# Whether two one-dimensional Brownian motions yield a two-dimensional Brownian motion

Let $$X_t = \displaystyle\int_{0}^{t}sgn(W_{1s})dW_{1s}$$ and $$Y_t=W_{2s}$$, where $$(W_{1s})_s$$ and $$(W_{2s})_s$$ are one-dimensional standard Brownian motions with correlation coefficient $$\rho$$, that is, $$corr(W_{1s},W_{2s})=\rho$$ for each $$s$$. Then, $$X$$ and $$Y$$ are clearly standard Brownian motions, because $$X$$ is a martingale with quadratic variation $$[X]_t=t$$, for each $$t$$. Now, I am trying to make sure whether $$(X_s,Y_s)_s$$ becomes a two-dimensional Brownian motion. My approach was to show whether $$(X_t,Y_t)=_d normal$$ with covariance varying proportionately with time $$t$$, in which I am now stuck due in particular to algebra. Any comment will be welcome.

\begin{align*} \langle W^{1},W^{2} \rangle_{t} &={\mathbb E}[W_{t}^{1}W_{t}^{2}] \\ &={\rm Corr}[W_{t}^{1},W_{t}^{2}]\sqrt[]{{\rm Var}[W_{t}^{1}]}\sqrt[]{{\rm Var}[W_{t}^{2}]} \\ &=\rho t \end{align*} and \begin{align*} \langle X,Y \rangle_{t} &=\langle \int_{0}^{\bullet}{\rm sgn}(W_{s}^{1}){\rm d}W_{s}^{1},\int_{0}^{\bullet}{\rm d}W_{s}^{2} \rangle_{t} \\ &=\int_{0}^{t}{\rm sgn}(W_{s}^{1}){\rm d}\langle W^{1},W^{2} \rangle_{s} \\ &=\int_{0}^{t}{\rm sgn}(W_{s}^{1})\rho{\rm d}s \\ &\not\equiv 0. \end{align*}