I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group. In doing so i'm trying to define a formula equivalent to the Ito's lemma. I tried looking in the Book of Hsu https://www.amazon.com/Stochastic-Analysis-Manifolds-Graduate-Mathematics/dp/0821808028, but there he defines only the Stratonovich integral of a diffusion process on a manifold. For what i have understood it is impossible to define an Ito integral in the general case, but i was thinking that in the case of a map from a Lie algebra there can be some hope. The problem can be formulate in this way:
Given a compact semisimple matrix Lie group $G$ endowed with a $G$-invariant metric $g$ and let $\mathfrak{g}$ its Lie algebra. Given a Wiener process on $G$ Define the SDE
$dX_t=G^0_t X_t dt + G^1_t X_t dW_t$, with $G^0_t, G^1_t$ left invariant vector fields.
Assuming that $X_t$ can be written as $X_t=e^{Y_t}$ where $Y_t=\mu_t dt + \sigma_t dB_t $ is a process on the Lie algebra. If i fix a set of generators for the Lie algebra it is possible to write a local version of the Itô's lemma and/or of the Itô isometry?
In other words it is possible to write
$de^{Y_t}=(d\exp_{Y_t}(\mu_t)+\mathcal{L})e^{Y_t} dt + d\exp_{Y_t}(\sigma_t)e^{Y_t} dW_t$
where $d\exp_{Y_t}$ is the differential of the exponential map and $\mathcal{L}$ is a second order differential operator (e.g the Laplace-Beltrami operator) which makes this expression a martingale.
More generally, for stochastic processes on the Lie algebra it is still true that
$\mathbb{E}\left[(\int_0^t Y_t dW_t)^2\right | \mathcal{F}_0]=\mathbb{E}\left[\int_0^t (Y_t)^2 d \operatorname{vol}_g\right | \mathcal{F}_0]$
if $t$ is small enough?
