We consider a mixed moving average (MMA) process $X$ driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of $X$ have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.
References
[1] Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes., J. Appl. Probab. 21 930–934. 0552.60049 10.2307/3213710[1] Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes., J. Appl. Probab. 21 930–934. 0552.60049 10.2307/3213710
[2] Bardet, J. M., Doukhan, P. and León, J. R. (2008). A uniform central limit theorem for the periodogram and its applications to Whittle parametric estimation for weakly dependent time series., J. Time Series Anal. 29 906–945. 1198.62087 10.1111/j.1467-9892.2008.00588.x[2] Bardet, J. M., Doukhan, P. and León, J. R. (2008). A uniform central limit theorem for the periodogram and its applications to Whittle parametric estimation for weakly dependent time series., J. Time Series Anal. 29 906–945. 1198.62087 10.1111/j.1467-9892.2008.00588.x
[3] Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes., Theory Probab. Appl. 45 175–194. 1003.60039 10.1137/S0040585X97978166[3] Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes., Theory Probab. Appl. 45 175–194. 1003.60039 10.1137/S0040585X97978166
[4] Barndorff-Nielsen, O. E, Benth, F. and Veraart, E. D. (2015). Cross-commodity modelling by multivariate ambit fields, in, Commodities, Energy and Environmental Finance, edited by R. Aïd, M. Ludkovski and R. Sircar. Springer, New York.[4] Barndorff-Nielsen, O. E, Benth, F. and Veraart, E. D. (2015). Cross-commodity modelling by multivariate ambit fields, in, Commodities, Energy and Environmental Finance, edited by R. Aïd, M. Ludkovski and R. Sircar. Springer, New York.
[5] Barndorff-Nielsen, O. E, Hedevang, E., Schmiegel, J. and Szozda, B. (2016). Some recent developments in ambit stochastics, in, Stochastics of Environmental and Financial Economics, edited by F. Benth and G. Di Nunno. Springer, Cham. 1382.60003[5] Barndorff-Nielsen, O. E, Hedevang, E., Schmiegel, J. and Szozda, B. (2016). Some recent developments in ambit stochastics, in, Stochastics of Environmental and Financial Economics, edited by F. Benth and G. Di Nunno. Springer, Cham. 1382.60003
[6] Barndorff-Nielsen, O. E., Nicolato, E. and Shepard, N. (2002). Some recent developments in stochastic volatility modeling. Special issue on volatility modelling., Quant. Finance 2 11–23. 1405.91583 10.1088/1469-7688/2/1/301[6] Barndorff-Nielsen, O. E., Nicolato, E. and Shepard, N. (2002). Some recent developments in stochastic volatility modeling. Special issue on volatility modelling., Quant. Finance 2 11–23. 1405.91583 10.1088/1469-7688/2/1/301
[7] Barndorff-Nielsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, methodology and computing., Methodol. Comput. Appl. Probab. 7 335–352. 1089.60014 10.1007/s11009-005-4521-0[7] Barndorff-Nielsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, methodology and computing., Methodol. Comput. Appl. Probab. 7 335–352. 1089.60014 10.1007/s11009-005-4521-0
[8] Barndorff-Nielsen, O. E. and Schmiegel, J. (2007). Ambit processes: with applications to turbulence and cancer growth, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.[8] Barndorff-Nielsen, O. E. and Schmiegel, J. (2007). Ambit processes: with applications to turbulence and cancer growth, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.
[9] Barndorff-Nielsen, O. E. and Shepard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics., J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241. 0983.60028 10.1111/1467-9868.00282[9] Barndorff-Nielsen, O. E. and Shepard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics., J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241. 0983.60028 10.1111/1467-9868.00282
[10] Barndorff-Nielsen, O. E. and Stelzer, R. (2011). Multivariate supOU processes., Ann. Appl. Probab. 21 140–182. 1235.60032 10.1214/10-AAP690 euclid.aoap/1292598030[10] Barndorff-Nielsen, O. E. and Stelzer, R. (2011). Multivariate supOU processes., Ann. Appl. Probab. 21 140–182. 1235.60032 10.1214/10-AAP690 euclid.aoap/1292598030
[11] Barndorff-Nielsen, O. E. and Stelzer, R. (2013). The multivariate supOU stochastic volatility model., Math. Finance 23 275–296. 1262.91139 10.1111/j.1467-9965.2011.00494.x[11] Barndorff-Nielsen, O. E. and Stelzer, R. (2013). The multivariate supOU stochastic volatility model., Math. Finance 23 275–296. 1262.91139 10.1111/j.1467-9965.2011.00494.x
[12] Barndorff-Nielsen, O. E. and Veraart, A. E. D. (2013). Stochastic volatility of volatility and variance risk premia., J. Fin. Econometrics 11 1–46.[12] Barndorff-Nielsen, O. E. and Veraart, A. E. D. (2013). Stochastic volatility of volatility and variance risk premia., J. Fin. Econometrics 11 1–46.
