Stochastic Covariant Calculus With Jumps And Stochastic Calculus ...
Stochastic covariant calculus with jumps and
Stochastic calculus with covariant jumps
Laurence Maillard-Teyssier
Laboratoire LAMA, Universit´e de Versailles Saint Quentin en Yvelines
maillard@math.uvsq.fr
Summary. We propose a stochastic covariant calculus for c`
adl`
ag semimartingales
in the tangent bundle T M over a manifold M . In ordinary differential geometry, a
connection on M is needed to define the covariant derivative of a C1 curve in T M ; by
applying the transfer principle, Norris has defined a stochastic covariant integration
along a continuous semimartingale in T M . We extend this to the case when the
semimartingale jumps, using Norris’ work and Cohen’s results on stochastic calculus
with jumps on manifolds. Depending on the order in which the function giving the
jumps and the connection are composed, one obtains a “stochastic covariant calculus
with jumps” or a “stochastic calculus with covariant jumps”, which are in general
not equivalent. Under suitable conditions, Norris’ results for the continuous case are
recovered. This case can be described by a covariant continuous calculus of order
two, which involves the notion of a connection of order two.
References
[Coh96-1] Cohen, S.: G´eom´etrie diff´erentielle stochastique avec sauts I. Stoch. Stoch.
Rep. 56 (1996), no. 3-4, 179–203.
[Coh96-2] Cohen,
S.:
G´eom´etrie
diff´erentielle
stochastique
avec
sauts
II.
Discr´etisation et applications des EDS avec sauts. Stoch. Stoch. Rep. 56 (1996),
no. 3-4, 205–225.
[Elw82] Elworthy, K.D.: Stochastic Differential Equations on Manifolds. LMS Lec-
ture Note Series 70. Cambridge University Press, Cambridge-New York, 1982.
[Em89] Emery, M.: Stochastic Calculus in Manifolds. Universitext. Springer-Verlag,
Berlin, 1989.
[Em90] Emery, M.: On two transfer principles in stochastic differential geometry,
S´eminaire de Probabilit´es XXIV, 1988/89, 407–441, Lecture Notes in Math. 1426,
Springer, Berlin, 1990.
[KN63] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol I.
Interscience Publishers, John Wiley & Sons, New York-Lond, 1963.
[Ma03] Maillard-Teyssier, L.: Calcul Stochastique Covariant `
a Sauts et Cal-
cul Stochastique `
a Sauts Covariants . Th`ese de doctorat (16 d´ecembre 2003)
- Universit´e de Versailles Saint Quentin en Yvelines. “th`eses-EN-ligne”:
http://tel.ccsd.cnrs.fr/
2
Laurence Maillard-Teyssier
[Mey81] Meyer, P.A.: G´eom´etrie stochastique sans larmes. S´eminaire de Probabilit´es
XV, 44–102, Lecture Notes in Math. 850, Springer, Berlin-New York, 1981.
[Mey82] Meyer, P.A.: G´eometrie diff´erentielle stochastique II. S´eminaire de Proba-
bilit´es XVI, Suppl´ement G´eom´etrie Diff´erentielle Stochastique, 165–207, Lecture
Notes in Math. 921, Springer, Berlin-New York, 1982.
[Nor92] Norris, J.R.: A complete differential formalism for stochastic calculus in
manifolds. S´eminaire de Probabilit´es XXVI, 189–209, Lecture Notes in Math.
1526, Springer, Berlin, 1992.
[Pic91] Picard, J.: Calcul stochastique avec sauts sur une vari´et´e. S´eminaire de Prob-
abilit´es XXV, 196–219, Lecture Notes in Math. 1485, Springer, Berlin, 1991.
[Sch82] Schwartz, L.: G´eom´etrie diff´erentielle du 2`eme ordre, semi-martingales et
´
equations diff´erentielles stochastiques sur une vari´et´e diff´erentielle. S´eminaire de
Probabilit´es XVI, Suppl´ement G´eom´etrie Diff´erentielle Stochastique, 1–148, Lec-
ture Notes in Math. 921, Springer, Berlin-New York, 1982.
[Spi79] Spivak, M.: A Comprehensive Introduction to Differential Geometry. Vol. II.
Second edition. Publish or Perish, Inc., Wilmington, Del., 1979.
[YI73] Yano, K, Ishihara, S.: Tangent and Cotangent Bundles: Differential Geom-
etry. Pure and Applied Mathematics, No. 16. Marcel Dekker, Inc., New York,
1973.
