Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode $$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$ Where $b$ is a bounded continuous function (not Lipchitz). For fixed $\omega_{\cdot}$, the collection of solutions to above equation donated by $X(\omega)$. My question is: can we pick up a adapted solution $x_{\cdot}(\omega)\in X(\omega)$ (w.r.t the filtration $F_t=\sigma\{\omega_s;s\leq t\}$) using measurable selection theorem or something like that.
Writing $y_t=x_t+w_t$, your equation is $dy_t=b(y_t)dt+dw_t$. With $b$ bounded, at least in the Brownian motion case there is a unique strong solution by a result of Zvonkin: A transformation of the phase space of a diffusion process that removes the drift. 1974 Math. USSR Sb. 22 129 (see Theorem 4). Knowing $y_t$ you can then reconstruct $x_t$.