Abstract:
The resolution of stochastic differential equations put a very complex problem. For this
raison, we have presented a deep study (Continuous Generalized Auto Regressive Conditioned Heteroscedastic) (COGARCH), largely applied in financial mathematics and in economy.
In the first part of this memoir, we have presented necessary definitions and properties of all processes used in this study. Then, we have sweep the computing of stochastic integral using Itô method. In this context, the existence, the unique and the stability of solutions of
stochastic differential equations have been studied. In the second part of this memoir, we have presented in general manner the differential equation of COGARCH process. Therefore, a deep study of volatility process has been illustrated. This study is based on the stationary and the non negativity of this process. The autocorrelation function of squares increases is presented to estimate the process parameters of COGARCH. In the third part, we have presented two different methods of estimation of process parameters only for the particularly case COGARCH (1,1) and its consistencies. It’s acted of moment method and quasi-maximum of vraisemblance. Finally, we arrived to push the sail from statistical and probabilistic properties of COGARCH processes
