Skip to main content Accessibility help
×
Home

Limit theorems for multivariate Brownian semistationary processes and feasible results

  • Riccardo Passeggeri (a1) and Almut E. D. Veraart (a1)

Abstract

In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

Copyright

Corresponding author

* Postal address: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK.

References

Hide All
[1]Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Prob. 6, 325331.
[2]Barndorff-Nielsen, O. E. and Schmiegel, J. (2009). Brownian semistationary processes and volatility/intermittency. In Advanced Financial Modelling, eds Albrecher, H., Runggaldier, W., and Schachermayer, W., Walter de Gruyter, Berlin, pp. 125.
[3]Barndorff-Nielsen, O. E. and Shephard, N. (2004). Econometric analysis of realized covariation: high frequency based covariance, regression, and correlation in financial economics. Econometrica 72, 885925.
[4]Barndorff-Nielsen, O. E., Benth, F. E. and Veraart, A. E. D. (2013). Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes. Bernoulli 19, 803845.
[5]Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2009). Multipower variation for Brownian semistationary processes (full version). CREATES research paper 2009-21, Aarhus University.
[6]Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2009). Power variation for Gaussian processes with stationary increments. Stoch. Process. Appl. 119, 18451865.
[7]Barndorff-Nielsen, O. E., Corcuera, J. M. and Podolskij, M. (2011). Multipower variation for Brownian semistationary processes. Bernoulli 17, 11591194.
[8]Barndorff-Nielsen, O. E., Pakkanen, M. S. and Schmiegel, J. (2014). Assessing relative volatility/intermittency/energy dissipation. Electron. J. Statist. 8, 19962021.
[9]Bateman, H. (1954). Tables of Integral Transforms. McGraw-Hill, New York.
[10]Bennedsen, M. (2017). A rough multi-factor model of electricity spot prices. Energy Econom. 63, 301313.
[11]Bennedsen, M., Lunde, A. and Pakkanen, M. (2017). Decoupling the short- and long-term behavior of stochastic volatility. Preprint. Available at https://arxiv.org/abs/1610.00332v2.
[12]Bennedsen, M., Lunde, A. and Pakkanen, M. (2017). Hybrid scheme for Brownian semistationary processes Finance Stoch. 21, 931965.
[13]Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[14]Corcuera, J. M. (2012). New central limit theorems for functionals of Gaussian processes and their applications. Methodology Comput. Appl. Prob. 14.3, 477500.
[15]Corcuera, J. M., Hedevang, E., Podolskij, M. S. and Pakkanen, M. (2013). Asymptotic theory for Brownian semi-stationary processes with application to turbulence. Stoch. Process. Appl. 123, 25522574.
[16]Granelli, A. (2017). Limit theorems and stochastic models for dependence and contagion in financial markets. Doctoral Thesis, Imperial College London.
[17]Granelli, A. and Veraart, A. E. D. (2019). A central limit theorem for the realised covariation of a bivariate Brownian semistationary process. To appear in Bernoulli.
[18]Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités XXXI (Lecture Notes Math. 1655), Springer, Berlin, pp. 232246.
[19]Jacod, J. (2008), Asymptotic properties of realized power variations and related functionals of semimartingales. Stoch. Process. Appl. 118, 517559.
[20]Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus. From Stein’s Method to Universality (Camb. Tracts Math. 92). Cambridge University Press.
[21]Podolskij, M. and Vetter, M. (2010). Understanding limit theorems for semimartingales: a short survey. Statist. Neerlandica 64, 329351.
[22]Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics, Vol. II. Academic Press, New York.
[23]Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.
[24]Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: with Applications to Statistics. Springer, New York.

Keywords

MSC classification

Related content

Powered by UNSILO

Limit theorems for multivariate Brownian semistationary processes and feasible results

  • Riccardo Passeggeri (a1) and Almut E. D. Veraart (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.