[13] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987)., Regular Variation. Cambridge University Press, Cambridge. 0617.26001[13] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987)., Regular Variation. Cambridge University Press, Cambridge. 0617.26001
[14] Brandes, D-P. and Curato I. V. (2018). On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence., arxiv:1804.02254v1.[14] Brandes, D-P. and Curato I. V. (2018). On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence., arxiv:1804.02254v1.
[16] Brockwell, P.J. (2001). Lévy driven CARMA processes. Non-linear non-Gaussian models and related filtering methods (Tokyo, 2000)., Ann. Inst. Statist. Math. 53 113–124.[16] Brockwell, P.J. (2001). Lévy driven CARMA processes. Non-linear non-Gaussian models and related filtering methods (Tokyo, 2000)., Ann. Inst. Statist. Math. 53 113–124.
[17] Cont, R. and Tankov, P. (2004)., Financial Modeling With Jump Processes. Chapman & Hall/CRC, Boca Raton. 1052.91043[17] Cont, R. and Tankov, P. (2004)., Financial Modeling With Jump Processes. Chapman & Hall/CRC, Boca Raton. 1052.91043
[18] Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications., Stochastic Process Appl. 106 63–80. 1075.60513 10.1016/S0304-4149(03)00040-1[18] Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications., Stochastic Process Appl. 106 63–80. 1075.60513 10.1016/S0304-4149(03)00040-1
[19] Dedecker, J., Doukhan, P., Lang, G., Léon, J. R., Louhichi, S. and Prieur, C. (2008)., Weak dependence: with examples and applications. Springer.[19] Dedecker, J., Doukhan, P., Lang, G., Léon, J. R., Louhichi, S. and Prieur, C. (2008)., Weak dependence: with examples and applications. Springer.
[20] Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes., Ann. Inst. H. Poincaré Probab. Statist. 36 1–34. 0949.60049 10.1016/S0246-0203(00)00111-4[20] Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes., Ann. Inst. H. Poincaré Probab. Statist. 36 1–34. 0949.60049 10.1016/S0246-0203(00)00111-4
[21] Doukhan, P. (1994)., Mixing: properties and examples. Lecture Notes in Statistics, 85. Springer-Verlag, New York. 0801.60027[21] Doukhan, P. (1994)., Mixing: properties and examples. Lecture Notes in Statistics, 85. Springer-Verlag, New York. 0801.60027
[22] Doukhan, P., Fokianos, K. and Li, X. (2012). On weak dependence conditions: the case of discrete valued processes., Statist. Probab. Lett. 82 1941–1948. 1254.60036 10.1016/j.spl.2012.06.020[22] Doukhan, P., Fokianos, K. and Li, X. (2012). On weak dependence conditions: the case of discrete valued processes., Statist. Probab. Lett. 82 1941–1948. 1254.60036 10.1016/j.spl.2012.06.020
[23] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities., Stochastic Process. Appl. 84 313–342. 0996.60020 10.1016/S0304-4149(99)00055-1[23] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities., Stochastic Process. Appl. 84 313–342. 0996.60020 10.1016/S0304-4149(99)00055-1
[24] Doukhan, P. and Wintenberger, O. (2007). An invariance principle for weakly dependent stationary general models., Probab. Math. Statist. 27 45–73. 1124.60031[24] Doukhan, P. and Wintenberger, O. (2007). An invariance principle for weakly dependent stationary general models., Probab. Math. Statist. 27 45–73. 1124.60031
[25] Fasen, V. and Klüppelberg, C. (2007). Extremes of supOU processes, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.[25] Fasen, V. and Klüppelberg, C. (2007). Extremes of supOU processes, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.
