I have the problem: Let $\displaystyle (B^1_t,B^2_t)_{t\ge0}$ be a 2-dimensional Brownian motion. Show that the process $\displaystyle X_t=(B^1_t,B_t^2,\frac{1}{2}\int_0^tB_s^1dB_s^2-B_s^2dB_s^1)_{t\ge0}$ solves the stochastic differential equation that may be written $\displaystyle dX_t=V_1(X_t)\circ dB^1_t+V_2(X_t)\circ dB_t^2$ where $\displaystyle V_1,V_2$ are 2 vector fields to be computed.

Any idea?