GOBET Emmanuel

In the medical field, the use of features extracted from images is more and more widespread. These measures can be real numbers (volume, cognitive score), organ meshes or the image itself. In these last two cases, a Euclidean space cannot describe the space of measures and it is necessary to place oneself on a Riemannian variety. Using this Riemannian framework and mixed effects models, it is then possible to estimate a representative object of the population as well as the inter-individual variability. In the longitudinal case (subjects observed repeatedly over time), these models allow to create an average trajectory representative of the global evolution of the population. In this thesis, we propose to generalize these models in the case of a mixed population. Each sub-population can follow different dynamics over time and their representative trajectory can be the same or differ from one time interval to another. This new model allows for example to model the onset of a disease as a deviation from normal aging.We are also interested in the detection of anomalies (e.g. tumors) in a population. With an object representing a control population, we define an anomaly as what cannot be reconstructed by diffeomorphic deformation of this representative object. Our method has the advantage of requiring neither a large dataset nor annotation by physicians and can be easily applied to any organ.Finally, we focus on different theoretical properties of the estimation algorithms used. In the context of nonlinear mixed effects models, the MCMC-SAEM algorithm is used. We will discuss two theoretical limitations. First, we will lift the geometric ergodicity assumption by replacing it with a sub-geometric ergodicity assumption. Furthermore, we will focus on a method to apply the SAEM algorithm when the joint distribution is not exponentially curved. We will show that this method introduces a bias in the estimate that we will measure. We will also propose a new algorithm to reduce it.

Over the last few years, a new paradigm of generative models based on neural networks have shown impressive results to simulate – with high fidelity – objects in high-dimension, while being fast in the simulation phase. In this work, we focus on the simulation of continuous-time processes (infinite dimensional objects) based on Generative Adversarial Networks (GANs) setting. More precisely, we focus on fractional Brownian motion, which is a centered Gaussian process with specific covariance function. Since its stochastic simulation is known to be quite delicate, having at hand a generative model for full path is really appealing for practical use. However, designing the architecture of such neural networks models is a very difficult question and therefore often left to empirical search. We provide a high confidence bound on the uniform approximation of fractional Brownian motion B^H(t), with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a Z-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction of B^H and in the general fractional Brownian motion case ( H ≠ 1/2 ), on the Lemarié-Meyer wavelet representation of B^H. This work gives theoretical support to use, and guidelines to construct, new generative models based on neural networks for simulating stochastic processes. It may well open the way to handle more complicated stochastic models written as a Stochastic Differential Equation driven by fractional Brownian motion.

In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of beta-mixing coefficients associated to the training sample. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression to derive corresponding generalization bounds for regression schemes in which the training sample may not be independent. These results provide a framework to analyze the error associated to regression schemes whose training sample comes from a large class of β−mixing sequences, including geometrically ergodic Markov samples, using only the independent case. More generally, they permit a meaningful extension of the Vapnik-Chervonenkis and similar theories for independent training samples to this class of β−mixing samples.

Feedforward neural networks based on Rectified linear units (ReLU) cannot efficiently approximate quantile functions which are not bounded, especially in the case of heavy-tailed distributions. We thus propose a new parametrization for the generator of a Generative adversarial network (GAN) adapted to this framework, basing on extreme-value theory. We provide an analysis of the uniform error between the extreme quantile and its GAN approximation. It appears that the rate of convergence of the error is mainly driven by the second-order parameter of the data distribution. The above results are illustrated on simulated data and real financial data.

In this study, we propose a new parametrization for the generator of a Generative adversarial network (GAN) adapted to data from heavy-tailed distributions. We provide an analysis of the uniform error between an extreme quantile and its GAN approximation. Numerical experiments are conducted both on real and simulated data.

We provide a large probability bound on the uniform approximation of fractional Brownian motion $(B^H(t) : t ∈ [0,1])$ with Hurst parameter $H$, by a deep-feedforward ReLU neural network fed with a $N$-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of $\mathcal{O}(N^{-H})$ is possible with log$(N)$ hidden layers and $\mathcal{O} (N )$ parameters. Our analysis relies, in the standard Brownian motion case $(H = 1/2)$, on the Levy construction of $B^H$ and in the general fractional Brownian motion case $(H \ne 1/2)$, on the Lemarié-Meyer wavelet representation of $B^H$. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.

We establish a new concentration-of-measure inequality for the sum of independent random variables with β-heavy tail. This includes exponential of Gaussian distributions (a.k.a. log-normal distributions), or exponential of Weibull distributions, among others. These distributions have finite polynomial moments at any order but many not have finite α-exponential moments. We exhibit a new Orlicz norm adapted to this setting of β-heavy tails, we prove a new Talagrand inequality for the sum and a new maximal inequality. As consequence, a deviation probability of the sum from its mean is obtained.

We develop a new iterative method based on Pontryagin principle to solve stochastic control problems. This method is nothing else than the Newton method extended to the framework of stochastic controls, where the state dynamics is given by an ODE with stochastic coefficients. Each iteration of the method is made of two ingredients: computing the Newton direction, and finding an adapted step length. The Newton direction is obtained by solving an affine-linear Forward-Backward Stochastic Differential Equation (FBSDE) with random coefficients. This is done in the setting of a general filtration. We prove that solving such an FBSDE reduces to solving a Riccati Backward Stochastic Differential Equation (BSDE) and an affine-linear BSDE, as expected in the framework of linear FBSDEs or Linear-Quadratic stochastic control problems. We then establish convergence results for this Newton method. In particular, sufficient regularity of the second-order derivative of the cost functional is required to obtain (local) quadratic convergence. A restriction to the space of essentially bounded stochastic processes is needed to obtain such regularity. To choose an appropriate step length while fitting our choice of space of processes, an adapted backtracking line-search method is developed. We then prove global convergence of the Newton method with the proposed line-search procedure, which occurs at a quadratic rate after finitely many iterations. An implementation with regression techniques to solve BSDEs arising in the computation of the Newton step is developed. We apply it to the control problem of a large number of batteries providing ancillary services to an electricity network.

In this thesis, we use tools from stochastic optimal control and stochastic and convex optimization to develop mechanisms to drive energy storage systems to manage the production uncertainty of intermittent energy sources (solar and wind).First, we introduce a mechanism in which a consumer commits to follow a consumption profile on the grid, and then controls its storage systems to follow this profile in real time. We model this situation by a mean-field control problem, for which we obtain theoretical and numerical results. Then, we introduce a problem of controlling a large number of thermal storage units subject to a common noise and providing services to the network. We show that this control problem can be replaced by a stochastic differential Stackelberg problem. This allows a decentralized control scheme with performance guarantees, while preserving the privacy of the consumers' data and limiting the telecommunication requirements. Next, we develop a Newton method for stochastic control problems. We show that the Newton step can be computed by solving Stochastic Retrograde Differential Equations, then we propose an appropriate linear search method, and prove the global convergence of the obtained Newton method in a suitable space. Its numerical performance is illustrated on a problem of controlling a large number of batteries providing services to the network. Finally, we study the extension of the "Alternating Current Optimal Power Flow" problem to the stochastic multistage case in order to control an electrical network equipped with storage systems. For this problem, we give realistic and verifiable a priori conditions guaranteeing the absence of relaxation jumps, as well as an a posteriori bound on the latter. In the broader framework of non-convex multistage problems with a generic structure, we also establish a priori bounds on the duality jump, based on results related to the Shapley-Folkman Theorem.

