of solutions of E. D. S. which are related to an O. F. D. whose coefficients are discontinuous. The statistical aspects of the models involved are also studied.

The first part of this thesis deals with the approximation of solutions of one-dimensional stochastic differential equations with non-Lipschitzian coefficients. Our attention is focused on two classes of equations widely used in finance. We first consider a generalization of the Cox-Ingersoll-Ross and Hull & White models . the drift coefficient is boundedly derivative, while the diffusion coefficient is of the type σ (x) = xα, with ½ ≤ α < 1. We then consider the SDE verified by a Bessel process . the drift coefficient is of type C on x, with C > 0 and thus has a singularity in zero. We place ourselves under assumptions that ensure the existence and uniqueness of solutions with strictly positive trajectories almost surely and propose discretization schemes that preserve the positivity of the approximated processes. On the one hand, we obtain the weak convergence speed of the schemes for a class of regular test functions and, on the other hand, we analyze by a time change method the strong convergence speed of the scheme in the case where the diffusion coefficient is of the type σ (x) = xα. The second part of the thesis addresses the problem of the asymptotic behavior of the solution of a parabolic partial differential equation (PDE) with a discontinuous first order coefficient when the viscosity tends to zero. We show that under an assumption of monotonicity on the first order coefficient, the solution converges weakly to the "measure solution" of the associated transport equation.

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This thesis is divided into three parts. The first two parts are devoted to model risk in finance (valuation, management). The third part is devoted to backward stochastic differential equations with random terminal time and some of their applications. In the first part, we study the speed of convergence of the numerical approximation of quantiles of the law of a component of (X_t), when (X_t) is a diffusion process and when we use a Monte-Carlo method combined with the Euler time discretization scheme of the process. The speed of convergence is obtained under two different assumptions: either (X_t) has a uniformly hypoelliptic generator, or the inverse of the Malliavin covariance matrix of the component of X_t considered satisfies a certain condition (M). We then show that this condition (M) is satisfied in various contexts in finance. In the second part, we focus on model risk control. We study a strategy that, in a sense, guarantees good performance regardless of the (unknown) model of the underlying assets used in the hedge portfolio. We consider the model risk control problem as a two-player (trader vs. market) zero-sum stochastic game problem corresponding to a 'worst case' protection. We prove that the corresponding value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman-Isaacs equation. The third part of the thesis deals with various issues related to backward stochastic differential equations with random terminal time, their relations with Dynkin sets and viscosity solutions of various elliptic problems.

This thesis is devoted to Monte-Carlo methods and more particularly to variance reduction. In the first part, we study a probabilistic algorithm, based on an iterative use of the method of control variables, allowing the calculation of quadratic approximations. Its use in dimension one for regular functions using the Fourier basis after periodization, the bases of orthogonal Legendre and Tchebychef polynomials, provides estimators with an increased order of convergence for Monte-Carlo integration. It is then extended to the multidimensional setting by a judicious choice of basis functions, allowing to attenuate the dimensional effect. Numerical validation is performed on many examples and applications. The second part is devoted to the study of the critical regime in neutron transport. The method developed consists in numerically calculating the principal eigenvalue of the neutron transport operator by combining the asymptotic development of the solution of the associated evolution problem with the calculation of its probabilistic interpretation by a Monte-Carlo method. Different techniques of variance reduction are implemented in the study of many homogeneous and inhomogeneous models. A probabilistic interpretation of the principal eigenvalue is given for a particular homogeneous model.

Partial differential equations (PDE) with random initial condition are used in the modeling of some complex physical phenomena such as turbulence. The characterization of the solution law, or statistical solution, has been the subject of much theoretical work. However, it is often difficult to estimate the accuracy of the usual methods of simulation of the average solutions of the e. D. P, or moments of the statistical solution. This thesis consists of two parts: we start by presenting the theory of statistical solutions, in particular in the case of the vortex equation of an incompressible fluid in the plane. This example leads us to consider, in the second part of this thesis, the model problem of a Mckean-Vlasov equation with random initial condition. Assuming that the coefficients of the equation are lipschitzian and bounded, we show that it admits a unique statistical solution whose moments can be represented using a nonlinear diffusion process. We deduce from this interpretation a stochastic particle method for the simulation of the moments. Its originality is that the interaction weights between the particles are random variables, defined from non-parametric estimators of a regression function. Finally, we study the convergence speed (theoretical and numerical) of the method for different families of weights.

The first part of this thesis aims at building a probabilistic algorithm for numerically solving stochastic backward differential equations (sdDEs) in the Markovian case, where the equation is associated with a forward process solution of a sdDE. We describe a first algorithm that relies on a double discretization of the equation, in time and in space, and uses simulations of trajectories of the forward process. The discretization in time is an extension of Euler's scheme for eds, where we replace the Brownian motion by a random walk. We then introduce an additional approximation by projecting at each discretization time the forward process on the set of simulated trajectories. This avoids an algorithmic complexity that would be exponential. We show a speed of convergence for this algorithm in dimension 1. We also present a variant of this algorithm, adapted to edsr whose parameters are less regular, by replacing in particular the euler scheme in the discretization of the forward process by the milshtein scheme. This allows us to write an algorithm for the discretization of reflected edsr. In a second part, we analyze the macmillan, and barone-adesi and whaley approximation, used in finance to estimate the price of an American option. By writing the price of the American option as the solution of a certain reflected backward stochastic differential equation, we obtain a general bound for the error of the approximation and show that the approximation converges to the exact price when the volatility of the underlying asset tends to zero. We then propose a second, more elementary, demonstration of this asymptotic result, using the price of a perpetual put.

The aim of this thesis is the numerical analysis of probabilistic algorithms for the solution of transportation equations and for the computation of complex option prices in financial mathematics. In the part concerning the transport equations, we have constructed an algorithm for approximating the solution and we have obtained an estimate of the speed of convergence of the error as a function of the discretization step in time. We then validated this algorithm on test cases related to industrial problems. Our results are comparable to those provided by determinist methods used at the e. The financial mathematics part deals with the problem of approximating the expectation of functionals depending at most on a diffusion process. This problem is related to the evaluation of exotic options. We first give convergence speed results for particular functions. Then, in order to generalize these results for a large class of functions and diffusions, we study the regularity of the solution of a degenerate parabolic edp with neumann condition and we obtain accurate estimates of the derivatives.

The convergence of the random vortex method for the navier-stokes equation has not yet been established in a fully satisfactory sense. This problem has strongly motivated the study of particle algorithms for some nonlinear P.D.E.'s, in particular, the burger equation which we present in this paper. The objective of this work is to give new results of convergence speed of stochastic particle methods, using the probabilistic interpretation of nonlinear e. D. P in terms of a system of interacting particles. The theory of stochastic processes allows us to interpret nonlinear e. D. P of the mckean-vlasov type as limit equations for systems of interacting particles. We derive a simple and natural algorithm based on the simulation of the underlying particle system. We obtain the speed of convergence of the algorithm, when the interaction kernels are lipschitzian and bounded. We then give a new probabilistic interpretation of the Burgers equation in terms of a system of interacting particles (the corresponding interaction kernel is discontinuous) and show that the system of particles possesses the propagation property of chaos. We study the convergence (theoretical and numerical) of the algorithm. The convergence speed we obtain seems to be what can be expected for this family of algorithms and gives a new theoretical insight to the random vortex method.