a.k.a. “Fancy ARIMA”.
Classically, you estimate statistics from many i.i.d. realisations from a presumed generating process.
What if your data are realisations of sequentially dependent time series? How do you estimate parameters from a single time series realisation?
By being a flashy quant!
Bonus points: How do you do this with many time series, whose parameters themselves have a distribution you wish to estimate?
See Mark Podolskij who explains “high frequency asymptotics” well. I think that the original framework is due to Jacod. (i.e. when you don’t have an asymptotic limit in number of data points, but in how densely you sample a single time series.)
This feels contrived for me, but it is probably interesting if you are not working with a multivariate Brownian motion, but a rather general Lévy process or something with interesting jumps AND continuous movement, and can sample with arbitrary density but not arbitrarily long. AFAICT this is little outside finance.
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Bibby, Bo Martin, and Michael Sørensen. 1995. “Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli 1 (1/2): 17–39. https://doi.org/10.2307/3318679.
Duembgen, Moritz, and Mark Podolskij. 2015. “High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications 125 (4): 1195–1217. https://doi.org/10.1016/j.spa.2014.08.007.
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Heinrich, Claudio, and Mark Podolskij. 2014. “On Spectral Distribution of High Dimensional Covariation Matrices,” October. http://arxiv.org/abs/1410.6764.
Heyde, C. C., and E. Seneta. 2010. “Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. Selected Works in Probability and Statistics. Springer New York. http://link.springer.com/chapter/10.1007/978-1-4419-5823-5_31.
Jacod, Jean. 1997. “On Continuous Conditional Gaussian Martingales and Stable Convergence in Law.” In Séminaire de Probabilités XXXI, edited by Jacques Azéma, Marc Yor, and Michel Emery, 232–46. Lecture Notes in Mathematics 1655. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0119308.
Jacod, Jean, Mark Podolskij, and Mathias Vetter. 2010. “Limit Theorems for Moving Averages of Discretized Processes Plus Noise.” The Annals of Statistics 38 (3): 1478–1545. https://doi.org/10.1214/09-AOS756.
Podolskij, Mark, and Mathias Vetter. 2010. “Understanding Limit Theorems for Semimartingales: A Short Survey: Limit Theorems for Semimartingales.” Statistica Neerlandica 64 (3): 329–51. https://doi.org/10.1111/j.1467-9574.2010.00460.x.