The Annals of Probability

Stochastic integration with respect to cylindrical Lévy processes

Adam Jakubowski and Markus Riedle

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A cylindrical Lévy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Lévy processes in Hilbert spaces. The space of admissible integrands consists of càglàd, adapted stochastic processes with values in the space of Hilbert–Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semimartingale with càdlàg trajectories.

Article information

Source
Ann. Probab. Volume 45, Number 6B (2017), 4273-4306.

Dates
Received: December 2015
Revised: October 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069260

Digital Object Identifier
doi:10.1214/16-AOP1164

Subjects
Primary: 60H05: Stochastic integrals
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60G20: Generalized stochastic processes 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
Cylindrical Lévy processes stochastic integration decoupled tangent sequence cylindrical Brownian motion random measures

Citation

Jakubowski, Adam; Riedle, Markus. Stochastic integration with respect to cylindrical Lévy processes. Ann. Probab. 45 (2017), no. 6B, 4273--4306. doi:10.1214/16-AOP1164. https://projecteuclid.org/euclid.aop/1513069260


Export citation

References

  • [1] Applebaum, D. and Riedle, M. (2010). Cylindrical Lévy processes in Banach spaces. Proc. Lond. Math. Soc. (3) 101 697–726.
  • [2] Beśka, M., Kłopotowski, A. and Słomiński, L. (1982). Limit theorems for random sums of dependent $d$-dimensional random vectors. Z. Wahrsch. Verw. Gebiete 61 43–57.
  • [3] Brzeźniak, Z. and Zabczyk, J. (2010). Regularity of Ornstein–Uhlenbeck processes driven by a Lévy white noise. Potential Anal. 32 153–188.
  • [4] Daletskij, Y. L. (1967). Infinite-dimensional elliptic operators and parabolic equations connected with them. Russian Math. Surveys 22 1–53.
  • [5] de la Peña, V. H. and Giné, E. (1999). Decoupling. from Dependence to Independence, Randomly Stopped Processes. $U$-Statistics and Processes. Martingales and Beyond. Springer, New York.
  • [6] Gaveau, B. (1973). Intégrale stochastique radonifiante. C. R. Acad. Sci. Paris Sér. A-B 276 A617–A620.
  • [7] Jakubowski, A. (1980). On limit theorems for sums of dependent Hilbert space valued random variables. In Mathematical Statistics and Probability Theory (Proc. Sixth Internat. Conf., WisłA, 1978). Lecture Notes in Statist. 2 178–187. Springer, New York.
  • [8] Jakubowski, A. (1982). Twierdzenia graniczne dla sum zależnych zmiennych losowych o wartościach w przestrzeni Hilberta. Ph.D. thesis, Nicolaus Copernicus University in Toruń.
  • [9] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. Henri Poincaré Probab. Stat. 22 263–285.
  • [10] Jakubowski, A. (1986). Principle of conditioning in limit theorems for sums of random variables. Ann. Probab. 14 902–915.
  • [11] Jakubowski, A. (1988). Tightness criteria for random measures with application to the principle of conditioning in Hilbert spaces. Probab. Math. Statist. 9 95–114.
  • [12] Jakubowski, A., Kwapień, S., de Fitte, P. R. and Rosiński, J. (2002). Radonification of cylindrical semimartingales by a single Hilbert–Schmidt operator. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 429–440.
  • [13] Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case. Probabilistic models for nonlinear partial differential equations. In Lect. Notes Math. 1627 197–285. Springer, Berlin.
  • [14] Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
  • [15] Lépingle, D. and Ouvrard, J.-Y. (1973). Martingales browniennes hilbertiennes. C. R. Acad. Sci. Paris Sér. A-B 276 A1225–A1228.
  • [16] Métivier, M. and Pellaumail, J. (1976). Cylindrical stochastic integral. Sém. de L’Université de Rennes 63 1–28.
  • [17] Métivier, M. and Pellaumail, J. (1980). Stochastic Integration. Academic Press, New York.
  • [18] Mikulevičius, R. and Rozovskiǐ, B. L. (1998). Normalized stochastic integrals in topological vector spaces. Séminaire de Probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686 137–165.
  • [19] Mikulevicius, R. and Rozovskii, B. L. (1999). Martingale problems for stochastic PDE’s. In Stochastic Partial Differential Equations: Six Perspectives (R. A. Carmona, ed.). Math. Surveys Monogr. 64 243–325. Amer. Math. Soc., Providence, RI.
  • [20] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York, NY.
  • [21] Peszat, S. and Zabczyk, J. (2013). Time regularity of solutions to linear equations with Lévy noise in infinite dimensions. Stochastic Process. Appl. 123 719–751.
  • [22] Priola, E. and Zabczyk, J. (2011). Structural properties of semilinear SPDEs driven by cylindrical stable processes. Probab. Theory Related Fields 149 97–137.
  • [23] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Springer, Heidelberg.
  • [24] Riedle, M. (2011). Infinitely divisible cylindrical measures on Banach spaces. Studia Math. 207 235–256.
  • [25] Riedle, M. (2014). Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An $L^{2}$ approach. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 1450008, 19.
  • [26] Riedle, M. (2015). Ornstein–Uhlenbeck processes driven by cylindrical Lévy processes. Potential Anal. 42 809–838.
  • [27] Vakhaniya, N. N., Tarieladze, V. I. and Chobanyan, S. A. (1987). Probability Distributions on Banach Spaces. Reidel, Dordrecht.