Study and modeling of stochastic differential equations.

Summary 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.