Metatimes, random measures and cylindrical random variables
Volume 8, Issue 3 (2021), pp. 349–371
Pub. online: 20 May 2021
Type: Research Article
Open Access
Open Access
Received
5 February 2021
5 February 2021
Revised
30 April 2021
30 April 2021
Accepted
3 May 2021
3 May 2021
Published
20 May 2021
20 May 2021
Abstract
Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.
References
Barndorff-Nielsen, O.E.: Stationary infinitely divisible processes. Braz. J. Probab. Stat. 25, 294–322 (2011). MR2832888. https://doi.org/10.1214/11-BJPS140
Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.: Ambit Stochastics. Springer, Cham (2018). 2018. MR3839270. https://doi.org/10.1007/978-3-319-94129-5
Barndorff-Nielsen, O.E., Pedersen, J.: Meta-times and extended subordination. Theory Probab. Appl. 56(2), 319–327 (2012). MR3136481. https://doi.org/10.4213/tvp4384
Barndorff-Nielsen, O.E., Lunde, A., Shephard, N., Veraart, A.: Integer-valued trawl processes: a class of stationary infinitely divisible processes. Scand. J. Stat. 41(3), 693–724 (2014). MR3249424. https://doi.org/10.1111/sjos.12056
Barndorff-Nielsen, O.E., Shiryaev, A.: Change of Time and Change of Measure. World Scientific, Singapore (2015). MR3363697. https://doi.org/10.1142/9609
Folland, G.B.: Real Analysis. Wiley-Interscience, New York (1984). MR0767633
Metivier, M., Pellaumail, J.: Stochastic Integration. Academic Press, New York (1980). MR0578177
Molchanov, I.: Theory of Random Sets. Springer, London (2005). MR2132405
Pedersen, G.K.: Analysis Now. Springer, New York (1989). MR0971256. https://doi.org/10.1007/978-1-4612-1007-8
Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82(3), 451–487 (1989). 1989. MR1001524. https://doi.org/10.1007/BF00339998
Shiryaev, A.N.: Probability. Springer, New York (1996). MR1368405. https://doi.org/10.1007/978-1-4757-2539-1
Walsh, J.: An introduction to stochastic partial differential equations. In: Carmona, R., Kesten, H., Walsh, J. (eds.) Ecole d’Ete de Probabilites de Saint-Flour XIV. Lecture Notes in Mathematics, vol. 1180. Springer, New York (1984). 1986. MR0876085. https://doi.org/10.1007/BFb0074920
