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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 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

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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 2612

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

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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 group

Nature: Original

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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 group

Nature: Original

Retrieve article from Numdam

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 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

Retrieve article from Numdam

XX: 23, 352-374, LNM 1204 (1986)

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

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

Retrieve article from Numdam

XXVI: 12, 146-154, LNM 1526 (1992)

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 group

Nature: Original

Retrieve article from Numdam

XXVI: 13, 155-156, LNM 1526 (1992)

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 group

Nature: Original

Retrieve article from Numdam