In dynamic minimum variance portfolio, we study the impact of the sequence of covariance matrices taken in inputs, on the realized variance of the portfolio computed along a sample market path. The allocation of the portfolio is adjusted on a regular basis (every H days) using an updated covariance matrix estimator. In a modelling framework where the covariance matrix of the asset returns evolves as an ergodic process, we quantify the probability of observing an underperformance of the optimal dynamic covariance matrix compared to any other choice. The bounds depend on the tails of the returns, on the adjustment period H, and on the total number of rebalancing times N. These results provide asset managers with new insights into the optimality of their choice of covariance matrix estimators, depending on the depth of the backtest N H and the investment period H. Experiments based on GARCH modelling support our theoretical results.

Feedforward neural networks based on rectified linear units (ReLU) cannot efficiently approximate quantile functions which are not bounded in the Fréchet maximum domain of attraction. We thus propose a new parametrization for the generator of a generative adversarial network (GAN) adapted to this framework of heavy-tailed distributions. We provide an analysis of the uniform error between the extreme quantile and its GAN approximation. It appears that the rate of convergence of the error is mainly driven by the second-order parameter of the data distribution. The above results are illustrated on simulated data and real financial data.

The treatment of uncertainties is a fundamental problem in the financial context. The variables studied are often time-dependent, with thick distribution tails. In this thesis, we are interested in tools to take into account uncertainties in its main forms: statistical uncertainties, parametric uncertainties and model error, keeping in mind that we wish to apply them to this context. The first part is devoted to the establishment of concentration inequalities in the context of variables with thick tails. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on a finite size of observations. In this thesis, we establish new concentration inequalities, which cover in particular the case of estimators with lognormal distribution.In the second part, we deal with the impact of the model error for the estimation of the covariance matrix on stock returns, under the assumption that there is an instantaneous covariance process between the returns whose present value depends on its past value. We can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and show that it provides the smallest realized variance with high probability in the minimum variance portfolio framework.In the third part, we propose an approach to estimate the Sharpe ratio and the portfolio allocation when they depend on parameters considered uncertain. Our approach involves the adaptation of a stochastic approximation technique for the computation of the polynomial decomposition of the quantity of interest.Finally, in the last part of this thesis, we focus on portfolio optimization with target distribution. This technique can be formalized without any model assumptions on the returns. We propose to find these portfolios by minimizing divergence measures based on kernel functions and optimal transport theory.

The estimation of extreme quantiles requires adapted methods to extrapolate beyond the largest observation of the sample. Extreme-value theory provides a mathematical framework to tackle this problem together with statistical procedures based on the estimation of the so-called tail-index describing the distribution tail. We focus on heavy-tailed distributions and consider the case where the shape of the distribution tail depends on unknown auxiliary variables. As a consequence, one has to deal with observations from a mixture of heavytailed distributions, and it is shown that, in such a situation, usual extreme-value estimators suffer from a strong bias. We propose several methods to mitigate this bias. Their asymptotic properties are established and their finite sample performance is illustrated both on simulated and real financial data This is joint work with Emmanuel Gobet.

We propose a stochastic control problem to control cooperatively Thermostatically Controlled Loads (TCLs) to promote power balance in electricity networks. We develop a method to solve this stochastic control problem with a decentralized architecture, in order to respect privacy of individual users and to reduce both the telecommunications and the computational burden compared to the setting of an omniscient central planner. This paradigm is called federated learning in the machine learning community, see [YFY20], therefore we refer to this problem as a federated stochastic control problem. The optimality conditions are expressed in the form of a high-dimensional Forward-Backward Stochastic Differential Equation (FBSDE), which is decomposed into smaller FBSDEs modeling the optimal behaviors of the aggregate population of TCLs of individual agents. In particular, we show that these FBSDEs fully characterize the Nash equilibrium of a stochastic Stackelberg differential game. In this game, a coordinator (the leader) aims at controlling the aggregate behavior of the population, by sending appropriate signals, and agents (the followers) respond to this signal by optimizing their storage system locally. A mean-field-type approximation is proposed to circumvent telecommunication constraints and privacy issues. Convergence results and error bounds are obtained for this approximation depending on the size of the population of TCLs. A numerical illustration is provided to show the interest of the control scheme and to exhibit the convergence of the approximation. An implementation which answers practical industrial challenges to deploy such a scheme is presented and discussed.

In financial risk management, modelling dependency within a random vector X is crucial, a standard approach is the use of a copula model. Say the copula model can be sampled through realizations of Y having copula function C: had the marginals of Y been known, sampling X^(i) , the i-th component of X, would directly follow by composing Y^(i) with its cumulative distribution function (c.d.f.) and the inverse c.d.f. of X^(i). In this work, the marginals of Y are not explicit, as in a factor copula model. We design an algorithm which samples X through an empirical approximation of the c.d.f. of the Y marginals. To be able to handle complex distributions for Y or rare-event computations, we allow Markov Chain Monte Carlo (MCMC) samplers. We establish convergence results whose rates depend on the tails of X, Y and the Lyapunov function of the MCMC sampler. We present numerical experiments confirming the convergence rates and also revisit a real data analysis from financial risk management.

Discrete time hedging produces a residual risk, namely, the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing asymmetrically profits and losses. After deriving the asymptotics within a discrete time risk measurement for a large number of trading dates, we derive the optimal strategies minimizing the asymptotic risk in the continuous time setting. We characterize the optimality through a class of fully nonlinear Partial Differential Equations (PDE). Numerical experiments show that the optimal strategies associated with discrete and asymptotic approach coincides asymptotically.

In this thesis, we examine several stochastic approximation methods for both the computation of financial risk measures and the pricing of derivatives.Since explicit formulas are rarely available for such quantities, the need for fast, efficient and reliable analytical approximations is of paramount importance to financial institutions.In the first part, we study several multilevel Monte Carlo approximation methods and apply them to two practical problems: the estimation of quantities involving nested expectations (such as initial margin) and the discretization of integrals appearing in rough models for the forward variance for VIX option pricing.In both cases, we analyze the asymptotic optimality properties of the corresponding multilevel estimators and numerically demonstrate their superiority over a classical Monte Carlo method.In the second part, motivated by the numerous examples from credit risk modeling, we propose a general metamodeling framework for large sums of weighted Bernoulli random variables, which are conditionally independent with respect to a common factor X. Our generic approach is based on the polynomial decomposition of the chaos of the common factor and on a Gaussian approximation. L2 error estimates are given when the factor X is associated with classical orthogonal polynomials.Finally, in the last part of this thesis, we focus on the short-time asymptotics of U.S. implied volatility and U.S. option prices in local volatility models. We also propose a law approximation of the VIX index in rough models for forward variance, expressed in terms of lognormal proxies, and derive expansion results for VIX options with explicit coefficients.

