Inférence statistique dans les équations différentielles stochastiques

Abstract

In this thesis, we are studying a class of continuous-time bilinear processes (COBL(1,1)) generated by some stochastic differential equations where we have investigate some probabilistic properties and statistical inference. We use Itô approach for studying the L2 structure of the COBL(1,1) process and its powers for any order with time varying coefficients. Furthermore we prove that these results can be obtained by using the transfer functions approach, moreover, by the spectral representation of the process, we give also conditions for the stability of moments, in particular the moments of the quadratic process provide us to checking the presence of the so called Taylor property for COBL(1,1) process. In a second part of this thesis, we use the results of the first part and we propose some methods of estimation for involving unknown parameters, so, we starting by the moments method (MM) to estimate the parameters by two methods, taking into consideration the relation that exists between the moments of the process and its quadratic version and those associate with the incremented processes where we have showed that the resulting estimators are strongly consistent and asymptotically normal under certain conditions. Using the linear representation of COBL(1,1) process, we are able to propose three other methods, one is in frequency domain and the rest are in time domain and we prove the asymptotic properties of the proposed estimators. Simulation studies are presented in order to illustrate the performances of the different estimators, furthermore, this methods are used to model some real data such as the exchanges rate of the Algerian Dinar against the US-dollar and against the single European currency and the electricity consumption sampled each 15mn in Algeria.