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Weak convergence and distributional assumptions for a general class of nonlinear ARCH models. (English) Zbl 0896.62111
Summary: We study the continuous time behavior of a class of nonlinear ARCH models. Such a class was proposed by Z. Ding, C. Granger and R. Engle [J. Empir. Finance 1, 83-106 (1993)] and is an encompassing formulation of several ARCH equations proposed so far in the literature. We obtain a general diffusion process according to which a sort of Box-Cox power transform has to be applied to the local dynamics of the conditional standard deviation. This process provides a fairly wide range of choices concerning the modeling of stochastic volatility in continuous time. The results can be of interest to both theoretical fields such as mathematical finance and more applied domains such as Monte Carlo-ARCH estimation of contingent claim models with stochastic volatility.
Throughout the paper, we also consider the case in which innovations processes are not conditionally normal, thus generalizing the first approximation results obtained by D. Nelson [J. Econ. 45, No. 1/2, 7-38 (1990; Zbl 0719.60089)]. More specifically, assuming that the innovations process is conditionally general error distributed enables us to find its stationary distribution. We find solutions in closed form such as the generalized Student’s \(t\) distribution.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B84 Economic time series analysis
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
91B28 Finance etc. (MSC2000)
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References:
[1] Arnold L., Theory and Applications · Zbl 0278.60039
[2] Billingsley P., Convergence of Probability Measures · Zbl 0944.60003 · doi:10.1002/9780470316962
[3] DOI: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062
[4] Black F., Business and Economic Statistics Section 81 pp 177– (1976)
[5] DOI: 10.1016/0304-4076(86)90063-1 · Zbl 0616.62119 · doi:10.1016/0304-4076(86)90063-1
[6] DOI: 10.2307/1925546 · doi:10.2307/1925546
[7] DOI: 10.1016/0304-4076(92)90064-X · Zbl 0825.90057 · doi:10.1016/0304-4076(92)90064-X
[8] Bollerslev B., Forthcoming in the Handbook of Econometrics 4 (1993)
[9] DOI: 10.1016/0927-5398(93)90006-D · doi:10.1016/0927-5398(93)90006-D
[10] DOI: 10.2307/1912773 · Zbl 0491.62099 · doi:10.2307/1912773
[11] DOI: 10.1080/07474938608800095 · Zbl 0619.62105 · doi:10.1080/07474938608800095
[12] DOI: 10.1016/0304-4076(92)90074-2 · Zbl 04506593 · doi:10.1016/0304-4076(92)90074-2
[13] Fornari F., Statistica 54 pp 293– (1994)
[14] DOI: 10.2307/2329067 · doi:10.2307/2329067
[15] Granger C., Some Properties of Absolute Return, and Alternative Measure of Risk, unpublished 48 (1993)
[16] Granger C., Stylized Facts on the Temporal and Distributional Properties of Daily Data from Speculative Markets, unpublished 48 (1994)
[17] DOI: 10.1016/0022-0531(79)90043-7 · Zbl 0431.90019 · doi:10.1016/0022-0531(79)90043-7
[18] DOI: 10.2307/2297980 · Zbl 0805.90026 · doi:10.2307/2297980
[19] DOI: 10.1093/rfs/6.2.327 · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[20] DOI: 10.2307/2526988 · Zbl 0744.62152 · doi:10.2307/2526988
[21] DOI: 10.1111/j.1467-9965.1992.tb00027.x · Zbl 0900.90095 · doi:10.1111/j.1467-9965.1992.tb00027.x
[22] Hsieh D., Journal of Business and Economic Statistics 7 pp 307– (1988)
[23] DOI: 10.2307/2328253 · doi:10.2307/2328253
[24] Karatzas I., Brownian Motion and Stochastic Calculus · Zbl 0734.60060
[25] Kushner H., with Applications to Stochastic Systems Theory · Zbl 0126.33304
[26] DOI: 10.1093/rfs/6.2.293 · doi:10.1093/rfs/6.2.293
[27] DOI: 10.2307/2328939 · doi:10.2307/2328939
[28] DOI: 10.1007/BF00135033 · Zbl 0442.62010 · doi:10.1007/BF00135033
[29] DOI: 10.1016/0304-4076(90)90092-8 · Zbl 0719.60089 · doi:10.1016/0304-4076(90)90092-8
[30] DOI: 10.2307/2938260 · Zbl 0722.62069 · doi:10.2307/2938260
[31] DOI: 10.1016/0304-4076(92)90065-Y · Zbl 0761.62169 · doi:10.1016/0304-4076(92)90065-Y
[32] DOI: 10.2307/2951474 · Zbl 0804.62085 · doi:10.2307/2951474
[33] DOI: 10.2307/2328636 · doi:10.2307/2328636
[34] DOI: 10.2307/2330793 · doi:10.2307/2330793
[35] DOI: 10.1016/0304-4076(94)90043-4 · Zbl 0800.62807 · doi:10.1016/0304-4076(94)90043-4
[36] Sims c., Martingale-Like Behavior of Prices and Interest Rates
[37] Stroock D., Multidimensional Diffusion Processes · Zbl 0426.60069 · doi:10.1007/3-540-28999-2
[38] Taylor S., Modeling Financial Time Series · Zbl 1130.91345
[39] DOI: 10.1111/j.1467-9965.1994.tb00057.x · Zbl 0884.90054 · doi:10.1111/j.1467-9965.1994.tb00057.x
[40] DOI: 10.1016/0304-405X(87)90009-2 · doi:10.1016/0304-405X(87)90009-2
[41] Wong E., Sixteen symposia in Applied mathematics. Stochastic processes in mathematical Physics and Engineering pp 264–
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