The Multilevel Monte-Carlo (MLMC) method developed by Giles [Gil08] has a natural application to the evaluation of nested expectation of the form E [g(E [f (X, Y)|X])], where f, g are functions and (X, Y) a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of Initial Margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotical optimality. at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal/dual algorithms for stochastic control problems.

The Multilevel Monte-Carlo (MLMC) method developed by Giles [Gil08] has a natural application to the evaluation of nested expectation of the form E [g(E [f (X, Y)|X])], where f, g are functions and (X, Y) a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of Initial Margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotical optimality. at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal/dual algorithms for stochastic control problems.

The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.

Power producers are interested in valuing their power plant production. By trading into forward contracts, we propose to reduce the contingency of the associated income considering the fixed costs and using an asymmetric risk criterion. In an asymptotic framework, we provide an optimal hedging strategy through a solution of a nonlinear partial differential equation. As a numerical experiment, we analyze the impact of the fixed costs structure on the hedging policy and the value of the assets.

In this thesis, we make some contributions to the theoretical and numerical study of some stochastic control problems, as well as their applications to financial mathematics and financial risk management. These applications concern problems of valuation and weak hedging of financial products, as well as regulatory issues. We propose numerical methods to efficiently compute these quantities for which no explicit formula exists. Finally, we study backward stochastic differential equations related to new switching problems with cost uncertainty.

Kernel techniques are among the most widely-applied and influential tools in machine learning with applications at virtually all areas of the field. To combine this expressive power with computational efficiency numerous randomized schemes have been proposed in the literature, among which probably random Fourier features (RFF) are the simplest and most popular. While RFFs were originally designed for the approximation of kernel values, recently they have been adapted to kernel derivatives, and hence to the solution of large-scale tasks involving function derivatives. Unfortunately, the understanding of the RFF scheme for the approximation of higher-order kernel derivatives is quite limited due to the challenging polynomial growing nature of the underlying function class in the empirical process. To tackle this difficulty, we establish a finite-sample deviation bound for a general class of polynomial-growth functions under α-exponential Orlicz condition on the distribution of the sample. Instantiating this result for RFFs, our finite-sample uniform guarantee implies a.s. convergence with tight rate for arbitrary kernel with α-exponential Orlicz spectrum and any order of derivative.

Stochastic Approximation is an iterative procedure for computing a zero θ of a function that is not explicit but defined as an expectation. It is, for example, a numerical tool for computing maximum likelihood in "regular" latent data models. If the definition of the statistical model is tainted with an uncertainty τ , of which only an a priori dπ(τ ) is known, then the zeros depend on τ and the natural question is to explore their distribution when τ ∼ dπ. In this paper, we propose an iterative algorithm based on a Stochastic Approximation scheme that,in the limit, computes θ (τ) for any τ and produces a characterization of its distribution. and weenounce sufficient conditions for the convergence of this algorithm.

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This paper is devoted to the study of the deviation of the (random) average $L^{2}-$error associated to the least--squares regressor over a family of functions ${\cal F}_{n}$ (with controlled complexity) obtained from $n$ independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average $L^{2}-$error associated to this family of functions, and to some of the corresponding consequences for the problem of consistency. In the i.i.d. case, this specializes as classical questions on least--squares regression problems, but in more general cases, this setting permits a precise investigation in the direction of the study of nonasymptotic errors for least--squares regression schemes in nonstationary settings, which we motivate providing background and examples. More precisely, we prove first two nonasymptotic deviation inequalities that generalize and refine corresponding known results in the i.i.d. case. We then explore some consequences for nonasymptotic bounds of the error both in the weak and the strong senses. Finally, we exploit these estimates to shed new light into questions of consistency for least--squares regression schemes in the distribution--free, nonparametric setting. As an application to the classical theory, we provide in particular a result that generalizes the link between the problem of consistency and the Glivenko-Cantelli property, which applied to regression in the i.i.d. setting over non--decreasing families $({\cal F}_{n})_{n}$ of functions permits to create a scheme which is strongly consistent in $L^{2}$ under the sole (necessary) assumption of the existence of functions in $\cup_{n}{\cal F}_{n}$ which are arbitrarily close in $L^{2}$ to the corresponding regressor.

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solution to a system of non-linear PDEs of the same kind. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We experiment the methodology in the context of swing contract (energy contract with flexibility in purchasing energy power), this allows to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.

In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial differential equation (PDE) obtained through the well known Feynman-Kac representation. For the sake of enriching the algorithm with high order convergence a weighted approximation of the solution is computed and appropriate conditions on the parameters of the method are inferred. With the challenge of tackling problems in high dimensions we propose suitable projections of the solution and efficient parallelizations of the algorithm taking advantage of powerful many core processors such as graphics processing units (GPUs).

We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. He controls the use of the storage facilities in order to diminish the random fluctuations of his residual load on the public grid, so that intermittent renewable energy is better used leading globally to a much greener carbon footprint. The optimization problem is described in terms of an extended McKean-Vlasov stochastic control problem. Using the Pontryagin principle, we characterize the optimal storage control as solution of a certain McKean-Vlasov Forward Backward Stochastic Differential Equation (possibly with jumps), for which we prove existence and uniqueness. Quasi-explicit solutions are derived when the cost functions may not be linear-quadratic, using a perturbation approach. Numerical experiments support the study.

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

This paper is devoted to the study of the deviation of the (random) average $L^{2}-$error associated to the least--squares regressor over a family of functions ${\cal F}_{n}$ (with controlled complexity) obtained from $n$ independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average $L^{2}-$error associated to this family of functions, and to some of the corresponding consequences for the problem of consistency. In the i.i.d. case, this specializes as classical questions on least--squares regression problems, but in more general cases, this setting permits a precise investigation in the direction of the study of nonasymptotic errors for least--squares regression schemes in nonstationary settings, which we motivate providing background and examples. More precisely, we prove first two nonasymptotic deviation inequalities that generalize and refine corresponding known results in the i.i.d. case. We then explore some consequences for nonasymptotic bounds of the error both in the weak and the strong senses. Finally, we exploit these estimates to shed new light into questions of consistency for least--squares regression schemes in the distribution--free, nonparametric setting. As an application to the classical theory, we provide in particular a result that generalizes the link between the problem of consistency and the Glivenko-Cantelli property, which applied to regression in the i.i.d. setting over non--decreasing families $({\cal F}_{n})_{n}$ of functions permits to create a scheme which is strongly consistent in $L^{2}$ under the sole (necessary) assumption of the existence of functions in $\cup_{n}{\cal F}_{n}$ which are arbitrarily close in $L^{2}$ to the corresponding regressor.

