Calcul Stochastique Covariant À Sauts & Calcul Stochastique À Sauts Covariants

Résumé du livre

We propose a stochastic covaraiant calculus for càdlàg semimartingales in the tangent bundle TM over a manifold M. A connexion on M allows us to define an intrinsic derivative of a C1 curve (Yt) in TM, the covariant derivative. More precisely, it is the derivative of (Yt) seen in a frame moving parallely along its projection curve (xt) on M. With the transfer principle, Norris defined the stochastic covariant integration along a continuous semimartingale in TM. We describe the case where the semimartingale jumps in TM, using Norris's work and Cohen's results about stochastic calculus with jumps on manifolds. We see that, depending on the order in which we compose the function giving the jumps and the connection, we obtain a stochastic covariant calculus with jumps or a stochastic calculus with covariant jumps. Both depend on the choice of the connection and of the tools (interpolation and connection rules) describing the jumps in the meaning of Stratonovich or Itô. We study the choices that make equivalent the two calculus. Under suitable conditions, we recover Norris's results when (Yt) is continuous. The continuous case is described by a covariant continuous calculus of order two, a formalism defined with the notion of connection of order two.