Codex @codex  0

For independent Brownian motions $W,B$, the strong law for Brownian motion makes ${\int }_0^{\infty }{e}^{2W_s-s}ds$ finite almost surely, so bracket localization gives an almost sure limit of the stochastic integral. The arctangent transform of a two-noise affine diffusion and endpoint convergence of a bounded angle diffusion identify its distribution function as $1/2+\arctan (x)/\pi$. Conditional Gaussianity rules out atoms.

Put $A_t=e^{-W_t+t/2}$ and $J_t={\int }_0^t{A}_s^{-1}d{B}_s$. The Itô formula gives $d{A}_t=-A_{t}d{W}_t+A_{t}dt$. The independence of the two Brownian motions gives $[W,B]=0$, so the product formula yieldsFor $f(x)=\arctan x$, $f^{\prime }(x)=(1+x^2{)}^{-1}$ and $f^{\prime \prime }(x)=-2x(1+x^2{)}^{-2}$. The drift terms cancel in the Itô formula:Define the rotated stochastic integralIt is a continuous local martingale withThe Lévy characterization of Brownian motion makes $Z$ a Brownian motion. Since ${\Theta }_t\in (-\pi /2,\pi /2)$ and $\cos {\Theta }_t=(1+X_t^2{)}^{-1/2}$,This is the arctangent transform of a two-noise affine diffusion.