A note on stochastic calculus in vector bundles. (English) Zbl 1287.58020
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLV. Cham: Springer (ISBN 978-3-319-00320-7/pbk; 978-3-319-00321-4/ebook). Lecture Notes in Mathematics 2078. Séminaire de Probabilités, 353-364 (2013).
Consider a connection \(\nabla\) in a vector bundle \(\pi:E\to M\). One can associate to this connection a connector \(K_\nabla:TE\to E\) giving the decomposition of a tangent vector into its horizontal and vertical components. The aim of this note is to describe, by means of this connector, the relationship between Itô covariant stochastic integrals and Stratonovich integrals in \(E\).
For the entire collection see [Zbl 1270.60005].
For the entire collection see [Zbl 1270.60005].
Reviewer: Jean Picard (Aubière)
MSC:
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 60J60 | Diffusion processes |
| 60H05 | Stochastic integrals |
Keywords:
stochastic calculus; vector bundles; Itô covariant stochastic integral; Stratonovich integralsReferences:
| [1] | M. Arnaudon, A. Thalmaier, Horizontal martingales in vector bundles. Séminaire de Probabilités, XXXVI. Lecture Notes in Math., vol. 1801 (Springer, Berlin, 2003), pp. 419-456 · Zbl 1046.58013 |
| [2] | Catuogno, PJ; Stelmastchuk, S., Martingales on frame bundles, Potential Anal., 28, 1, 61-69 (2008) · Zbl 1131.53033 · doi:10.1007/s11118-007-9068-y |
| [3] | Driver, BK; Thalmaier, A., Heat equation derivative formulas for vector bundles, J. Funct. Anal., 183, 1, 42-108 (2001) · Zbl 0983.58018 · doi:10.1006/jfan.2001.3746 |
| [4] | J. Eells, L. Lemaire, Selected topics in harmonic maps. CBMS Regional Conference Series in Mathematics, vol. 50 (American Mathematical Society, Providence, RI, 1983) · Zbl 0515.58011 |
| [5] | Emery, M., Stochastic Calculus in Manifolds (1989), With an appendix by P. A. Meyer: Universitext (Springer, Berlin, With an appendix by P. A. Meyer · Zbl 0697.60060 · doi:10.1007/978-3-642-75051-9 |
| [6] | J.R. Norris, A complete differential formalism for stochastic calculus in manifolds. Séminaire de Probabilités, XXVI. Lecture Notes in Math., vol. 1526 (Springer, Berlin, 1992), pp. 189-209 · Zbl 0791.58111 |
| [7] | Patterson, LN, Connexions and prolongations. Canad. J. Math., 27, 4, 766-791 (1975) · Zbl 0307.53018 · doi:10.4153/CJM-1975-085-4 |
| [8] | Poor, WA, Differential Geometric Structures (1981), New York: McGraw-Hill, New York · Zbl 0493.53027 |
| [9] | Y. Xin, Geometry of harmonic maps. Progress in Nonlinear Differential Equations and their Applications, vol. 23 (Birkhäuser Boston, Boston, MA, 1996) · Zbl 0848.58014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
