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**Horizontal lift of a càdlàg semimartingale.
(Relèvement horizontal d’une semi-martingale càdlàg.)**
*(French)*
Zbl 0764.60047

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 127-145 (1992).

[For the entire collection see Zbl 0754.00008.]

Everyone who has been interested in manifold-valued continuous semimartingales knows how useful it is to consider the stochastic horizontal lift in the frame bundle. This was introduced by many authors in the continuous case but very few worked with discontinuous processes.

The purpose of this paper is to construct a stochastic horizontal lift for Riemannian manifold-valued discontinuous semimartingales. The construction is quite similar to Shigekawa’s one and therefore the main tool is the integral of a one-form along a cadlag semimartingale. Some geometric notions are recalled; then an integral of forms along discontinuous semimartingales is defined as in the continuous case between two jumps of the driving semimartingale, and at jump times a geodesic is used to connect one point and the next one. Next, by classical method, the horizontal lift of such a semimartingale is introduced as the action on a given orthonormal frame of a parallel transport above the semimartingale. The last section deals with the stochastic development, presented as a vectorial lecture of a manifold semimartingale.

This processes (horizontal lift and stochastic development) are used in another paper to solve filtering problems where the observation is a manifold-valued discontinuous process.

Everyone who has been interested in manifold-valued continuous semimartingales knows how useful it is to consider the stochastic horizontal lift in the frame bundle. This was introduced by many authors in the continuous case but very few worked with discontinuous processes.

The purpose of this paper is to construct a stochastic horizontal lift for Riemannian manifold-valued discontinuous semimartingales. The construction is quite similar to Shigekawa’s one and therefore the main tool is the integral of a one-form along a cadlag semimartingale. Some geometric notions are recalled; then an integral of forms along discontinuous semimartingales is defined as in the continuous case between two jumps of the driving semimartingale, and at jump times a geodesic is used to connect one point and the next one. Next, by classical method, the horizontal lift of such a semimartingale is introduced as the action on a given orthonormal frame of a parallel transport above the semimartingale. The last section deals with the stochastic development, presented as a vectorial lecture of a manifold semimartingale.

This processes (horizontal lift and stochastic development) are used in another paper to solve filtering problems where the observation is a manifold-valued discontinuous process.

Reviewer: M.Pontier (Orléans)

### MSC:

60G48 | Generalizations of martingales |

58J65 | Diffusion processes and stochastic analysis on manifolds |

60H05 | Stochastic integrals |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |