In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.

Dans cette thèse, nous nous intéressons à des noeuds critiques du calcul du risque de contrepartie, la valorisation rapide des produits dérivées et de leurs sensibilités. Nous proposons plusieurs méthodes mathématiques et informatiques pour répondre à cette problématique. Nous contribuons à quatre domaines différents: une extension de la méthode Vibrato et l'application des méthodes multilevel Monte Carlo pour le calcul des grecques à ordre élevé n>1 avec une technique de différentiation automatique. La troisième contribution concerne l'évaluation des produits Américain, ici nous nous servons d'un schéma pararéel pour l'accélération du processus de valorisation et nous faisons également une application pour la résolution d'une équation différentielle stochastique rétrograde. La quatrième contribution est la conception d'un moteur de calcul performant à architecture parallèle. / In this thesis, we will focus on the critical node of the computation of counterparty credit risk, the fast evaluation of financial derivatives and their sensitivities. We propose several mathematical and computer-based methods to address this issue. We have contributed to four areas: an extension of the Vibrato method and an application of the weighted multilevel Monte Carlo for the computation of the greeks for high order derivatives n>1 with automatic differentiation. The third contribution concerns the evaluation of American style option, here we use a parareal scheme to speed up the assessing process and we made an application for solving backward stochastic differential equations. The last contribution is the conception of an efficient computation engine for financial derivatives with a parallel architecture.

Monte Carlo path simulations are common in mathematical and computational finance as a way of estimating the expected values of a quantity such as a European put option, which is functional to the solution of a stochastic differential equation (SDE). The computational complexity of the standard Monte Carlo (MC) method grows quite large quickly, so in this thesis we focus on the Multilevel Monte Carlo (MLMC) method by Giles, which uses multigrid ideas to reduce the computational complexity. We use a Euler-Maruyama time discretisation for the approximation of the SDE and investigate how the convergence rate of the MLMC method improves the computational times and cost in comparison with the standard MC method. We perform a numerical analysis on the computational times and costs in order to achieve the desired accuracy and present our findings on the performance of the MLMC method on a European put option compared to the standard MC method.

In biochemically reactive systems with small copy numbers of one or more reactant molecules, stochastic effects dominate the dynamics. In the first part of this thesis, we design novel efficient simulation techniques for a reliable and fast estimation of various statistical quantities for stochastic biological and chemical systems under the framework of Stochastic Reaction Networks. In the first work, we propose a novel hybrid multilevel Monte Carlo (MLMC) estimator, for systems characterized by having simultaneously fast and slow timescales. Our hybrid multilevel estimator uses a novel split-step implicit tau-leap scheme at the coarse levels, where the explicit tau-leap method is not applicable due to numerical instability issues. In a second work, we address another challenge present in this context called the high kurtosis phenomenon, observed at the deep levels of the MLMC estimator. We propose a novel approach that combines the MLMC method with a pathwise-dependent importance sampling technique for simulating the coupled paths. Our theoretical estimates and numerical analysis show that our method improves the robustness and complexity of the multilevel estimator, with a negligible additional cost. In the second part of this thesis, we design novel methods for pricing financial derivatives. Option pricing is usually challenging due to: 1) The high dimensionality of the input space, and 2) The low regularity of the integrand on the input parameters. We address these challenges by developing different techniques for smoothing the integrand to uncover the available regularity. Then, we approximate the resulting integrals using hierarchical quadrature methods combined with Brownian bridge construction and Richardson extrapolation. In the first work, we apply our approach to efficiently price options under the rough Bergomi model. This model exhibits several numerical and theoretical challenges, implying classical numerical methods for pricing being either inapplicable or computationally expensive. In a second work, we design a numerical smoothing technique for cases where analytic smoothing is impossible. Our analysis shows that adaptive sparse grids’ quadrature combined with numerical smoothing outperforms the Monte Carlo approach. Furthermore, our numerical smoothing improves the robustness and the complexity of the MLMC estimator, particularly when estimating density functions.