[26] Fechner, Z. and Stelzer, R. (2018). Limit behavior of the truncated pathwise Fourier-transformation of Lévy driven CARMA processes for non-equidistant discrete time observations., Statist. Sinica 28 1633–1650. 1394.62119[26] Fechner, Z. and Stelzer, R. (2018). Limit behavior of the truncated pathwise Fourier-transformation of Lévy driven CARMA processes for non-equidistant discrete time observations., Statist. Sinica 28 1633–1650. 1394.62119
[27] Fuchs, F. and Stelzer, R. (2013). Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model., ESAIM Probab. Stat. 17 455–471. 1311.60023 10.1051/ps/2011158[27] Fuchs, F. and Stelzer, R. (2013). Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model., ESAIM Probab. Stat. 17 455–471. 1311.60023 10.1051/ps/2011158
[28] Giraitis, L., Koul, H. L. and Surgailis, D. (2012)., Large Sample Inference for Long Memory Processes. Imperial College Press, London. 1279.62016[28] Giraitis, L., Koul, H. L. and Surgailis, D. (2012)., Large Sample Inference for Long Memory Processes. Imperial College Press, London. 1279.62016
[29] Grahovac, D., Leonenko, N., Sikorskii, A. and Taqqu, M. S. (2018). The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes., Bernoulli, to appear. 07066248 10.3150/18-BEJ1044 euclid.bj/1560326436[29] Grahovac, D., Leonenko, N., Sikorskii, A. and Taqqu, M. S. (2018). The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes., Bernoulli, to appear. 07066248 10.3150/18-BEJ1044 euclid.bj/1560326436
[30] Grahovac, D., Leonenko, N., Sikorskii, A. and Tešnjak, I. (2016). Intermittency of Superposition of Ornstein-Uhlenbeck type processes., J. Stat. Phys. 165 390–408. 1356.60121 10.1007/s10955-016-1616-7[30] Grahovac, D., Leonenko, N., Sikorskii, A. and Tešnjak, I. (2016). Intermittency of Superposition of Ornstein-Uhlenbeck type processes., J. Stat. Phys. 165 390–408. 1356.60121 10.1007/s10955-016-1616-7
[31] Grahovac, D., Leonenko, N. and Taqqu, M. S. (2017). Limit theorems,scaling of moments and intermittency for integrated finite variance supOU processes., ArXiv:1711.09623v1 .[31] Grahovac, D., Leonenko, N. and Taqqu, M. S. (2017). Limit theorems,scaling of moments and intermittency for integrated finite variance supOU processes., ArXiv:1711.09623v1 .
[32] Griffin, J. E. and Steel, M. F. (2010). Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes., Comput. Statist. Data Anal. 11 2594–2608. 1284.91586 10.1016/j.csda.2009.06.008[32] Griffin, J. E. and Steel, M. F. (2010). Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes., Comput. Statist. Data Anal. 11 2594–2608. 1284.91586 10.1016/j.csda.2009.06.008
[34] Jacod, J. and Shiryaev, A. N. (2003)., Limit theorems for stochastic processes, 2nd ed. Springer, Berlin. 1018.60002[34] Jacod, J. and Shiryaev, A. N. (2003)., Limit theorems for stochastic processes, 2nd ed. Springer, Berlin. 1018.60002
[35] Kelly, B. C., Treu, T., Malkan, M., Pancoast, A. and Woo, J-H. (2013). Active Galactic Nucleus Black Hole Mass Estimates in the Era of Time Domain Astronomy., Astrophys. J. 779 1–18.[35] Kelly, B. C., Treu, T., Malkan, M., Pancoast, A. and Woo, J-H. (2013). Active Galactic Nucleus Black Hole Mass Estimates in the Era of Time Domain Astronomy., Astrophys. J. 779 1–18.
