On Itô's formula for certain fields of geometric objects

Hiroshi AKIYAMA

doi:10.2969/jmsj/03910079

Hiroshi AKIYAMA

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JOURNAL FREE ACCESS

1987 Volume 39 Issue 1 Pages 79-91

DOI https://doi.org/10.2969/jmsj/03910079

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Published: 1987 Received: October 20, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : On Ito's formula for certain fields of geometric objects
Right : On Itô's formula for certain fields of geometric objects

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) H. Akiyama, Differential geometric approach to stochastic differential equations on vector bundles, I, Math. Japon., 26 (1981), 421-432.
2) J.-M. Bismut, Mécanique Aléatoire, Lecture Notes in Math., 866, Springer, 1981.
3) M. Ferraris, M. Francaviglia and C. Reina, A constructive approach to bundles of geometric objects on a differentiable manifold, J. Math. Phys., 24 (1983), 120-124.
4) M. Ferraris, M. Francaviglia and C. Reina, Sur les fibrés d'objets géométriques etleurs applications physiques, Ann. Inst. Henri Poincaré Sect. A Phys. Théor., 38 (1983), 371-383.
5) V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys No. 14, Amer. Math. Soc., Providence, R. I., 1977.
6) N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha/North-Holland, Tokyo/Amsterdam, 1981.
7) N. Ikeda and S. Watanabe, Stochastic flows of diffeomorphisms, Stochastic Analysis and Applications (Ed. by M. A. Pinsky), Adv. Probab. Related Topics, 7, Marcel Dekker, New York, 1984, pp. 179-198.
8) K. Ito, Stochastic parallel displacement, Probabilistic Methods in Differential Equations, Lecture Notes in Math., 451, Springer, 1975, pp. 1-7.
9) S. Kobayashi, Transformation Groups in Differential Geometry, Springer, Berlin, 1972.
10) S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, Interscience, New York, 1963.
11) H. Kunita, Some extensions of Ito's formula, Séminaire des Probabilités, XV (ed. by J. Azéma and M. Yor), Lecture Notes in Math., 850, Springer, 1981, pp. 118-141.
12) H. Kunita, On the decomposition of solutions of stochastic differential equations, Stochastic Integrals, Proc. LMS Durham Symp. 1980 (ed. by D. Williams), Lecture Notes in Math., 851, Springer, 1981, pp. 213-255.
13) A. Nijenhuis, Geometric aspects of formal differential operators on tensor fields, Proc. Internat. Congr. Math., 1958, Cambridge Univ. Press, Cambridge, 1960, pp. 463-469.
14) S. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.
15) K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland, Amsterdam, 1955.

Right : [1] H. Akiyama, Differential geometric approach to stochastic differential equations on vector bundles, I, Math. Japon., 26 (1981), 421-432.
[2] J. -M. Bismut, Mécanique Aléatoire, Lecture Notes in Math., 866, Springer, 1981.
[3] M. Ferraris, M. Francaviglia and C. Reina, A constructive approach to bundles of geometric objects on a differentiable manifold, J. Math. Phys., 24 (1983), 120-124.
[4] M. Ferraris, M. Francaviglia and C. Reina, Sur les fibrés d'objets géométriques etleurs applications physiques, Ann. Inst. Henri Poincaré Sect. A Phys. Théor., 38 (1983), 371-383.
[5] V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys No. 14, Amer. Math. Soc., Providence, R. I., 1977.
[6] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha/North-Holland, Tokyo/Amsterdam, 1981.
[7] N. Ikeda and S. Watanabe, Stochastic flows of diffeomorphisms, Stochastic Analysis and Applications (Ed. by M. A. Pinsky), Adv. Probab. Related Topics, 7, Marcel Dekker, New York, 1984, pp. 179-198.
[8] K. Itô, Stochastic parallel displacement, Probabilistic Methods in Differential Equations, Lecture Notes in Math., 451, Springer, 1975, pp. 1-7.
[9] S. Kobayashi, Transformation Groups in Differential Geometry, Springer, Berlin, 1972.
[10] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, Interscience, New York, 1963.
[11] H. Kunita, Some extensions of Itô's formula, Séminaire des Probabilités, XV (ed. by J. Azéma and M. Yor), Lecture Notes in Math., 850, Springer, 1981, pp. 118-141.
[12] H. Kunita, On the decomposition of solutions of stochastic differential equations, Stochastic Integrals, Proc. LMS Durham Symp. 1980 (ed. by D. Williams), Lecture Notes in Math., 851, Springer, 1981, pp. 213-255.
[13] A. Nijenhuis, Geometric aspects of formal differential operators on tensor fields, Proc. Internat. Congr. Math., 1958, Cambridge Univ. Press, Cambridge, 1960, pp. 463-469.
[14] S. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.
[15] K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland, Amsterdam, 1955.

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