We have a well-defined SDE: $$ {\rm d}X_t=\mu(X_t){\rm d}t+\sigma(X_t) {\rm d}B_t, $$ where the initial condition $X_0$ is a known r.v., and $B_t$ is a standard Brownian motion.
Can we say the above SDE is a "continuous"-mapping $f:\mathbb{R}\times C[0,\infty)\to C[0,\infty)$ which maps $(X_0,B_t)$ into $X_t$ ?
What conditions should we need to realize the above statement? Could you provide some literature?
Thank you very much!