Currently, I am working on a project that involves the estimation of the diffusion matrix $\Sigma(t)$ of a continuous time Markov chain. The model can be described by the stochastic differential equation $$ \mathrm{d}X(t)=\mu(X(t),t)\mathrm{d}t+\sigma(X(t),t)\mathrm{d}W(t) $$ where $W(t)$ is the standard $m$-dimensional Brownian motion. If we assume that the process is sampled with a constant interval $\Delta$, then a simple estimator for the diffusion matrix would be $$ \hat{\Sigma}(t)=\frac{1-\lambda}{1-\lambda^n}\sum_{k=1}^{n}\lambda^{k-1}Y_{t-k}Y_{t-k}^{\top}, $$ where $\lambda\in(0,1)$ is the smoothing factor and $Y_k=(X(t_{k})-X(t_{k-1}))/\sqrt{\Delta}$. For some reference, this is the estimator that is used by Fan, Fan and Lv in their work Aggregation of Nonparametric Estimators for Volatility Matrix. Yet, one of their assumptions is that each dimension of the process $X(t)$ is described by $$ \mathrm{d}X^i(t)=\mu_i(f_t)\mathrm{d}t+\sum_{j=1}^m\sigma_{ij}(f_t)\mathrm{d}W_j(t) $$ where $f_t$ is itself a stochastic process which is equal for each of the processes $X_i(t)$ for $i\in(1,...,m)$. The main difference here is that, unlike $X(t)$, the process $f_t$ is a scalar process, that is, it is a one-dimensional path.
My question is if, given the scalar process $f_t$, the estimator $\hat{\Sigma}(t)$ would remain equal if we left out the assumption that the process $X(t)$ is governed by a single scalar process $f_t$? And if not, is there a paper that describes the asymptotics of the multivariate extension, i.e. asymptotic normality and consistency? Additionally, in the case that the estimator would remain equal, would this imply that the asymptotic properties (mainly its asymptotic distribution) would also remain equal?
Thank you in advance!