XVII: 22, 198-204, LNM 986 (1983)
KARANDIKAR, Rajeeva L.
Girsanov type formula for a Lie group valued Brownian motion (
Brownian motion,
Stochastic differential geometry)
A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential
Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle
2023 and Arnaudon
2612Keywords: Changes of measure,
Brownian motion in a manifold,
Lie groupNature: Original
Retrieve article from Numdam
XX: 23, 352-374, LNM 1204 (1986)
HAKIM-DOWEK, M.;
LÉPINGLE, Dominique
L'exponentielle stochastique des groupes de Lie (
Stochastic differential geometry)
Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$
Comment: See also Karandikar
Ann. Prob. 10 (1982) and
1722. For a sequel, see Arnaudon
2612Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXVI: 12, 146-154, LNM 1526 (1992)
ARNAUDON, Marc
Connexions et martingales dans les groupes de Lie (
Stochastic differential geometry)
The stochastic exponential of Hakim-Dowek-Lépingle
2023 is interpreted in terms of second-order geometry, studied in details and generalized
Keywords: Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXVI: 13, 155-156, LNM 1526 (1992)
ARNAUDON, Marc;
MATTHIEU, Pierre
Appendice : Décomposition en produit de deux browniens d'une martingale à valeurs dans un groupe muni d'une métrique bi-invariante (
Stochastic differential geometry)
Using
2612, it is shown that in a Lie group with a bi-invariant Riemannian structure, every martingale is a time-changed product of two Brownian motions
Keywords: Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
