 Solution

Define the radial martingale and its clock byThe Itô formula gives $R_t^2=2H_t+2t$. Independence of the coordinate Brownian motions makes their cross variation zero, soThe last identity is orthogonality of the radial martingale and planar Brownian area.

The clock is adapted and continuous, and it is strictly increasing almost surely. Otherwise the two coordinate paths would both vanish throughout a nontrivial interval. Such an interval contains a rational subinterval, while a Brownian increment over each fixed rational subinterval is a nondegenerate Gaussian and cannot be zero with positive probability.

Also $A(\infty )=\infty$ almost surely. If it were finite, the finite-bracket convergence lemma would make $H_t$ converge to a finite limit. Then $R_t^2=2t+O(1)$ on that event, forcing ${\int }_0^{\infty }{R}_t^{2}dt=\infty$, a contradiction. Thus no finite-lifetime extension is needed here.

Use the same inverse clock $T_u=\inf \{t:A(t)>u\}$ for both martingales, and set $W_u=H_{T_u}$, $B_u=Z_{T_u}$. Their bracket matrix isThe vector characterization proved in part (a) makes $(W,B)$ a two-dimensional Brownian motion; in particular its two coordinate processes are independent. Reversing the common clock givesThis is a common-clock Brownian representation of radius and area. Independence follows from the joint time change and identity bracket matrix; no independence of either Brownian motion from $A$ is asserted.