A link between complete models with stochastic volatility and ARCH models
The author proposes a heteroscedastic model in discrete time which converges, when the sampling interval goes to zero, towards the special complete model with stochastic volatility in continuous time. The aim of the paper is to investigate the link between ARCH models and the complete model with stochastic volatility of \textit{David G. Hobson} and \textit{L. C. G. Rogers} [Math. Finance 8, No. 1, 27--48 (1998; Zbl 0908.90012)]. Under a simple assumption on the rate of convergence of the parameters, the discrete-time model, which is conditionally heteroscedastic, converges to the diffusion process, that is a complete model with stochastic volatility. This diffusion approximation result provides a numerical scheme, to approximate the Hobson and Rogers model. The stationary and moment properties of the model are studied. The consistency of the pseudo conditional likelihood maximum estimates for this specific model are proved.
scientific article; zbMATH DE number 2118691