We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods.

We design a meta-model for the loss distribution of a large credit portfolio in the Gaussian copula model. Using both the Wiener chaos expansion on the systemic economic factor and a Gaussian approximation on the associated truncated loss, we significantly reduce the computational time needed for sampling the loss and therefore estimating risk measures on the loss distribution. The accuracy of our method is confirmed by many numerical examples.

This thesis consists of two parts that can be read independently. In the first part of the thesis, we study hedging and option pricing problems related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDEs).In the first chapter, we focus on the valuation and hedging of European options. We consider the problem of optimizing the residual risk generated by a discrete-time hedge in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution of the associated discrete problem, we study the asymmetric residual risk measure integrated in a Markovian framework. In this context, we show the existence of this asymptotic risk measure. We then describe an asymptotically optimal hedging strategy via the solution of a totally nonlinear PDE. The second chapter applies this hedging method to the problem of valuing the output of a power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with the uncertain revenues of this power plant by hedging with futures contracts. In the second part of the thesis, we consider several control problems related to economics and finance.The third chapter is dedicated to the study of a class of McKean-Vlasov (MKV) type problem with common noise, called conditional polynomial MKV. We reduce this polynomial class by Markov folding to finite dimensional control problems.We compare three different probabilistic techniques for numerically solving the reduced problem: quantization, control randomization regression, and delayed regression. We provide many numerical examples, such as portfolio selection with uncertainty about an underlying trend.In the fourth chapter, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated model for the underlyings is not available and that a small sample obtained from historical data is accessible.Moreover, in this context, we assume that futures contracts are often governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the historical log price of observable futures contracts.

This thesis is devoted to the theoretical and numerical study of two nonlinear problems in the McKean sense in finance. In the first part, we address the problem of calibrating a model with local and stochastic volatility to take into account the prices of European vanilla options observed on the market. This problem results in the study of a nonlinear stochastic differential equation (SDE) in the McKean sense because of the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor with respect to the SDE solution. We obtain the existence of the process in the particular case where the stochastic volatility factor is a jump process with a finite number of states. We also obtain the weak convergence at order 1 of the time discretization of the nonlinear DHS in the McKean sense for general stochastic volatility factors. In the industry, the calibration is efficiently performed using a regularization of the conditional expectation by a Nadaraya-Watson type kernel estimator, as proposed by Guyon and Henry-Labordère in [JGPHL]. We also propose a half-time numerical scheme and study the associated particle system that we compare to the algorithm proposed by [JGPHL]. In the second part of the thesis, we focus on a problem of contract valuation with margin calls, a problem that appeared with the application of new regulations since the financial crisis of 2008. This problem can be modeled by an anticipatory stochastic differential equation (SDE) with dependence on the law of the solution in the generator. We show that this equation is well-posed and propose an approximation of its solution using standard linear SRDEs when the liquidation time of the option in case of default is small. Finally, we show that the computation of the solutions of these standard EDSRs can be improved using the multilevel Monte Carlo method introduced by Giles in [G].

The increase in the share of intermittent renewable energies in the energy mix is generating problems related to the predictability of electricity production. In particular, on a seasonal basis, transmission system operators are forced to project the availability of generation resources and forecast demand. This allows them to guarantee supply for the next winter or summer. Nevertheless, current projections are mainly based on historical data (climatology) of temperatures (consumption), winds (wind production), or solar radiation (photovoltaic production). The thesis presents four works: three in the framework of seasonal forecasting, and one study on the realism of surface wind as modeled by the European Center's weather forecasting model.If wind energy forecasting at short time scales ranging from minutes to days as well as wind tendency at climatic scales have been widely studied, wind production forecasting at intermediate time scales ranging from a fortnight to the season has received little attention. The predictability of mid-latitude weather at these distant horizons is indeed still an open question. However, several studies have shown that seasonal numerical forecasting models are able to provide information on the variability of large-scale atmospheric circulation through the prediction of large-scale circulation oscillations, such as ENSO in the Pacific, or the NAO in the North Atlantic. It has also been shown that these oscillations have a strong impact on precipitation, temperature, and surface wind. Building the relationship between these indicators of large-scale atmospheric circulation and surface wind in France can therefore take into account the interannual variability of surface wind, which is not capable by definition climatology. This is the idea developed in the 3 studies concerning seasonal forecasting. In order to forecast the wind resource and production on a seasonal scale, two probabilistic models are developed. One is parametric, based on the prediction of the seasonal distribution of surface wind at different locations in France. The other is non-parametric, based on the estimation of the probability density of daily surface wind conditional on the state of the atmosphere. The third study proposes to reconstruct the joint probability of French national consumption and production, thus allowing to measure the risk of imbalance between supply and demand.

This thesis contains two parts that study two different topics. Chapters 1-4 are devoted to problems of discretization of processes with stopping times. In Chapter 1 we study the optimal discretization error for stochastic integrals with respect to a continuous multidimensional Brownian semimartingale. In this framework we establish a trajectory lower bound for the renormalized quadratic variation of the error. We provide a sequence of stopping times that gives an asymptotically optimal discretization. This sequence is defined as the output time of random ellipsoids by the semimartingale. Compared to the previous results we allow a rather large class of semimartingales. We prove that the lower bound is exact. In Chapter 2 we study the adaptive version of the model of the optimal discretization of stochastic integrals. In Chapter 1 the construction of the optimal strategy uses the knowledge of the diffusion coefficient of the considered semimartingale. In this work we establish an asymptotically optimal discretization strategy that is adaptive to the model and does not use any information about the model. We prove the optimality for a rather general class of discretization grids based on kernel techniques for adaptive estimation. In Chapter 3 we study the convergence of renormalized discretization error laws of Itô processes for a concrete and rather general class of discretization grids given by stopping times. Previous works on the subject consider only the case of dimension 1. Moreover they concentrate on particular cases of grids, or prove results under abstract assumptions. In our work the boundary distribution is given explicitly in a clear and simple form, the results are shown in the multidimensional case for the process and for the discretization error. In Chapter 4 we study the parametric estimation problem for diffusion processes based on time-lapse observations. Previous works on the subject consider deterministic, strongly predictable or random observation times independent of the process. Under weak assumptions, we construct a suite of consistent estimators for a large class of observation grids given by stopping times. An asymptotic analysis of the estimation error is performed. Furthermore, for the parameter of dimension 1, for any sequence of estimators that verifies an unbiased LCT, we prove a uniform lower bound for the asymptotic variance. We show that this bound is exact. Chapters 5-6 are devoted to the uncertainty quantification problem for stochastic approximation bounds. In Chapter 5 we analyze the uncertainty quantification for stochastic approximation limits (SA). In our framework the limit is defined as a zero of a function given by an expectation. This expectation is taken with respect to a random variable for which the model is supposed to depend on an uncertain parameter. We consider the limit of SA as a function of this parameter. We introduce an algorithm called USA (Uncertainty for SA). It is a procedure in increasing dimension to compute the basic chaos expansion coefficients of this function in a basis of a well chosen Hilbert space. The convergence of USA in this Hilbert space is proved. In Chapter 6 we analyze the convergence rate in L2 of the USA algorithm developed in Chapter 5. The analysis is non-trivial because of the infinite dimension of the procedure. The rate obtained depends on the model and the parameters used in the USA algorithm. Its knowledge allows to optimize the rate of growth of the dimension in USA.

