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?


A good and general reference on stochastic integrals on manifolds is

An Invitation to Second-Order Stochastic Differential Geometry

See in particular Theorem 1, page 14. The survey also contains a list of references with some specific pointers to the literature. This is the classical probabilist approach using an isometry property.

There is also (very new) deterministic approach to stochastic integrals on manifolds that relies on rough paths theory.

Rough paths on manifolds

The class of integrators goes then far beyond Brownian motion and semimartingales.

  • $\begingroup$ The first article you linked is very nice. Sadly, it rarely provides reverences... $\endgroup$ – Matthias Ludewig Mar 1 '15 at 15:43

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