In this paper we construct and study an integral of operator-valued functions with respect to Hilbert space-valued measures generated by a resolution of identity. Our integral generalizes the Itô stochastic integral with respect to normal martingales and the Itô integral on a Fock space.
@article{CML_2010__2_1_135_0,
author = {Volodymyr Tesko},
title = {Hilbert space-valued integral of operator-valued functions},
journal = {Confluentes Mathematici},
pages = {135--157},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {1},
year = {2010},
doi = {10.1142/S1793744210000144},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000144/}
}
TY - JOUR AU - Volodymyr Tesko TI - Hilbert space-valued integral of operator-valued functions JO - Confluentes Mathematici PY - 2010 SP - 135 EP - 157 VL - 2 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000144/ DO - 10.1142/S1793744210000144 LA - en ID - CML_2010__2_1_135_0 ER -
%0 Journal Article %A Volodymyr Tesko %T Hilbert space-valued integral of operator-valued functions %J Confluentes Mathematici %D 2010 %P 135-157 %V 2 %N 1 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744210000144/ %R 10.1142/S1793744210000144 %G en %F CML_2010__2_1_135_0
Volodymyr Tesko. Hilbert space-valued integral of operator-valued functions. Confluentes Mathematici, Tome 2 (2010) no. 1, pp. 135-157. doi : 10.1142/S1793744210000144. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000144/
[1] S. Albeverio, Yu. M. Berezansky and V. Tesko, Ukr. Mat. Zh. 59, 588 (2007).
[2] S. Attal , Quantum Noises, Open Quantum Systems. II. The Markovian Approach , Lect. Notes in Math. 1881 ( Springer-Verlag , 2006 ) .
[3] S. Attal and A. Belton, Probab. Th. Relat. Fields 139, 543 (2007), DOI: 10.1007/s00440-006-0052-z .
[4] S. Attal and J. M. Lindsay, Ann. Probab. 32, 488 (2004).
[5] M. Berezansky, N. W. Zhernakov and G. F. Us, Rep. Math. Phys. 28, 347 (1989).
[6] C. Dellacherie, B. Maisonneuve and P. A. Meyer, Probabilites et Potentiel. Ch XVII a XXIV. Processus de Markov (fin), Complements de calcul stochastique (Paris, 1992) .
[7] M. Hitsuda, Formula for Brownian partial derivatives, Second Japan-USSR Symp. Probab.2 (1972) pp. 111-114.
[8] M. Emery , On the Azema Martingales , Lect. Notes in Math. 1372 ( Springer , 1989 ) .
[9] M. Emery, Probability Theory and Mathematical Statistics, eds. I. A. Ibragimov and A. Yu. Zaitsev (Gordon and Breach, 1996) pp. 155-166.
[10] N. A. Kachanovsky and V. A. Tesko, Ukr. Math. J. 61, 733 (2009), DOI: 10.1007/s11253-009-0257-2 .
[11] J. Ma, P. Protter and J. S. Martin, Bernoulli 4, 81 (1998), DOI: 10.2307/3318533 .
[12] P. Malliavin , Stochastic Analysis ( Springer-Verlag , 1997 ) .
[13] P. A. Meyer , Quantum Probability for Probabilists , Lect. Notes in Math. 1538 ( Springer-Verlag , 1993 ) .
[14] D. Nualart , The Malliavin Calculus and Related Topics ( Springer-Verlag , 1995 ) .
[15] B. Simon , The Euclidian (Quantum) Field Theory ( Princeton Univ. Press , 1974 ) .
[16] A. V. Skorohod, Teor. Verojatnost. i Primenen. 20, 223 (1975).
[17] V. Tesko, Methods Funct. Anal. Topology 14, 132 (2008).
[18] I. I. Gikhman and A. V. Skorokhod , The Theory of Random Processes 3 ( Nauka , 1975 ) .
[19] Ph. E. Protter , Stochastic Integration and Differential Equations ( Springer-Verlag , 2004 ) .
[20] S. Watanabe , Lecture on Stochastic Differential Equations and Malliavin Calculus ( Springer-Verlag , 1994 ) .
Cité par Sources :