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a proba-bilistic forecast: the input parameters of the SDE model are the Arome deterministic forecast computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement.

No summary available.

In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial differential equation (PDE) obtained through the well known Feynman-Kac representation. For the sake of enriching the algorithm with high order convergence a weighted approximation of the solution is computed and appropriate conditions on the parameters of the method are inferred. With the challenge of tackling problems in high dimensions we propose suitable projections of the solution and efficient parallelizations of the algorithm taking advantage of powerful many core processors such as graphics processing units (GPUs).

No summary available.

In this paper we study the problem of parametric inference for multidimensional diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject (such as [Doh87, GJ93, Gob01, AM04] among others) consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids (studied in [GL14, GS18]). Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp.

No summary available.

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. " Propagation of chaos " results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

Discrete time hedging produces a residual risk, namely, the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing asymmetrically profits and losses. After deriving the asymptotics within a discrete time risk measurement for a large number of trading dates, we derive the optimal strategies minimizing the asymptotic risk in the continuous time setting. We characterize the optimality through a class of fully nonlinear Partial Differential Equations (PDE). Numerical experiments show that the optimal strategies associated with discrete and asymptotic approach coincides asymptotically.

This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solution to a system of non-linear PDEs of the same kind. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We experiment the methodology in the context of swing contract (energy contract with flexibility in purchasing energy power), this allows to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.

We study the convergence in distribution of the renormalized error arising from the discretization of a Brownian semimartingale sampled at stopping times. Our mild assumptions on the form of stopping times allow the time grid to be a combination of hitting times of stochastic domains and of Poisson-like random times. Remarkably, a Functional Central Limit Theorem holds under great generality on the semimartingale and on the form of stopping times. Furthermore, the asymptotic characteristics are quite explicit. Along the derivation of such results, we also establish some key estimates related to approximations and sensitivities of hitting time/position with respect to model and domain perturbations.

No summary available.

The subject of this thesis is the statistical inference of multidimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs defined as binary observations of trajectories. We then show that it is possible to deduce the dynamics of the underlying trajectory from the binary observations. For this, we construct statistics from the graph and show new convergence properties in the context of a long time and high frequency observation. We also analyze the properties of stochastic graphs from the point of view of evolving networks. In a second part, we work under the assumption of complete information and continuous time and add a sparsity assumption concerning the textit{drift} parameter of the Ornstein-Uhlenbeck process. We then show sharp oracle properties of the Lasso estimator, prove a lower bound on the estimation error in the minimax sense and show asymptotic optimality properties of the Adaptive Lasso estimator. We then apply these methods to estimate the speed of return at the average of daily returns of US stocks as well as the prices of dividend futures for the EURO STOXX 50 index.

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.

The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $\targetfn$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\param$. We aim at deriving the function $\targetfn$, as well as the probabilistic distribution of $\targetfn(\param)$ given a probability distribution $\pi$ for $\param$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\param \mapsto \targetfn(\param)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{\targetfn_K}(\cdot)$ of $\targetfn(\cdot)$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{\targetfn_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{\targetfn_K}(\param)$ when $\param$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically.

In this paper, we develop the reversible shaking transformation methods on path space of Gobet and Liu [GL15] to estimate the rare event statistics arising in different financial risk settings which are embedded within a unified framework of isonormal Gaussian process. Namely, we combine splitting methods with both Interacting Particle System (IPS) technique and ergodic transformations using Parallel-One-Path (POP) estimators. We also propose an adaptive version for the POP method and prove its convergence. We demonstrate the application of our methods in various examples which cover usual semi-martingale stochastic models (not necessarily Markovian) driven by Brownian motion and, also, models driven by fractional Brownian motion (non semi-martingale) to address various financial risks. Interestingly, owing to the Gaussian process framework, our methods are also able to efficiently handle the important problem of sensitivities of rare event statistics with respect to the model parameters.

No summary available.

We design and analyze an algorithm for estimating the mean of a function of a conditional expectation, when the outer expectation is related to a rare-event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non asymptotic bounds for the L2-empirical risks associated to this least-squares regression. this generalizes the error bounds usually obtained in the case of i.i.d. observations. Global error bounds are also derived for the nested expectation problem. Numerical results in the context of financial risk computations illustrate the performance of the algorithms.

Monte-Carlo methods are widely used by the financial industry to price derivatives, estimate risks, or to calibrate/estimate models. They can also be used to handle big data, in machine learning, to perform online optimization, to study the propagation of uncertainty in fluid mechanics or geophysics. Under the same label Monte-Carlo, one actually finds very different techniques and communities that evolve in different directions. The thematic cycle that we organized from october 2015 to July 2016 aimed at confronting the different viewpoints of these communities and at contributing to a general thinking on how these techniques can be used by the financial industry and the economic world in general. It benefited from the financial support of the Louis Bachelier Institute, the Chaire Risques Financiers, the Chaire Finance et D´eveloppement durable, the Chaire Economie des nou- ´ velles donn´ees, the Chaire March´es en mutation, the ANR program ISOTACE ANR-12-MONU-0013 and the Institut Henri Poincar´e. Three topics were covered by academic lectures followed by a one-day workshop: propagation of uncertainty, particle methods for the management of risks, stochastic algorithms and big data. We thank Areski Cousin, Virginie Ehrlacher, Romuald Elie, Gersende Fort, St´ephane Gaiffas and Gilles Pag`es for having coordinated these workshops. The cycle was concluded by a one week closing conference with twelve plenary talks and sixteen minisymposia: see the website https://montecarlo16.sciencesconf.org Of course the six papers in these proceedings cannot account for all the topics addressed during the cycle. But they give qualitative spotlights on some of the active fields of research on stochastic methods in finance. We thank their authors for these valuable contributions.

We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretiza- tion stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.

We establish general moment estimates for the discrete and continuous exit times of a general Itô process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the L1 norm with an order 1/2 with respect to the mesh size.

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a proba-bilistic forecast: the input parameters of the SDE model are the Arome deterministic forecast computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement.

No summary available.