In numerical multiscale methods, one relies on a coupling between macroscopic model and a microscopic model. The macroscopic model does not include the microscopic properties that the microscopic model offers and that are vital for the desired solution. Such microscopic properties include parameters like material coefficients and fluxes which may variate microscopically in the material. The effective values of this data can be computed by running local microscale simulations while averaging the microscopic data. One desires the effect of the microscopic coefficients on a macroscopic scale, and this can be done using classical homogenisation theory. One method in the homogenization theory is to use local elliptic cell problems in order to compute the homogenized constants and this results in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clambda%20/R" data-classname="equation_inline" data-title="" /> error where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clambda" data-classname="equation" /> is the wavelength of the microscopic variations and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R" data-classname="mimetex" data-title="" /> is the size of the simulation domain. However, one could greatly improve the accuracy by a slight modification in the homogenisation elliptic PDE and use a filter in the averaging process to get much better orders of error. The modification relates the elliptic PDE to a parabolic one, that could be solved and integrated in time to get the elliptic PDE's solution.   In this thesis I apply the modified elliptic cell homogenization method with a qth order filter to compute the homogenized diffusion constant in a 2d Poisson equation on a rectangular domain. Two cases were simulated. The diffusion coefficients used in the first case was a deterministic 2d matrix function and in the second case I used stochastic 2d matrix function, which results in a 2d stochastic differential equation (SDE). In the second case two methods were used to determine the expected value of the homogenized constants, firstly the multi-level Monte Carlo (MLMC) and secondly its generalization multi-index Monte Carlo (MIMC). The performance of MLMC and MIMC is then compared when used in the process of the homogenization.   In the homogenization process the finite element notations in 2d were used to estimate a solution of the Poisson equation. The grid spatial steps were varied in a first order differences in MLMC (square mesh) and first order mixed differences in MIMC (which allows for rectangular mesh).

We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.

A standard problem in mathematical finance is the calculation of the price of some financial derivative such as various types of options. Since there exists analytical solutions in only a few cases it will often boil down to estimating the price with Monte Carlo simulation in conjunction with some numerical discretization scheme. The upside of using what we can call standard Monte Carlo is that it is relative straightforward to apply and can be used for a wide variety of problems. The downside is that it has a relatively slow convergence which means that the computational cost or complexity can be very large. However, this slow convergence can be improved upon by using Multilevel Monte Carlo instead of standard Monte Carlo. With this approach it is possible to reduce the computational complexity and cost of simulation considerably. The aim of this thesis is to introduce the reader to the Multilevel Monte Carlo method with applications to European and Asian call options in both the Black-Scholes-Merton (BSM) model and in the Heston model. To this end we first cover the necessary background material such as basic probability theory, estimators and some of their properties, the stochastic integral, stochastic processes and Ito’s theorem. We introduce stochastic differential equations and two numerical discretizations schemes, the Euler–Maruyama scheme and the Milstein scheme. We define strong and weak convergence and illustrate these concepts with examples. We also describe the standard Monte Carlo method and then the theory and implementation of Multilevel Monte Carlo. In the applications part we perform numerical experiments where we compare standard Monte Carlo to Multilevel Monte Carlo in conjunction with the Euler–Maruyama scheme and Milsteins scheme. In the case of a European call in the BSM model, using the Euler–Maruyama scheme, we achieved a cost O(ε-2(log ε)2) to reach the desired error in accordance with theory in comparison to the O(ε-3) cost for standard Monte Carlo. When using Milsteins scheme instead of the Euler–Maruyama scheme it was possible to reduce the cost in terms of the number of simulations needed to achieve the desired error even further. By using Milsteins scheme, a method with greater order of strong convergence than Euler–Maruyama, we achieved the O(ε-2) cost predicted by the complexity theorem compared to the standard Monte Carlo cost of order O(ε-3). In the final numerical experiment we applied the Multilevel Monte Carlo method together with the Euler–Maruyama scheme to an Asian call in the Heston model. In this case, where the coefficients of the Heston model do not satisfy a global Lipschitz condition, the study of strong or weak convergence is much harder. The numerical experiments suggested that the strong convergence was slightly slower compared to what was found in the case of a European call in the BSM model. Nevertheless, we still achieved substantial savings in computational cost compared to using standard Monte Carlo.

