# Ito lemma for SDEs on a Lie group

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?

• how about first working on the tangent space as here mathoverflow.net/questions/437864/… to get an Ito-isometry and then maybe Taylor expanding the exponential map? So that in a sense you capture the idea of t-small enough plus the errors from the Taylor expansion. Commented Mar 19, 2023 at 23:42
• There are not many semisimple Lie groups which do not embed as matrix groups. I think they embed as matrix groups if and only, up to finite covering, they have no factor of the universal covering group of $SL_2\mathbb{R}$. The compact ones always embed. So maybe not worth worrying about. Commented Mar 20, 2023 at 11:38
• @BenMcKay My problem is actually with matrix Lie groups. If i have for example a SDE on $SO_n(\mathbb{R})$, for which i know there is nothing similar to an Ito's lemma for the Ito integral. There is a lot about Stratonovich integral, but it is not a martingale and i don't know if the Ito isometry works on the manifold. I'm trying to understand if using the Dirinchlet form of the Lie algebra (as suggested above) will lead to something like that. If instead you means to embed in $GL_n(\mathbb{R})$ there are still problems when you compute the Laplacian (unless it is an isometric embedding) Commented Mar 20, 2023 at 16:02