ARCH MODELS AS DIFFUSION APPROXIMATIONS

This paper investigates the convergence of stochastic difference equations (e.g., ARCH) to stochastic differential equations as the length of the discrete time intervals between observations goes to zero. These results are applied to the GARCH(1,1) model of Bollerslev (1986) and to the AR(1) Exponential ARCH model of Nelson (1989). In their continuous time limits, the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal, respectively. In addition, a class of diffusion approximations based on the Exponential ARCH model is developed. © 1990.