Cette thèse est consacrée à l'étude des propriétés de convergence forte du schéma de Ninomiya et Victoir. Les auteurs de ce schéma proposent d'approcher la solution d'une équation différentielle stochastique (EDS), notée $X$, en résolvant $d+1$ équations différentielles ordinaires (EDOs) sur chaque pas de temps, où $d$ est la dimension du mouvement brownien. Le but de cette étude est d'analyser l'utilisation de ce schéma dans une méthode de Monte-Carlo multi-pas. En effet, la complexité optimale de cette méthode est dirigée par l'ordre de convergence vers $0$ de la variance entre les schémas utilisés sur la grille grossière et sur la grille fine. Cet ordre de convergence est lui-même lié à l'ordre de convergence fort entre les deux schémas. Nous montrons alors dans le chapitre $2$, que l'ordre fort du schéma de Ninomiya-Victoir, noté $X^{NV,eta}$ et de pas de temps $T/N$, est $1/2$. Récemment, Giles et Szpruch ont proposé un estimateur Monte-Carlo multi-pas réalisant une complexité $Oleft(epsilon^{-2}right)$ à l'aide d'un schéma de Milstein modifié. Dans le même esprit, nous proposons un schéma de Ninomiya-Victoir modifié qui peut-être couplé à l'ordre fort $1$ avec le schéma de Giles et Szpruch au dernier niveau d'une méthode de Monte-Carlo multi-pas. Cette idée est inspirée de Debrabant et Rossler. Ces auteurs suggèrent d'utiliser un schéma d'ordre faible élevé au niveau de discrétisation le plus fin. Puisque le nombre optimal de niveaux de discrétisation d'une méthode de Monte-Carlo multi-pas est dirigé par l'erreur faible du schéma utilisé sur la grille fine du dernier niveau de discrétisation, cette technique permet d'accélérer la convergence de la méthode Monte-Carlo multi-pas en obtenant une approximation d'ordre faible élevé. L'utilisation du couplage à l'ordre $1$ avec le schéma de Giles-Szpruch nous permet ainsi de garder un estimateur Monte-Carlo multi-pas réalisant une complexité optimale $Oleft( epsilon^{-2} right)$ tout en profitant de l'erreur faible d'ordre $2$ du schéma de Ninomiya-Victoir. Dans le troisième chapitre, nous nous sommes intéressés à l'erreur renormalisée définie par $sqrt{N}left(X - X^{NV,eta}right)$. Nous montrons la convergence en loi stable vers la solution d'une EDS affine, dont le terme source est formé des crochets de Lie entre les champs de vecteurs browniens. Ainsi, lorsqu'au moins deux champs de vecteurs browniens ne commutent pas, la limite n'est pas triviale. Ce qui assure que l'ordre fort $1/2$ est optimal. D'autre part, ce résultat peut être vu comme une première étape en vue de prouver un théorème de la limite centrale pour les estimateurs Monte-Carlo multi-pas. Pour cela, il faut analyser l'erreur en loi stable du schéma entre deux niveaux de discrétisation successifs. Ben Alaya et Kebaier ont prouvé un tel résultat pour le schéma d'Euler. Lorsque les champs de vecteurs browniens commutent, le processus limite est nul. Nous montrons que dans ce cas précis, que l'ordre fort est $1$. Dans le chapitre 4, nous étudions la convergence en loi stable de l'erreur renormalisée $Nleft(X - X^{NV}right)$ où $X^{NV}$ est le schéma de Ninomiya-Victoir lorsque les champs de vecteurs browniens commutent. Nous démontrons la convergence du processus d'erreur renormalisé vers la solution d'une EDS affine. Lorsque le champ de vecteurs dritf ne commute pas avec au moins un des champs de vecteurs browniens, la vitesse de convergence forte obtenue précédemment est optimale / This thesis is dedicated to the study of the strong convergence properties of the Ninomiya-Victoir scheme, which is based on the resolution of $d+1$ ordinary differential equations (ODEs) at each time step, to approximate the solution to a stochastic differential equation (SDE), where $d$ is the dimension of the Brownian. This study is aimed at analysing the use of this scheme in a multilevel Monte Carlo estimator. Indeed, the optimal complexity of this method is driven by the order of convergence to zero of the variance between the two schemes used on the coarse and fine grids at each level, which is related to the strong convergence order between the two schemes. In the second chapter, we prove strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,eta}$, with time step $T/N$, to the solution $X$ of the limiting SDE. Recently, Giles and Szpruch proposed a modified Milstein scheme and its antithetic version, based on the swapping of each successive pair of Brownian increments in the scheme, permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity $Oleft(epsilon^{-2}right)$ for the precision $epsilon$, as in a simple Monte Carlo method with independent and identically distributed unbiased random variables. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. This idea is inspired by Debrabant and R"ossler who suggest to use a scheme with high order of weak convergence on the finest grid at the finest level of the multilevel Monte Carlo method. As the optimal number of discretization levels is related to the weak order of the scheme used in the finest grid at the finest level, Debrabant and R"ossler manage to reduce the computational time, by decreasing the number of discretization levels. The coupling with the Giles-Szpruch scheme allows us to combine both ideas. By this way, we preserve the optimal complexity $Oleft(epsilon^{-2}right)$ and we reduce the computational time, since the Ninomiya-Victoir scheme is known to exhibit weak convergence with order 2. In the third chapter, we check that the normalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. This result ensures that the strong convergence rate is actually $1/2$ when at least two Brownian vector fields do not commute. To link this result to the multilevel Monte Carlo estimator, it can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived recently by Ben Alaya and Kebaier for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. We then prove strong convergence with order $1$ in this case. The fourth chapter deals with the convergence of the normalized error process $Nleft(X - X^{NV}right)$, where $X^{NV}$ is the Ninomiya-Victoir in the commutative case. We prove its stable convergence in law to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the Stratonovich drift vector field

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.

Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-Planck. / This work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.