differentiable and compactly supported $k$-forms on $\mathbb{R}^{d}$. A $k$*-dimensional current* is a linear continuous functional on $\mathcal{D}^{k}$. We denote by $\mathcal{D}_{k}$ the space of $k$-currents. In this paper we are only interested in the cases $k=1,2$, so we recall a few corresponding notations. Let $\varphi$ (resp. $\psi$) be an element of $\mathcal{D}^{1}$ (resp. $\mathcal{D}^{2}$). We shall write them as $$\varphi=\sum_{i=1}^{d}\varphi_{i}\,dx^{i},\qquad\psi=\sum_{i<j}\psi
_{ij}\,dx^{i}\wedge dx^{j}$$ where $\varphi_{i}$ and $\psi_{ij}$ are infinitely differentiable and compactly supported functions on $\mathbb{R}^{d}$. Typical examples of $1$-currents are those induced by regular curves $\left(  X_{t}\right)
_{t\in\left[  0,T\right]  }$ in $\mathbb{R}^{d}$: $$\label{map1}
\varphi\mapsto S(\varphi):=\int_{0}^{T}\left\langle \varphi\left(
X_{t}\right)  ,\dot{X}_{t}\right\rangle \,dt$$ while typical examples of $2$-currents are given by regular surfaces $\left(
f\left(  t,s\right)  \right)  _{(t,s)\in A}$ in $\mathbb{R}^{d}$, where $A$ is a Borel set in $\mathbb{R}^{2}$: $$\begin{aligned}
\label{map2}
\psi &&  \mapsto T(\psi):=\int_{A}\left\langle \psi\circ f,\frac{\partial
f}{\partial t}\wedge\frac{\partial f}{\partial s}\right\rangle \,dtds\nonumber \\
&&  =\int_{A}\sum_{i<j}\left(  \psi_{ij}\circ f\right)  \,\left\langle
dx^{i}\wedge dx^{j},\frac{\partial f}{\partial t}\wedge\frac{\partial
f}{\partial s}\right\rangle \,dtds \nonumber \\
&&  =\int_{A}\sum_{i<j}\left(  \psi_{ij}\circ f\right)  \,\left(
\frac{\partial f_{i}}{\partial t}\frac{\partial f_{j}}{\partial s}-\frac{\partial f_{j}}{\partial t}\frac{\partial f_{i}}{\partial s}\right)
\,dtds.\end{aligned}$$

The generalization of the previous example of $1$-current to the stochastic case has been the object of previous research in the recent literature. We refer to [@Gia], [@FG], [@FGR], [@FT] for the 