Les's say we have SDEs: $$\begin{aligned} dX=&f(X,t)dt+h(X,t)dw;\\ dY=&p(X,t)dt+q(X,t)dw. \end{aligned} $$
How can we represent $Y$ without the explicit dependence to $X$ like: $$ dY=m(Y,t)dt+n(Y,t)dw. $$
Edit:
I try to represent an intermediate variable $S(t)=\int_0^tp(X,s)ds$ and insert it into the SDE for $Y$. This gives back a very complicated SDE including both differential and integral terms.
I am aware that this can be represent as a vector SED with scalar noise: $$dK=A(K,t)dt+B(K,t)dW,$$ with $K=[X,Y]$ being a vector, $A,B$ being a matrix. and $dW$ being a scalar noise. We can solve this SDE using numeric methods.
However, I still want to know that if there is any analytical method to simplify this problem first.
I don't think this is possible in most cases. For example, if $q = f \equiv 0$ and $h = p \equiv 1$, then $X_t = W_t$ and $Y_t = \int_0^t W_s ds$. Then $Y$ is not a Markov process.
However, if there were $m,n$ such that $dY_t = m(Y_t,t)dt + n(Y_t,t)dW_t$, we would have $Y$ is an Ito diffusion, and in particular a Markov process. Thus no such $m$ and $n$ can exist.