No Access

UNIQUENESS OF INTEGRANDS IN QUANTUM STOCHASTIC INTEGRAL

Department of Mathematics, Research Institute of Mathematical Finance, Chungbuk National University, Cheongju 361-763, Korea

Search for more papers by this author

and

KALYAN B. SINHA

Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi 110016, India

Current address: Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P. O., Bangalore 560064, India.

Search for more papers by this author

https://doi.org/10.1142/S0219025706002561Cited by:2 (Source: Crossref)

Abstract

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.

Keywords:

AMSC: primary: 60H40, secondary: 46F25, secondary: 81S25

References

S. Attal, J. Funct. Anal. 124, 292 (1994), DOI: 10.1006/jfan.1994.1109. Crossref, Web of Science, Google Scholar
D. M. Chung, T. S. Chung and U. C. Ji, Bull. Korean Math. Soc. 34, 421 (1997), DOI: 10.1006/jfan.1994.1109. Web of Science, Google Scholar
D. M. Chung, U. C. Ji and N. Obata, Rev. Math. Phys. 14, 241 (2002), DOI: 10.1142/S0129055X0200117X. Link, Web of Science, Google Scholar
T. Hida , Analysis of Brownian Functionals , Carleton Math. Lect. Notes 13 ( Carleton University , 1975 ) . Google Scholar
R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301 (1984), DOI: 10.1007/BF01258530. Crossref, Web of Science, Google Scholar
R. L. Hudson and J. M. Lindsay, J. Funct. Anal. 61, 202 (1985), DOI: 10.1016/0022-1236(85)90034-5. Crossref, Web of Science, Google Scholar
R. L. Hudson, J. M. Lindsay and K. R. Parthasarathy, Stochastic integral representation of some quantum martingales in Fock space, From Local Times to Global Geometry, Control and Physics, Proc. Warwick Symposium 1984/1985 (Pitman RNM, 1986) pp. 121–131. Google Scholar
U. C. Ji, J. Funct. Anal. 201, 1 (2003), DOI: 10.1016/S0022-1236(02)00027-7. Crossref, Web of Science, Google Scholar
U. C. Ji and N. Obata, J. Math. Soc. Jpn. 56, 311 (2004), DOI: 10.1016/S0022-1236(02)00027-7. Web of Science, Google Scholar
U. C. Ji and K. B. Sinha, Inf. Dim. Anal. Quantum Probab. Rel. Topics 8, 55 (2005), DOI: 10.1142/S0219025705001858. Link, Web of Science, Google Scholar
H.-H. Kuo , White Noise Distribution Theory ( CRC Press , 1996 ) . Google Scholar
J. M. Lindsay, Probab. Th. Rel. Fields 71, 307 (1986), DOI: 10.1007/BF00332314. Crossref, Web of Science, Google Scholar
J. M. Lindsay, Quantum Probability and Related Topics VI (World Scientific, 1991) pp. 325–332. Link, Google Scholar
P.-A. Meyer , Quantum Probability for Probabilists , Lecture Notes in Math. 1538 ( Springer-Verlag , 1993 ) . Crossref, Google Scholar
P.-A. Meyer, Séminaire de Probabilités XXVII, Lecture Notes in Math. 1557 (Springer-Verlag, 1994) pp. 97–105. Google Scholar
N. Obata, J. Math. Soc. Jpn. 45, 421 (1993). Crossref, Web of Science, Google Scholar
N. Obata , White Noise Calculus and Fock Space , Lecture Notes in Math. 1577 ( Springer-Verlag , 1994 ) . Crossref, Google Scholar
N. Obata, Publ. RIMS, Kyoto Univ. 31, 667 (1995). Crossref, Google Scholar
K. R. Parthasarathy , An Introduction to Quantum Stochastic Calculus ( Birkhäuser , 1992 ) . Crossref, Google Scholar
K. R. Parthasarathy and K. B. Sinha, J. Funct. Anal. 67, 126 (1986), DOI: 10.1016/0022-1236(86)90047-9. Crossref, Web of Science, Google Scholar
K. R. Parthasarathy and K. B. Sinha, Quantum Probability and Applications III, Lecture Notes in Math. 1303, eds. L. Accardi and W. von Waldenfels (Springer-Verlag, 1988) pp. 232–250. Crossref, Google Scholar