Assume $X(t)$ and $Y(t)$ are two stochastic processes satisfying $$dX=\mu_1(X)dt+\sigma_1(X)dW_1$$ $$dY=\mu_2(Y)dt+\sigma_2(Y)dW_2$$ where $W_1$ and $W_2$ are independent standard Brownian motions.
Since $dX(t)dY(t)=0$, then $X(t)$ and $Y(t)$ are uncorrelated. My question is: if $X(t)$ and $Y(t)$ are uncorrelated, can we say they are also independent for any fixed $t$ ? (I don't need independence for $X(s)$ and $Y(t)$, only for the same $t$.) If the answer is no, is there any other condition I can check besides uncorrelated to guarantee independence?
My thought is: at any time t, $X$ and $Y$ are two random variables. For any two random variables $X$ and $Y$, if for any bounded measurable functions $f$ and $h$, $f(X)$ and $h(Y)$are uncorrelated, then $X$ and $Y$ are independent. But for stochastic processes, to verify $f(X)$ and $h(Y)$are uncorrelated, the only way I can think of is to use Ito's formula, which requires twice differentiability, which is more than bounded&measurable.
Thank you for any comments. Sorry if the answer is straightforward.