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A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}-\frac{1}{2}I(t-s)$ but the Stratonovich rough path is geometric.

Why the term "geometric" here? Is there some intuition I'm missing? When I think geometric, I think of geometric Brownian motion, (http://en.wikipedia.org/wiki/Geometric_Brownian_motion) or geometric series, something with an exponent. I don't see that here.

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    $\begingroup$ The definition I've seen goes like this: a smooth path has a canonical enhancement into a "smooth rough path". Then the geometric rough paths are the closure of the smooth rough paths, in the appropriate $p$-variation topology. So it could be that geometric rough paths have, in some sense, an enhancement that is geometrically reasonable, rather than arbitrary. $\endgroup$ – Nate Eldredge May 27 '15 at 3:44
  • $\begingroup$ Thanks for your response. I guess I don't understand, why is that condition geometrically reasonable? $\endgroup$ – user69208 May 27 '15 at 3:46
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    $\begingroup$ I don't really know either. Another guess is the fact that the Stratonovich integral is more "geometrically reasonable" than Ito: if you have a submanifold $M \subset \mathbb{R}^n$ and a vector field $V$ which is tangent to $M$, the process $dX_t = V(X_t) \cdot \delta B_t$ (Stratonovich) will lie in $M$, but $dX_t = V(X_t) \cdot dB_t$ (Ito) will not. I am not sure but there may be some similar property of geometric rough paths. Anyway, I think one should think of "geometric" meaning more like "differential geometry" than "geometric series". $\endgroup$ – Nate Eldredge May 27 '15 at 3:53
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Geometric rough paths have the property that if you want to solve an equation with values in a manifold, choose a coordinate chart, and write in local coordinates $$ dY^i = V_0^i(Y)\,dt + \sum_j V_j^i(Y)\,dX_j $$ for some vector fields $V_i$ (with the obvious abuse of notation that the solution actually depends on the choice of $\mathbb{X}$, not just on $X$), then the solution does not depend on the choice of chart. I believe that this is the reason why this terminology was chosen.

In other words, solutions to equations driven by "geometric" rough paths transform according to the usual chain rule, rather than some version of Itô's formula. This is also why people studying SDEs on manifolds tend to write everything in Stratonovich form, rather than in Itô form.

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