Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $0$.

Does then $X_1$ have a bounded pdf?

This interesting answer by James Martin shows that, without assuming that $a$ is bounded away from $0$, it is possible that $P(X_1=0)>0$. See also comments to that answer.

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $0$.

Does then $X_1$ have a bounded pdf?

This interesting answer by James Martin shows that, without assuming that $a$ is bounded away from $0$ and replacing $a(s,X_s)$ by $a\big(s,(X_u\colon 0\le u\le s)\big)$, it is possible that $P(X_1=0)>0$. See also comments to that answer.