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Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined on $T_xM$ into a semimartingale defined on $M$ by "rolling $T_xM$ on $M$ along $X$ without slipping". Let $\ell(V_y) \colon T_yM \to T_{V_y}TM$ denote horizontal lift and $FM$ the frame bundle of $M$. Then, following [E89, Ch.VIII] we can fix $\{{_k}B_0\}_k \in F_xM$ and define, for $V_x \in T_xM$ and $\{{_k}B_y\}_k = \mathscr B_y \in F_yM$ the field of linear maps $$e(\mathscr B_y) \colon T_{V_x}T_xM = T_xM \to T_{\mathscr B_y}FM, \quad T\pi_i \circ e(\mathscr B_y) U_x = U_x^j \ell({_i}B_y){_j}B_y$$ where $U_x = U^j_x {_0}B_j$ and $\pi_i \colon FM \to TM$ picks the $i$-th vector in a basis. This defines the Stratonovich equation $$\text{d} \mathscr B_t = e(\mathscr B_t) \circ \text{d} X_t, \quad \mathscr B_0 = \{{_k}B_0\}_k$$ whose solution is a $FM$-valued "horizontal" semimartingale (in the Riemannian case the $FM$ can be replaced with $OM$, the orthonormal frame bundle). This is then projected down onto $M$ to obtain the stochastic development of $X$.

Now, let $N$ and $P$ be smooth manifolds, $Z$ a semimartingale in $N$. [E89, Ch.VI] describes a way to define an "Ito-type" equation on $P$ driven by $Z$. The data needed for this consist of a field of Schwartz morphisms $\mathbb F(y,z) \colon \mathbb T_zN \to \mathbb T_yP$; a Schwartz morphism is a linear map between second order tangent bundles that behaves well w.r.t. the short exact sequences $0 \to TM \to \mathbb TM \to TM \odot TM \to 0$. Now, as explained in [E89, Ch.VII], a linear connection on the tangent bundle of the manifold splits this short exact sequence, so given connections on $N$ and $P$ we can consider fields of Schwartz morphisms of the form $$ \begin{bmatrix} F \ &0 \\ 0 & F \otimes F \end{bmatrix}(z,y) \colon T_zN \oplus (T_zN \odot T_zN) \to T_yP \oplus (T_yP \odot T_yP)$$ (the general block matrix form of a Schwartz morphism will still have a 0 on the bottom left, but the upper right hand term might not vanish). In this way the field $F$ alone defines the Ito SDE $\text{d} Y_t = F(Z_t,Y_t) \text{d} Z_t$ (when both manifolds are Euclidean spaces this defines a conventional Ito SDE).

Returning to stochastic development, an objection to considering the Ito counterpart to the Stratonovich equation for the horizontal process might be that, although $M$ possesses a connection, there is no canonical way to "lift" it to a connection on the tangent bundle of $TM$. However, assume that the connection on $M$ comes from a specified Riemannian metric $g$. In that case $TM$ carries a canonical Riemannian metric, the Sasaki metric (given by lifting $g$ vertically and horizontally, and then by declaring the vertical and horizontal bundles orthogonal). This lifted metric in turn yields a Riemannian connection on the tangent bundle of $TM$, and thus the SDE for $\mathscr B_t$ above can be canonically interpreted in the distinct Ito sense. So my question is

Does this Ito equation lead to a different version of stochastic development?

My guess is that the particular nature of the Sasaki metric causes the Ito equation to be equivalent to the Stratonovich one written above, perhaps due to some cancellations that occur in the Ito-Stratonovich correction term. Or perhaps the horizontal processes are indeed different, but their projections onto $M$ coincide. It has always been puzzling to me that stochastic development, which behaves well w.r.t. local martingales and Brownian motion, is defined using Stratonovich calculus, that is famous for its lack of nice properties concerning such processes. An answer to the above question could shed some light on this dilemma.

Any answers, comments, suggestions or corrections are very much appreciated. Many thanks for reading.

[E89] Michel Emery. Stochastic calculus in manifolds. Universitext. Springer-Verlag, Berlin, 1989.

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