A link between complete models with stochastic volatility and ARCH models

In this paper, we propose a heteroskedastic model in discrete time which converges, when the sampling interval goes to zero, towards the complete model with stochastic volatility in continuous time described in Hobson and Rogers (1998). Then, we study its stationarity and moment properties. In particular, we exhibit a specific model which shares many properties with the GARCH(1,1) model, establishing a clear link between the two approaches. We also prove the consistency of the pseudo conditional likelihood maximum estimates for this specific model.Received: December 2002Mathematics Subject Classification: 90A09, 60J60, 62M05JEL Classification: C32This work was supported in part by Dynstoch European network. Thanks to David Hobson for introducing me to these models, and to Valentine Genon-Catalot for numerous and very fruitful discussion on this work. The author is also grateful to Uwe Kuchler for various helpful suggestions, and to two referees and an associate editor for their comments and suggestions.