# zbMATH — the first resource for mathematics

ARCH models as diffusion approximations. (English) Zbl 0719.60089
Summary: 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
Full Text:
##### References:
 [1] Abel, A. B.: Stock prices under time-varying dividend risk: an exact solution in an infinite-horizon general equilibrium model. Journal of monetary economics 22, 375-393 (1988) [2] Arnold, L.: Stochastic differential equations: theory and applications. (1974) · Zbl 0278.60039 [3] Barsky, R. B.: Why don’t the prices of stocks and bonds move together?. American economic review 79, 1132-1146 (1989) [4] Bernanke, B. S.: Irreversibility, uncertainty and cyclical investment. Quarterly journal of economics 98, 85-106 (1983) [5] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201 [6] Black, F.: Studies of stock market volatility changes. Proceedings of the American statistical association, business and economic statistics section, 177-181 (1976) [7] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31, 307-327 (1986) · Zbl 0616.62119 [8] Clark, P. K.: A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135-155 (1973) · Zbl 0308.90011 [9] Cox, J.; Ingersoll, J.; Ross, S.: An intertemporal general equilibrium model of asset prices. Econometrica 53, 363-384 (1985) · Zbl 0576.90006 [10] Cox, J.; Ross, S.; Rubinstein, M.: Options pricing: A simplified approach. Journal of financial economics 7, 229-263 (1979) · Zbl 1131.91333 [11] Duffie, D.; Singleton, K. J.: Simulated moments estimation of Markov models of asset prices. (1989) · Zbl 0783.62099 [12] Engle, R. F.: Autoregressive conditional heteroskedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099 [13] Engle, R. F.; Bollerslev, T.: Modelling the persistence of conditional variances. Econometric reviews 5, 1-50 (1986) · Zbl 0619.62105 [14] Ethier, S. N.; Kurtz, T. G.: Markov processes: characterization and convergence. (1986) · Zbl 0592.60049 [15] French, K. R.; Schwert, G. W.; Stambaugh, R. F.: Expected stock returns and volatility. Journal of financial economics 19, 3-30 (1987) [16] Gennotte, G. and T.A. Marsh, Variations in economic uncertainty and risk premiums on capital assets, Mimeo. (University of California, Berkeley, CA). [17] Geweke, J.: Modelling the persistence of conditional variances: A comment. Econometric reviews 5, 57-61 (1986) [18] Hosking, J. R. M.: Fractional differencing. Biometrika 68, 165-176 (1981) · Zbl 0464.62088 [19] Johnson, N. L.; Kotz, S.: Distributions in statistics: continuous univariate distributions - I. (1970) · Zbl 0213.21101 [20] Kushner, H. J.: Approximation and weak convergence methods for random processes, with applications to stochastic systems theory. (1984) · Zbl 0551.60056 [21] Liptser, R. S.; Shiryayev, A. N.: Statistics of random processes. (1977) · Zbl 0364.60004 [22] Lo, A.: Maximum likelihood estimation of generalized Itô processes with discretely sampled data. Econometric theory 4, 231-247 (1988) [23] Merton, R. C.: An intertemporal capital asset pricing model. Econometrica 41, 867-888 (1973) · Zbl 0283.90003 [24] Nelson, D. B.: The time series behavior of stock market volatility and returns. Unpublished doctoral dissertation (1988) [25] Nelson, D. B.: Stationarity and persistence in the $$GARCH(1,1)$$ model. Graduate school of business working paper series in economics and econometrics no. 88-68 (1988) [26] Nelson, D. B.: Conditional heteroskedasticity in asset returns: A new approach. Graduate school of business working paper series in economics and econometrics no. 89-73 (1989) [27] Officer, R. R.: The variability of the market factor of the New York stock exchange. Journal of business 46, 434-453 (1973) [28] Pantula, S. G.: Modelling the persistence of conditional variances: A comment. Econometric reviews 5, 71-73 (1986) [29] Pardoux, E.; Talay, D.: Discretization and simulation of stochastic differential equations. Acta applicandae Mathematica 3, 23-47 (1985) · Zbl 0554.60062 [30] Priestley, M. B.: Spectral analysis and time series. (1981) · Zbl 0537.62075 [31] Sampson, M.: A stationarity condition for the $$GARCH(1,1)$$ model. (1988) [32] Sandburg, C.: The people, yes. The complete poems of carl sandburg (1969) [33] Stroock, D. W.; Varadhan, S. R. S.: Multidimensional diffusion processes. (1979) · Zbl 0426.60069 [34] Wong, E.: The construction of a class of stationary Markov processes. Sixteenth symposia in applied mathematics - stochastic processes in mathematical physics and engineering, 264-276 (1964)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.