Distribution and SDE of a ratio of two Brownian motions

Let $W$ and $Z$ be independent standard Brownian motions and $X_t=W_t/Z_t$. What is the distribution of $X_t$ for fixed $t>0$, and what SDE does $X$ satisfy?

Standard Cauchy at every $t$. $W_t$ and $Z_t$ are independent $N(0,t)$, and $t$ cancels in the ratio of two i.i.d. centred normals. Using $d(1/Z)=-Z^{-2}dZ+Z^{-3}dt$ and $dW\,dZ=0$, Ito gives $dX_t=\tfrac{X_t}{Z_t^2}dt+\tfrac{1}{Z_t}dW_t-\tfrac{X_t}{Z_t}dZ_t$ (valid while $Z\neq0$). The noise can be written as $\tfrac{\sqrt{1+X_t^2}}{Z_t}\,dB_t$ for a single Brownian motion $B$.