We present our contribution on the control and optimization of electricity production. The first part concerns the optimization of the management of a micro grid. We formulate the management program as an optimal control problem in continuous time, then we solve this problem by dynamic programming using a solver developed for this purpose: BocopHJB. We show that this type of formulation can be extended to a stochastic modeling. We end this part with the adaptive weights algorithm, which allows a management of the micro network battery integrating its aging. The algorithm exploits the two time scale structure of the control problem. The second part concerns networked market models, and in particular those of electricity. We introduce an incentive mechanism to decrease the market power of energy producers, to the benefit of the consumer. We study some mathematical properties of the optimization problems faced by market agents (producers and regulators). The last chapter studies the existence and uniqueness of Nash equilibria in pure strategies of a class of Bayesian games to which some network market models belong. For some simple cases, an equilibrium computation algorithm is proposed. An appendix gathers a documentation on the numerical solver BocopHJB.

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. " Propagation of chaos " results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

This is a companion paper based on our previous work [ADGL15] on rare event simulation methods. In this paper, we provide an alternative proof for the ergodicity of shaking transformation in the Gaussian case and propose two variants of the existing methods with comparisons of numerical performance. In numerical tests, we also illustrate the idea of extreme scenario generation based on the convergence of marginal distributions of the underlying Markov chains and show the impact of the discretization of continuous time models on rare event probability estimation.

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. " Propagation of chaos " results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths.

No summary available.

We consider a random map x → F (ω, x) and a random variable Θ(ω), and we denote by F^N (ω, x) and Θ^N (ω) their approximations: We establish a strong convergence result, in Lp-norms, of the compound approximation F^N (ω, Θ^N (ω)) to the compound variable F (ω, Θ(ω)), in terms of the approximations of F and Θ. We then apply this result to the composition of two Stochastic Differential Equations through their initial conditions, which can give a way to solve some Stochastic Partial Differential Equations.

We consider a multidimensional stochastic differential equation Y written as a drift-perturbation of an ergodic Ornstein-Uhlenbeck process X. Under the condition of time-reversibility of X, we derive a first and second order expansion of the stationary distribution µ^Y of Y in terms of X. Error estimates are established. These approximations are then turned into a simulation scheme for sampling approximately according to µ Y. Numerical experiments support the theoretical error estimates.

In this thesis, we propose a progressive (forward) approach for the probabilistic representation of nonlinear and nonconservative Partial Differential Equations (PDEs), allowing to develop a particle-based algorithm to numerically estimate their solutions. The Nonlinear Stochastic Differential Equations of McKean type (NLSDE) studied in the literature constitute a microscopic formulation of a phenomenon modeled macroscopically by a conservative PDE. A solution of such a NLSDE is the data of a couple $(Y,u)$ where $Y$ is a solution of a stochastic differential equation (SDE) whose coefficients depend on $u$ and $t$ such that $u(t,cdot)$ is the density of $Y_t$. The main contribution of this thesis is to consider nonconservative PDEs, i.e. conservative PDEs perturbed by a nonlinear term of the form $Lambda(u,nabla u)u$. This implies that a pair $(Y,u)$ will be a solution of the associated probabilistic representation if $Y$ is still a stochastic process and the relation between $Y$ and the function $u$ will then be more complex. Given the law of $Y$, the existence and uniqueness of $u$ are proved by a fixed point argument via an original Feynmann-Kac formulation.

Given Y a graph process defined by an incomplete information observation of a multivariate Ornstein-Uhlenbeck process X, we investigate whether we can estimate the parameters of X. We define two statistics of Y. We prove convergence properties and show how these can be used for parameter inference. Finally, numerical tests illustrate our results and indicate possible extensions and applications.

We design and analyze an algorithm for estimating the mean of a function of a conditional expectation, when the outer expectation is related to a rare-event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non asymptotic bounds for the L2-empirical risks associated to this least-squares regression. this generalizes the error bounds usually obtained in the case of i.i.d. observations. Global error bounds are also derived for the nested expectation problem. Numerical results in the context of financial risk computations illustrate the performance of the algorithms.

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.

We design a numerical scheme for solving the Multi step-forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical least-squares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy. Numerical experiments are presented illustrating theoretical convergence estimates.

This thesis deals with two areas of stochastic analysis and financial mathematics: the Malliavin calculus for Markov chains (Part I) and counterparty risk (Part II). Part I aims at studying the Malliavin calculus for Markov chains in continuous time. Two points are presented: proving the existence of the density for the solutions of a stochastic differential equation and computing the sensitivities of derivatives. Part II deals with current topics in the field of market risk, namely XVA (price adjustments) and multi-curve modeling.

In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on multicore devices such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization.

In this paper we design a numerical scheme for approximating Backward Doubly Stochastic Differential Equations (BDSDEs for short) which represent solution to Stochastic Partial Differential Equations (SPDEs). We first use a time-discretization and then, we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion B. The coefficients are evaluated through a empirical regression scheme, which is performed conditionally to B. We establish non asymptotic error estimates, conditionally to B, and deduce how to tune parameters to obtain a convergence conditionally and unconditionally to B. We provide numerical experiments as well.

We design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares Monte Carlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization. secondly, we develop refined concentration-of-measure techniques to capture the variance of reduction. Our theoretical results are supported by numerical experiments.

This paper consists in providing and mathematically analyzing the expansion of an option price (with bounded Lipschitz payoff) for model combining local and stochastic volatility. The local volatility part has a general form, with appropriate growth and boundedness assumptions. For the stochastic part, we choose a square root process, which is widely used for modeling the behavior of the variance process (Heston model). We rigorously establish tight error estimates of our expansions, using Malliavin calculus, which requires a careful treatment because of the lack of weak differentiability of the model. this error analysis is interesting on its own. Moreover, in the particular case of Call-Put options, we also provide expansions of the Black-Scholes implied volatility which allows to obtain very simple and rapid formulas in comparison to the Monte Carlo approach while maintaining a very competitive accuracy.

We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explicit. We provide non asymptotic error bounds, which justifies the expansion accuracy as the time or the diffusion coefficients are small in a suitable sense.

In this thesis we apply Malliavin's calculus to obtain the local asymptotic normality (LAN) property from discrete observations of certain uniformly elliptic diffusion processes with jumps. In Chapter 2 we revise the proof of the local asymptotic mixed normality (LAMN) property for diffusion processes with jumps from continuous observations, and as a consequence we obtain the LAN property assuming the ergodicity of the process. In Chapter 3 we establish the LAN property for a simple Lévy process with unknown drift and diffusion parameters and intensity. In Chapter 4, using Malliavin's calculation and transition density estimates, we prove that the LAN property is verified for a jumping diffusion process whose drift coefficient depends on an unknown parameter. Finally, in the same direction we obtain in Chapter 5 the LAN property for a jump diffusion process where the two unknown parameters are involved in the drift and diffusion coefficients.

