Parametric estimation for a class of multidimensional affine processes

Résumé

This thesis deals with statistical inference of some particular affine diffusion processes in the state space $\R_+^m\times\R^n$, where $m,n\in\N$. Such subclass of diffusions, denoted by $\mathit{AD}(m,n)$, is applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross (CIR) and Heston models. In this thesis, we consider two different cases : the first one is when $m=1$ and $n\in\N$ and the second one is when $m=2$ and $n=1$. For the $\mathit{AD}(1,n)$ model, we introduce, in Chapter 2, a classification result where we distinguish three different cases : subcritical, critical and supercritical. Then, we study the stationarity and the ergodicity of its solution under some assumptions on the drift parameters. For the parameter estimation problem, we use two different methods: the maximum likelihood estimation (MLE) and the conditional least squares estimation (CLSE). In Chapter 2, we present the estimator obtained by the MLE method based on continuous time observations and we study its consistency and its asymptotic behavior in ergodic and particular non-ergodic cases. In Chapter 3, we present the estimator obtained by the CLSE method based on continuous then discrete time observations with high frequency and infinite horizon and we study its consistency and its asymptotic behavior in ergodic and particular non-ergodic cases. It is worth to note here that we obtain the same asymptotic results in both discrete and continuous sets under additional assumptions on the discretization step $\Delta_N$. In Chapter 4, we study the $\mathit{AD}(2,1)$ model, called also double Heston model, we introduce first its classification with respect to subcritical, critical and supercritical case and we establish the relative stationarity and ergodicity theorems. In the statistical part of this chapter, we study the MLE and the CLSE of the ergodic double Heston model based on continuous time observations and we introduce its consistency and asymtotic normality theorems for each estimation method.

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