Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral $$ I(X) = \int_0^T \langle X(t, B_t), \mathrm{d} B_t\rangle$$ and would like for a reference of the following method.

Denote by $M \bowtie M$ the set of pairs $(x, y) \in M \times M$ such that there exists a unique shortest geodesic $\gamma_{xy}$ connecting $x$ to $y$. This set has full measure in $M \times M$. Denote by $\mathbb{W}^x$ the Wiener measure on $C([0, T], M)$ corresponding to Brownian motion starting at $x$.

Now let $0 =\tau_0 < \tau_1 < \dots < \tau_N = T$ be a partition of the interval $[0, T]$. For a path $\omega \in C_0([0, T], M)$ such that $(\omega(\tau_{j-1}), \omega(\tau_j)) \in M \bowtie M$ define $$ \Delta^\tau_j B := \dot{\gamma}_{\omega(\tau_{j-1})\omega(\tau_{j})}(0)~~~~~( "\approx B_{\tau_j} - B_{\tau_{j-1}}\!")$$ and $$I_\tau = \sum_{j=1}^N \langle X(\tau_{j-1}, B_{\tau_{j-1}}), \Delta^\tau_j B\rangle$$ Because $M \bowtie M$ has full measure in $M \times M$, this is a well-defined bounded operator $$I_{\tau}: L^2(\mathbb{R} \times M, TM) \longrightarrow L^2(\mathbb{W}^x)$$ for each such partition. Now I claim that for each sequence of partitions with mesh $|\tau|$ going to zero, $I_\tau$ converges to an isometry $I$, the Itô integral.

**This should be a well-known intrinsic construction of the Itô integral on Riemannian manifolds. Is there a reference, where this method is explained?**