Quantum stochastic integral representations of Fock space operators

References

• Aase , K. , Øksendal , B. , Privault , N. and Ubøe , J. 2000 . White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance . Finance Stoch. , 4 : 465 – 496 .
   Google Scholar
• Attal , S. 1994 . An algebra of non-commutative bounded semimartingales: Square and angle quantum brackets . J. Funct. Anal. , 124 : 292 – 332 .
   Web of Science ®Google Scholar
• Belavkin , V.P. 1991 . A quantum nonadapted Ito formula and stochastic analysis in Fock scale . J. Funct. Anal. , 102 : 414 – 447 .
   Web of Science ®Google Scholar
• Chung , D.M. , Chung , T.S. and Ji , U.C. 1997 . A simple proof of analytic characterization theorem for operator symbols . Bull. Korean Math. Soc. , 34 : 421 – 436 .
   Google Scholar
• Chung , D.M. , Ji , U.C. and Obata , N. 2002 . Quantum stochastic analysis via white noise operators in weighted Fock space . Rev. Math. Phys. , 14 : 241 – 272 .
   Web of Science ®Google Scholar
• Hudson , R.L. and Parthasarathy , K.R. 1984 . Quantum Ito's formula and stochastic evolutions . Commun. Math. Phys. , 93 : 301 – 323 .
   Web of Science ®Google Scholar
• Ji , U.C. 2003 . Stochastic integral representation theorem for quantum semimartingales . J. Funct. Anal. , 201 : 1 – 29 .
   Web of Science ®Google Scholar
• Ji , U.C. and Obata , N. 2002 . “ Quantum white noise calculus ” . In Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads , Edited by: Obata , N. , Matsui , T. and Hora , A. 143 – 191 . Singapore : World Scientific .
   Google Scholar
• Ji , U.C. and Obata , N. 2005 . “ Admissible white noise operators and their quantum white noise derivatives ” . In Infinite Dimensional Harmonic Analysis III , Edited by: Heyer , H. , Hirai , T. , Kawazoe , T. and Saito , K. 213 – 232 . Singapore : World Scientific .
   Google Scholar
• Ji , U.C. and Obata , N. 2007 . Generalized white noise operators fields and quantum white noise derivatives . Semi. Congrès , 16 : 17 – 33 .
   Google Scholar
• Ji , U.C. and Obata , N. Quantum stochastic gradients . Interdisciplinary Inform. Sci. , in press
   Google Scholar
• Ji , U.C. and Obata , N. 2009 . Annihilation-derivative, creation-derivative and representation of quantum martingales . Commun. Math. Phys. , 286 : 751 – 775 .
   Web of Science ®Google Scholar
• Ji , U.C. and Sinha , K.B. 2005 . Integral representation of quantum martingales . Infin. Dimen. Anal. Quant. Probab. Rel. Top. , 8 : 55 – 72 .
   Web of Science ®Google Scholar
• Ji , U.C. and Sinha , K.B. 2006 . Uniqueness of integrands in quantum stochastic integral . Infin. Dimen. Anal. Quant. Probab. Rel. Top. , 9 : 607 – 616 .
   Web of Science ®Google Scholar
• Kuo , H.-H. 1996 . White Noise Distribution Theory , Boca Raton, FL : CRC Press .
   Google Scholar
• Lindsay , J.M. 1993 . Quantum and non-causal stochastic integral . Probab. Theory Relat. Fields , 97 : 65 – 80 .
   Google Scholar
• Lindsay , J.M. and Maassen , H. 1988 . “ An integral kernel approach to noise ” . In Quantum Probability and Applications III , Lecture Notes in Mathematics Edited by: Accardi , L. and von Waldenfels , W. Vol. 1303 , 192 – 208 . New York, NY : Springer-Verlag .
   Google Scholar
• Lindsay , J.M. and Parthasarathy , K.R. 1989 . Cohomology of power sets with applications in quantum probability . Commun. Math. Phys. , 124 : 337 – 364 .
   Web of Science ®Google Scholar
• Malliavin , P. 1997 . Stochastic Analysis , New York, NY : Springer-Verlag .
   Google Scholar
• Meyer , P.-A. 1993 . Quantum Probability for Probabilists , Lecture Notes in Mathematics Vol. 1538 , New York, NY : Springer-Verlag .
   Google Scholar
• Meyer , P.-A. 1994 . “ Représentation de martingales d'opérateurs ” . In Séminaire de Probabilités XXVII , Lecture Notes in Mathematics Vol. 1557 , 97 – 105 . New York, NY : Springer-Verlag .
   Google Scholar
• Obata , N. 1993 . An analytic characterization of symbols of operators on white noise functionals . J. Math. Soc. Jpn , 45 : 421 – 445 .
   Web of Science ®Google Scholar
• Obata , N. 1994 . White Noise Calculus and Fock Space , Lecture Notes in Mathematics Vol. 1577 , New York, NY : Springer-Verlag .
   Google Scholar
• Obata , N. 1995 . Generalized quantum stochastic processes on Fock space . Publ. RIMS Kyoto Univ. , 31 : 667 – 702 .
   Web of Science ®Google Scholar
• Obata , N. 1995 . Conditional expectation in classical and quantum white noise calculi . RIMS Kokyuroku , 923 : 154 – 190 .
   Google Scholar
• Obata , N. 1996 . “ White noise approach to quantum martingales ” . In Probability Theory and Mathematical Statistics , Edited by: Watanabe , S. 379 – 386 . Singapore : World Scientific .
   Google Scholar
• Obata , N. 1997 . Integral kernel operators on Fock space – Generalizations and applications to quantum dynamics . Acta Appl. Math. , 47 : 49 – 77 .
   Web of Science ®Google Scholar
• Parthasarathy , K.R. 1986 . “ A remark on the paper “Une martingale d'opérateurs bornés, non représentable en intégrale stochastique”, by J.L. Journe and P.A. Meyer ” . In Séminaire de Probabilités XX 1984/85 , Lecture Notes in Mathematics Edited by: Azéma , J. and Yor , M. Vol. 1204 , 317 – 320 . New York, NY : Springer–Verlag .
   Google Scholar
• Parthasarathy , K.R. 1992 . An Introduction to Quantum Stochastic Calculus , Basel : Birkhäuser .
   Google Scholar
• Parthasarathy , K.R. and Sinha , K.B. 1986 . Stochastic integral representation of bounded quantum martingales in Fock space . J. Funct. Anal. , 67 : 126 – 151 .
   Web of Science ®Google Scholar
• Parthasarathy , K.R. and Sinha , K.B. 1988 . “ Representation of a class of quantum martingales II ” . In Quantum Probability and Applications III , Vol. 1303 , Lecture Notes in Mathemetics Edited by: Accardi , L. and von Waldenfels , W. 232 – 250 . New York, NY : Springer-Verlag .
   Google Scholar
• Treves , F. 1967 . Topological Vector Spaces, Distributions and Kernels , New York : Academic Press .
   Google Scholar