[36] Kerss, A. D. J., Leonenko, N. N. and Sikorskii, A. (2014). Risky asset models with tempered stable fractal activity time., Stoch. Anal. Appl. 32 642–663. 1305.60039 10.1080/07362994.2014.913183[36] Kerss, A. D. J., Leonenko, N. N. and Sikorskii, A. (2014). Risky asset models with tempered stable fractal activity time., Stoch. Anal. Appl. 32 642–663. 1305.60039 10.1080/07362994.2014.913183
[37] Leonenko, N. N, Petherick, S. and Sikorskii, A. (2012). Fractal activity time models for risky asset with dependence and generalized hyperbolic distributions., Stoch. Anal. Appl. 30 476–492. 1251.91062 10.1080/07362994.2012.668443[37] Leonenko, N. N, Petherick, S. and Sikorskii, A. (2012). Fractal activity time models for risky asset with dependence and generalized hyperbolic distributions., Stoch. Anal. Appl. 30 476–492. 1251.91062 10.1080/07362994.2012.668443
[38] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes., Bernoulli 12 1099–1126. 1126.60038 10.3150/bj/1165269152 euclid.bj/1165269152[38] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes., Bernoulli 12 1099–1126. 1126.60038 10.3150/bj/1165269152 euclid.bj/1165269152
[39] Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA process., Stochastic Processes Appl. 117 96–120. 1115.62087 10.1016/j.spa.2006.05.014[39] Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA process., Stochastic Processes Appl. 117 96–120. 1115.62087 10.1016/j.spa.2006.05.014
[40] Masuda, H. (2007). Ergodicity and exponential $\beta $-mixing bounds for multidimensional diffusions with jumps., Stochastic Process. Appl. 117 35–56. 1118.60070 10.1016/j.spa.2006.04.010[40] Masuda, H. (2007). Ergodicity and exponential $\beta $-mixing bounds for multidimensional diffusions with jumps., Stochastic Process. Appl. 117 35–56. 1118.60070 10.1016/j.spa.2006.04.010
[42] Moser, M. and Stelzer, R. (2013). Functional regular variation of Lévy-driven multivariate mixed moving average processes., Extremes 16 351–382. 1284.60093 10.1007/s10687-012-0165-y[42] Moser, M. and Stelzer, R. (2013). Functional regular variation of Lévy-driven multivariate mixed moving average processes., Extremes 16 351–382. 1284.60093 10.1007/s10687-012-0165-y
[43] Pedersen, J. (2003). The Lévy-Itô decomposition of an independently scattered random measure., MaPhySto Research Paper 2, available at http://www.maphysto.dk.[43] Pedersen, J. (2003). The Lévy-Itô decomposition of an independently scattered random measure., MaPhySto Research Paper 2, available at http://www.maphysto.dk.
[44] Pigorsch, C. and Stelzer, R. (2009). A multivariate Ornstein-Uhlenbeck type stochastic volatility model., preprint available on www.uni-ulm.de/en/mawi/finmath/people/stelzer/publications.html. 1221.60074 10.3150/08-BEJ175 euclid.bj/1251463280[44] Pigorsch, C. and Stelzer, R. (2009). A multivariate Ornstein-Uhlenbeck type stochastic volatility model., preprint available on www.uni-ulm.de/en/mawi/finmath/people/stelzer/publications.html. 1221.60074 10.3150/08-BEJ175 euclid.bj/1251463280
[45] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes., Probab. Theory and Relat. Fields 82 451–487. 0659.60078 10.1007/BF00339998[45] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes., Probab. Theory and Relat. Fields 82 451–487. 0659.60078 10.1007/BF00339998
[46] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition., Proc. Nat. Acad. Sci. U.S.A. 42 43–47. 0070.13804 10.1073/pnas.42.1.43[46] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition., Proc. Nat. Acad. Sci. U.S.A. 42 43–47. 0070.13804 10.1073/pnas.42.1.43
[47] Sato, K. I. (1999)., Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.[47] Sato, K. I. (1999)., Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
[48] Stelzer, R., Tossdorf, T. and Wittilinger, M. (2015). Moment based estimation of supOU processes and a related stochastic volatility model., Stat. Risk Model. 32 1–24.[48] Stelzer, R., Tossdorf, T. and Wittilinger, M. (2015). Moment based estimation of supOU processes and a related stochastic volatility model., Stat. Risk Model. 32 1–24.
[49] Stelzer, R. and Zavĭsin, J. (2015). Derivative pricing under the possibility of long memory in the supOU stochastic volatility model, in, Innovations in Quantitative Risk Management edited by K. Glau, M. Scherer and R. Zagst. Springer.[49] Stelzer, R. and Zavĭsin, J. (2015). Derivative pricing under the possibility of long memory in the supOU stochastic volatility model, in, Innovations in Quantitative Risk Management edited by K. Glau, M. Scherer and R. Zagst. Springer.