XV: 05, 44-102, LNM 850 (1981)
MEYER, Paul-André
Géométrie stochastique sans larmes (
Stochastic differential geometry)
Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is
continuous semimartingales in manifolds, following L.~Schwartz (LN
780, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous
martingales in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions
Comment: A short introduction by the same author can be found in
Stochastic Integrals, Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery,
Stochastic Calculus on Manifolds Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Transfer principle,
Stochastic differential equations,
Stochastic integrals,
Stratonovich integralsNature: Original
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