I give a pedagogical introduction to Brownian motion, stochastic calculus introduced by Ito in the fifties, following the elementary (at least not too technical) approach by Follmer [Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 143–150. Springer, Berlin, 1981]. Based on this, I develop the connection with linear and semi-linear parabolic PDEs. Then, I provide and analyze some Monte Carlo methods to approximate the solution to these PDEs. This course is aimed at master students, Ph.D. students and researchers interesting in the connection of stochastic processes with PDEs and their numerical counterpart. The reader is supposed to be familiar with basic concepts of probability (say first chapters of the book Probability essentials by Jacod and Protter [Probability Essentials, 2nd edn. Springer, Berlin, 2003]), but no a priori knowledge on martingales and stochastic processes is required.

No summary available.

We design a new stochastic algorithm (called SALT) that sequentially approximates a given function in L2 w.r.t. a probability measure, using a finite sample of the distribution. By increasing the sets of approximating functions and the simulation effort, we compute a L2-approximation with higher and higher accuracy. The simulation effort is tuned in a robust way that ensures the convergence under rather general conditions. Then, we apply SALT to build efficient control variates for accurate numerical integration. Examples and numerical experiments support the mathematical analysis.

This short note corrects an error (a factor is missing) in two formulas related to L 2 -limits, established in “Discrete time hedging errors for options with irregular payoffs” by E. Gobet and E. Temam, Finance and Stochastics, 5, 357–367 ( 2001 ). Copyright Springer-Verlag Berlin Heidelberg 2014.

We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explicit. We provide non asymptotic error bounds, which justifies the expansion accuracy as the time or the diffusion coefficients are small in a suitable sense.

We introduce random transformations called reversible shaking transformations which we use to design two schemes for estimating rare event probability. One is based on interacting particle systems (IPS) and the other on time-average on a single path (POP) using ergodic theorem. We discuss their convergence rates and provide numerical experiments including continuous stochastic processes and jump processes. Our examples cover rather important situations related to insurance, queueing system and random graph for instance. Both schemes have good performance, with a seemingly better one for POP.

We provide and analyse analytical approximations of BSDEs in the limit of small non-linearity {and short time}, in the case of non-smooth drivers. We identify the first and the second order approximations within this asymptotics and consider two topical financial applications: the two interest rates problem and the Funding Value Adjustment. In high dimensional diffusion setting, we show how to compute explicitly the first order formula by taking advantage of recent proxy techniques. Numerical tests up to dimension 10 illustrate the efficiency of the numerical schemes.

We design a numerical scheme for solving the Multi step-forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical least-squares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy. Numerical experiments are presented illustrating theoretical convergence estimates.

We study the optimization of the joint $(p^Y,p^Z)-$variations of two continuous semimartingales $(Y,Z)$ driven by the same Itô process $X$. The $p$-variations are defined on random grids made of finitely many stopping times. We establish an explicit asymptotic lower bound for our criterion, valid in rather great generality on the grids, and we exhibit minimizing sequences of hitting time form. The asymptotics is such that the spatial increments of $X$ and the number of grid points are suitably converging to 0 and $+\infty$ respectively.

In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error.

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the Z-component by [Ma-Zhang 2002]. When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.

We provide and analyse analytical approximations of BSDEs in the limit of small non-linearity {and short time}, in the case of non-smooth drivers. We identify the first and the second order approximations within this asymptotics and consider two topical financial applications: the two interest rates problem and the Funding Value Adjustment. In high dimensional diffusion setting, we show how to compute explicitly the first order formula by taking advantage of recent proxy techniques. Numerical tests up to dimension 10 illustrate the efficiency of the numerical schemes.

This thesis has 8 chapters. Chapter 1 is an introduction to the problems encountered in energy markets: low intervention frequency, high transaction costs, and spread option pricing. Chapter 2 studies the convergence of the hedging error of a call option in the Bachelier model, for proportional transaction costs (Leland-Lott model) and when the intervention frequency becomes infinite. It is shown that this error is bounded by a random variable proportional to the transaction rate. However, proofs of convergence in probability require regularities on sensitivities that are quite restrictive in practice. The following chapters circumvent these obstacles by studying almost certain convergences. Chapter 3 first develops new tools for almost sure convergence. These results have many consequences on the almost safe control of martingales and their quadratic variation, as well as their increments between two general stopping times. These trajectory convergence results are known to be difficult to obtain without information about the laws. In the following, we apply these results to the almost certain minimization of the renormalized quadratic variation of the hedging error of a general payoff option (multidimensional framework, Asian payoff, lookback) over a large class of intervention times. A lower bound on our criterion is found and a minimizing sequence of optimal stopping times is exhibited: these are random ellipsoid reaching times, depending on the gamma of the option. Chapter 4 studies the convergence of the hedging error of a convex payoff option (dimension 1) taking into account Leland-Lott transaction costs. We decompose the hedging error into a martingale part and a negligible part, and then minimize the quadratic variation of this martingale over a class of general attainment times for Deltas satisfying a certain nonlinear PDE on the second derivatives. We also exhibit a sequence of stopping times reaching this bound. Numerical tests illustrate our approach against a series of known strategies in the literature. Chapter 5 extends Chapter 3 by considering a functional of discrete variations of order Y and Z of two real-valued Itô processes Y and Z, the minimization being over a large class of stopping times used to compute the discrete variations. Lower bound and minimizing sequence are obtained. A numerical study on the transaction costs is done. Chapter 6 studies the Euler discretization of a multidimensional process X driven by an Itô semi-martingale Y. We minimize on the discretization grid times a quadratic criterion on the error of the scheme. We find a lower bound and an optimal grid, depending only on the observable data. Chapter 7 gives a central limit theorem for stochastic integral discretizations on any adapted ellipsoidal reach time grids. The limit correlation is a consequence of fine asymptotics on Dirichlet problems. In Chapter 8, we focus on expansion formulas for spread options, for models with local volatility. The key to the approach is to retain the martingale property of the arithmetic mean and to exploit the payoff call structure. Numerical tests show the relevance of the approach.

This thesis is devoted to the approximation of the expectation of a functional (which can depend on the whole trajectory) applied to a diffusion process (which can be multidimensional). The motivation for this work comes from financial mathematics where the valuation of options is reduced to the calculation of such expectations. The speed of the price calculations and calibration procedures is a very strong operational constraint and we bring real-time tools (or at least more competitive than Monte Carlo simulations in the multidimensional case) to meet these needs. To obtain approximation formulas, we choose a proxy model in which analytical calculations are possible, then we use stochastic developments around this proxy model and the Malliavin calculation to approximate the quantities of interest. In the case where the Malliavin calculus cannot be applied, we develop an alternative methodology combining Itô calculus and PDE arguments. All approaches (ranging from PDEs to stochastic analysis) provide explicit formulas and accurate error estimates as a function of the model parameters. Although the end result is often the same, the explicit derivation of the development can be very different and we compare the approaches, both in terms of how the correction terms are made explicit and the assumptions required to obtain the error estimates. We consider different classes of models and functionals in the four parts of the thesis. In Part I, we focus on local volatility models and obtain new approximation formulas for prices, sensitivities (delta) and implied volatilities of vanilla products that surpass in accuracy the previously known formulas. We also present new results for the valuation of forward-looking options. Part II deals with the analytical approximation of vanilla prices in models combining local and stochastic volatility (Heston type). This model is very delicate to analyze because its moments are not all finite and it is not regular in the Malliavin sense. The error analysis is original and the idea is to work on an appropriate regularization of the payoff and on a skillfully modified model, regular in the Malliavin sense and from which we can control the distance from the initial model. Part III deals with the valuation of regular barrier options in the context of local volatility models. This is a case not considered in the literature, difficult because of the exit time indicator. We mix Itô's calculation, PDE arguments, martingale properties and time convolutions of densities in order to decompose the approximation error and to explain the corrective terms. We obtain explicit and very accurate approximation formulas under a martingale assumption. Part IV presents a new methodology (denoted SAFE) for the effective law approximation of multidimensional diffusions in a rather general framework. We combine the use of a Gaussian proxy to approximate the law of the multidimensional diffusion and a local interpolation of the terminal function by finite elements. We give an estimate of the complexity of our methodology. We show an improved efficiency compared to Monte Carlo simulations in small and medium dimensions (up to 10).

