  Let $G$ be a Lie Group with a left invariant connection $\nabla^{G}$. Denote
by $\g$ the Lie algebra of $G$, which is equipped with a connection
$\nabla^{\g}$. Our main is to introduce the concept of the It\^o exponential
and the It\^o logarithm, which take in account the geometry of the Lie group
$G$ and the Lie algebra $\g$. This definition characterize directly the
martingales in $G$ with respect to the left invariant connection $\nabla^{G}$.
Further, if any $\nabla^{\g}$ geodesic in $\g$ is send in a $\nabla^{G}$
geodesic we can show that the It\^o exponential and the It\^o logarithm are the
same that the stochastic exponential and the stochastic logarithm due to M.
Hakim-Dowek and D. L\'epingle in [10]. Consequently, we have a
Campbell-Hausdorf formula. From this formula we show that the set of affine
maps from $(M,\nabla^{G})$ into $(G,\nabla^{G})$ is a subgroup of the Loop
group. As in general, the Lie algebra is considered as smooth manifold with a
flat connection, we show a Campbell-Hausdorf formula for a flat connection on
$\g$ and a bi-invariant connection on $G$. To this main we introduce the
definition of the null quadratic variation property. To end, we use the
Campbell-Hausdorff formula to show that a product of harmonic maps with value
in $G$ is a harmonic map.
