Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what is the formula. Any reference would be appreciated.

Have you thought of adding all the areas for each two dimension Brownian motion area? You may choose partial combination as well. – Creator Nov 15 at 0:36

I don't know how relevant that is, but the quantity $\int_0^tX_s\otimes dY_s  \int_0^tY_s\otimes dX_s$ comes to mind, with $\otimes$ the tensor product. It looks a bit like things one encounters in rough path theory. – Pierre PC Nov 16 at 19:40

@PierrePC It is exactly required for "rough path" Imho. Interesting comment, do you know if anyone has used the way you are expressing? More importantly can you suggest how to evaluate the integral? Any suggestion would be helpful, I am very much interested in this question. – Creator Nov 19 at 21:15

I just saw that you were dealing with twodimensional Brownian motion, so yes, a central object in rough paths theory is $\int X_s\otimes dX_s  \int dX_s\otimes X_s$. However, I would argue than one of the lessons of RPT is that it is an interesting object on its own, and does not easily reduce to anything else... – Pierre PC Nov 19 at 22:01