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.

We design a variance reduction method to reduce the estimation error in regression problems. It is based on an appropriate use of other known regression functions. Theoretical estimates are supporting this improvement and numerical experiments are illustrating the efficiency of the method.

We design a variance reduction method to reduce the estimation error in regression problems. It is based on an appropriate use of other known regression functions. Theoretical estimates are supporting this improvement and numerical experiments are illustrating the efficiency of the method.

No summary available.

For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well.

We derive an analytical weak approximation of a multidimensional diffusion process as coefficients or time are small. Our methodology combines the use of Gaussian proxys to approximate the law of the diffusion and a Finite Element interpolation of the terminal function applied to the diffusion. We call this method Stochastic Approximation Finite Element (SAFE for short) method. We provide error bounds of our global approximation depending on the diffusion process coefficients, the time horizon and the regularity of the terminal function. Then we give estimates of the computational cost of our algorithm. This shows an improved efficiency compared to Monte-Carlo methods in small and medium dimensions (up to 10), which is confirmed by numerical experiments.

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No summary available.

Recent work in the field of financial mathematics has highlighted the importance of studying the regularity and fine-grained behavior of distribution tails for certain classes of diffusions with non-globally regular coefficients. In this thesis, we deal with problems arising from this context. We first study the existence, regularity and asymptotics in density space for solutions of stochastic differential equations by imposing only local conditions on the coefficients of the equation. Our analysis is based on the tools of Malliavin calculus and on estimates for Ito processes confined in a tube around a deterministic curve. We obtain significant estimates of the distribution function and density in classes of models including generalizations of the CIR and CEV and models with local-stochastic volatility: in the latter case, the estimates lead to the explosion of the moments of the underlying and thus have an impact on the asymptotic strike behavior of the implied volatility. The parametric modeling of the volatility surface, in turn, is the subject of the second part. We consider the SVI model of J. Gatheral, proposing a new quasi-explicit calibration strategy, whose performance on market data is illustrated. Then, we analyze the ability of SVI to generate approximations for symmetric smiles, by generalizing it to a time-dependent model. We test its application to a Heston model (without and with displacement), generating semi-closed approximations for the volatility smile.

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.

This thesis concerns three topics in numerical probability and financial mathematics. First, we study the L2 regularity modulus in time of the Z-component of a Markovian RDS with lipschitzian coefficients, but whose terminal function g is irregular. This module is related to the approximation error by Euler scheme. We show, in an optimal way, that the order of convergence is explicitly related to the fractional regularity of g. Next, we propose a sequential Monte Carlo method for efficient pricing of a CDO tranche, based on sequential control variables, in a setting with random recovery rates and i. I. D. Finally, we analyze the hedging error associated with the Delta-Gamma strategy. The fractional regularity of the payoff function plays a crucial role in the choice of rollover dates, in order to achieve optimal convergence speeds.

Two different topics of numerical probabilities and their financial applications are addressed in my thesis: one deals with the approximation and simulation of backward-looking stochastic differential equations (SRDEs), the other is related to American options and approaches them from the point of view of domain optimization and frontier perturbations. The first part of my thesis revisits the issue of convergence analysis in the time discretization of Markovian (Y,Z) RDSRs into a dynamic programming equation of n time steps. We establish a first-order limited expansion of the error on (Y,Z): precisely, the trajectory error on X is fully transferred to the EDSR and thus show that if X is accurately approximated or simulated, better speeds are possible (in 1/n). The second part of my thesis focuses on the resolution of EDSRs via the Picard process and sequential Monte Carlo methods. We have shown that the convergence of our algorithm takes place at geometric speed and with an accuracy independent at the 1st order of the number of simulations. The last part of my thesis gathers first results on the valuation of American options by optimization of the exercise frontier. The keystone of this type of approach is the ability to evaluate a gradient with respect to the frontier. Continuous time has been treated by Costantini et al (2006) and this thesis covers the discrete case of Bermuda options.

We present some contributions to the simulation and analysis of process discretization, with their applications in finance. We have grouped our work into 4 themes: 1. statistics of processes with discrete observations. 2. Discrete time coverage in finance. 3. Sensitivities of expectations. 4. discretization error analysis. The first chapter on process statistics is quite independent from the rest. On the other hand, the three other chapters correspond to a coherence and a progression in the questions raised. Nevertheless, as you read the book, you will notice links between the four parts: differentiation with respect to domains and improvement of output time simulation, expectation sensitivities and asymptotic statistics with Malliavin's calculus, expectation sensitivities and error analysis etc. The proofs of the results are based on the tools of Malliavin calculus, martingales, Partial Differential Equations and their links with Stochastic Differential Equations.

This thesis consists of two chapters, devoted to the approximation by euler diagrams of the expectation of a certain functional of the trajectory of a multidimensional diffusion process between time 0 and t. We are interested in the law in t of the diffusion killed at its exit from an open d : the functional in question is equal to a function f of the value of the process in t if the process remains in d between times 0 and t, and is 0 if the process has exited. The motivation for this work comes from financial mathematics, where the evaluation of the price of barrier options comes down to the calculation of this type of expectation. To obtain an approximate value of this expectation, we discretize the diffusion with an approximation scheme and evaluate the expectation of the functional for the scheme by a monte-carlo method. In the first chapter, we consider the euler scheme in continuous time, obtained from a regular subdivision of the time interval, and we analyze the approximation error as a function of the discretization step. The use of the malliavin calculus allows to reach the case where the function f is only measurable. The simulation by monte-carlo method is easy in the one-dimensional case, but becomes more delicate in higher dimension. In the second chapter, we consider the euler scheme in discrete time. In this case, the simulation is easy independently of the dimension, but the approximation error is more important than in the continuous case. The analysis of the error leads us in particular to explain a semimartingale decomposition of the orthogonal projection on the closure of d of a continuous semimartingale.