# Conditions for two processes to be continuous transformations

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 $$n=1$$.