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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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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