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Let $L$ be a fixed non-negative integer, $X_t$ and $Y_t$ be stochastic processes, with values in $\mathbb{R}^n$, adapted to a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$. Suppose moreover, that $X_t$ is a diffusion process whic his a strong solution to the Markovian SDE $$ dX_t = A_t X_t + B_t dW_t\qquad X_0=x $$ where $x \in \mathbb{R}^n$, and $A_t,B_t$ are continuous functions with values in $\mathbb{R}^n$ and $Mat_{n\times n}(\mathbb{R})$ respectively, and here $\mathcal{F}_t$ is the completion of the filtration generated by the Brownian motion $W_t$.

Under what conditions does there exist a continuous function $$ f(X_t,\dots,X_{t-L}) = Y_t, $$ for all $t\geq L$? Where the equality is $\mathbb{P}$ almost-sure?

Note: By this paper I doubt that in general $Y_t$ is Markovian, even in the case when $L=0$, $X_t=W_t$, and $n=1$.

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