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A note on maximal inequality for stochastic convolutions
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 785-790 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution \[ \int ^t_0 S(t-s)\psi (s)\mathrm{d}W(s) \] driven by a Wiener process $W$ in a Hilbert space in the case when the semigroup $S(t)$ is of contraction type.
Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution \[ \int ^t_0 S(t-s)\psi (s)\mathrm{d}W(s) \] driven by a Wiener process $W$ in a Hilbert space in the case when the semigroup $S(t)$ is of contraction type.
Classification : 60H05, 60H15
Keywords: infinite-dimensional Wiener process; stochastic convolution; maximal inequality 
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Hausenblas, Erika; Seidler, Jan. A note on maximal inequality for stochastic convolutions. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 785-790. http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a8/

[1] Z.  Brzeźniak: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 (1997), 245–295. | DOI | MR

[2] Z.  Brzeźniak and S.  Peszat: Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces. Stochastic Processes, Physics and Geometry: New Interplays, I (Leipzig, 1999), Amer.  Math.  Soc., Providence, 2000, pp. 55–64. | MR

[3] G. Da Prato, M.  Iannelli and L.  Tubaro: Semi-linear stochastic differential equations in Hilbert spaces. Boll. Un. Mat. Ital.  A (5) 16 (1979), 168–177. | MR

[4] G. Da Prato, S.  Kwapień and J.  Zabczyk: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23 (1987), 1–23. | MR

[5] G. Da Prato and J.  Zabczyk: A note on stochastic convolution. Stochastic Anal. Appl. 10 (1992), 143–153. | DOI | MR

[6] G.  Da Prato and J.  Zabczyk: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992. | MR

[7] E. B.  Davies: Quantum Theory of Open Systems. Academic Press, London, 1976. | MR | Zbl

[8] D. Gątarek: A note on nonlinear stochastic equations in Hilbert spaces. Statist. Probab. Lett. 17 (1993), 387–394. | DOI | MR

[9] A. Ichikawa: Some inequalities for martingales and stochastic convolutions. Stochastic Anal. Appl. 4 (1986), 329–339. | DOI | MR | Zbl

[10] P. Kotelenez: A submartingale type inequality with applications to stochastic evolution equations. Stochastics 8 (1982), 139–151. | DOI | MR | Zbl

[11] P. Kotelenez: A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations. Stochastic Anal. Appl. 2 (1984), 245–265. | DOI | MR | Zbl

[12] J. Seidler: Da Prato-Zabczyk’s maximal inequality revisited I. Math. Bohem. 118 (1993), 67–106. | MR | Zbl

[13] J. Seidler and T. Sobukawa: Exponential integrability of stochastic convolutions. Submitted.

[14] B. Sz.-Nagy: Sur les contractions de l’espace de Hilbert. Acta Sci. Math. Szeged 15 (1953), 87–92. | MR | Zbl

[15] B. Sz.-Nagy: Transformations de l’espace de Hilbert, fonctions de type positif sur un groupe. Acta Sci. Math. Szeged 15 (1954), 104–114. | MR

[16] B. Sz.-Nagy and C. Foiaş: Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam, 1970. | MR

[17] L. Tubaro: An estimate of Burkholder type for stochastic processes defined by the stochastic integral. Stochastic Anal. Appl. 2 (1984), 187–192. | DOI | MR | Zbl