Lévy area ($A_{s,t}^{ij}=\frac{1}{2}{\int }_s^t((B_r^i-B_s^i)\circ d{B}_r^j-(B_r^j-B_s^j)\circ d{B}_r^i)$)

Lévy area is the antisymmetric second-level integral of multidimensional Brownian motion. For distinct independent components the integrals are also Itô integrals. Combined with the symmetric tensor $\frac{1}{2}{B}_{s,t}^{\otimes 2}$, it gives the second level of enhanced Brownian motion. It is not continuously determined by the uniform norm of the Brownian path.