REY Clement

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In this paper, we propose a method to prove the total variation convergence of approximation of Markov semigroups with singularities. In particular our approach is adapted to the study of numerical schemes for Stochastic Differential Equation (SDE) with simply locally smooth coefficients. First we present this method and then, we apply it to the CIR process. In particular, we consider the weak second order scheme introduced in [2] (Alfonsi 2010) and we prove that it also converges towards the CIR diffusion process for the total variation distance. This convergence occurs with almost order two.

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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.

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In this paper, we propose a method to prove the total variation convergence of approximation of Markov semigroups with singularities. In particular our approach is adapted to the study of numerical schemes for Stochastic Differential Equation (SDE) with simply locally smooth coefficients. First we present this method and then, we apply it to the CIR process. In particular, we consider the weak second order scheme introduced in [2] (Alfonsi 2010) and we prove that it also converges towards the CIR diffusion process for the total variation distance. This convergence occurs with almost order two.

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This paper provides a general and abstract approach to approximate ergodic regimes of Markov and Feller processes. More precisely, we show that the recursive algorithm presented by Lamberton an Pagès in 2002, and based on simulation algorithms of stochastic schemes with decreasing step can be used to build invariant measures for general Markov and Feller processes. We also propose applications in three different configurations: Approximation of Markov switching Brownian diffusion ergodic regimes using Euler scheme, approximation of Markov Brownian diffusion ergodic regimes with Milstein scheme and approximation of general diffusions with jump components ergodic regimes.

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The first goal of this paper is to prove that, regularization properties of a Markov semigroup enable to prove convergence in total variation distance for approximation schemes for the semigroup. Moreover, using an interpolation argument we obtain estimates for the error in distribution sense (at the level of the densities of the semigroup with respect to the Lebesgue measure). In a second step, we build an abstract Malliavin calculus based on a splitting procedure, which turns out to be the suited instrument in order to prove the above mentioned regularization properties. Finally, we use these results in order to estimate the error in total variation distance for the Ninomiya Victoir scheme (which is an approximation scheme, of order 2, for diffusion processes).

In this paper, we study a third weak order scheme for diffusion processes which has been introduced by Alfonsi [1]. This scheme is built using cubature methods and is well defined under an abstract commutativity condition on the coefficients of the underlying diffusion process. Moreover, it has been proved in [1], that the third weak order convergence takes place for smooth test functions. First, we provide a necessary and sufficient explicit condition for the scheme to be well defined when we consider the one dimensional case. In a second step, we use a result from [3] and prove that, under an ellipticity condition, this convergence also takes place for the total variation distance with order 3. We also give an estimate of the density function of the diffusion process and its derivatives.

In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.

The development of technology and computer science in the last decades, has led the emergence of numerical methods for the approximation of Stochastic Differential Equations (SDE) and for the estimation of their parameters. This thesis treats both of these two aspects. In particular, we study the effectiveness of those methods. The first part will be devoted to SDE's approximation by numerical schemes while the second part will deal with the estimation of the parameters of the Wishart process. First, we focus on approximation schemes for SDE's. We will treat schemes which are defined on a time grid with size $n$. We say that the scheme $ X^n $ converges weakly to the diffusion $ X $, with order $ h in mathbb{N} $, if for every $ T> 0 $, $ vert mathbb{E} [f (X_T) -f (X_T^n)]vert leqslant C_f / h^n $. Until now, except in some particular cases (Euler and Victoir Ninomiya schemes), researches on this topic require that $ C_f$ depends on the supremum norm of $ f $ as well as its derivatives. In other words $C_f =C sum_{vert alpha vert leqslant q} Vert partial_{alpha} f Vert_{ infty}$. Our goal is to show that, if the scheme converges weakly with order $ h $ for such $C_f$, then, under non degeneracy and regularity assumptions, we can obtain the same result with $ C_f=C Vert f Vert_{infty}$. We are thus able to estimate $mathbb{E} [f (X_T)]$ for a bounded and measurable function $f$. We will say that the scheme converges for the total variation distance, with rate $h$. We will also prove that the density of $X^n_T$ and its derivatives converge toward the ones of $X_T$. The proof of those results relies on a variant of the Malliavin calculus based on the noise of the random variable involved in the scheme. The great benefit of our approach is that it does not treat the case of a particular scheme and it can be used for many schemes. For instance, our result applies to both Euler $(h = 1)$ and Ninomiya Victoir $(h = 2)$ schemes. Furthermore, the random variables used in this set of schemes do not have a particular distribution law but belong to a set of laws. This leads to consider our result as an invariance principle as well. Finally, we will also illustrate this result for a third weak order scheme for one dimensional SDE's. The second part of this thesis deals with the topic of SDE's parameter estimation. More particularly, we will study the Maximum Likelihood Estimator (MLE) of the parameters that appear in the matrix model of Wishart. This process is the multi-dimensional version of the Cox Ingersoll Ross (CIR) process. Its specificity relies on the square root term which appears in the diffusion coefficient. Using those processes, it is possible to generalize the Heston model for the case of a local covariance. This thesis provides the calculation of the EMV of the parameters of the Wishart process. It also gives the speed of convergence and the limit laws for the ergodic cases and for some non-ergodic case. In order to obtain those results, we will use various methods, namely: the ergodic theorems, time change methods or the study of the joint Laplace transform of the Wishart process together with its average process.

IN Single Photon Emission Computed Tomography (SPECT), attenuation and scatter introduce important artefacts in the reconstructed images biasing the diagnosis and the follow-up of the subject. Indeed, the presence of scatter results in a blurring and haziness of the observed projections, reduces reconstructed contrast and introduces significant uncertainty in quantification of the underlying activity distribution.

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

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