ARCH models as diffusion approximations.

*(English)*Zbl 0719.60089Summary: 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 T. Bollerslev [J. Econ. 31, 307–327 (1986; Zbl 0616.62119)] and to the AR(1) Exponential ARCH model of the author [Econometrica 59, No.2, 347–370 (1991; Zbl 0722.62069)]. 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.

##### MSC:

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

60H99 | Stochastic analysis |

62P20 | Applications of statistics to economics |

##### Keywords:

convergence of stochastic difference equations; stochastic differential equations; diffusion approximations
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##### References:

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