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The "yes" answer follows immediately from Theorem 2.5 in this paper by Kusuoka, which implies that $X_1$ has a normal-like pdf $p$, such that $$c_1 e^{-b_1x^2}\le p(x)\le c_2 e^{-b_2x^2}$$ for all real $x$, where $c_1,b_1,c_2,b_2$ are positive real constants depending only on $\inf_{t,x} a(t,x)>0$, $\sup_{t,x} a(t,x)<\infty$, and $\sup_{t,x,y\ne x}|(a(t,x)-a(t,y)|/|x-y|<\infty$.