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Let assume that $$ dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R} $$ Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion with Hurst index $H\in (0,1)$. I would like to ask if we can approximate the above diffusion by continuous time Markov chain $\alpha_t$ taking finite values? If we can, then how to build the generator matrix for $\alpha_t$ using $\mu(.) $ and $\sigma(.)$ of $X_t$ ?

I know the answer is yes for when $dW_t^H$ is replaced by a standard Brownian motion but not sure for $dW_t^H$ case.

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    $\begingroup$ As far as I am aware the fractional Brownion motion will not satisfy the Markov property, so even if you can make sense of the diffusion equation (how?), it will not be Markovian. Therefore I would guess it will not be the limit of a Markov chain, as such a limit would be Markovian. Unless, perhaps, you consider limits of 'higher order' Markov chains, e.g. a Markov chain in $\mathbb R^d$, and consider the limit of e.g. the first component. $\endgroup$
    – Joris Bierkens
    Commented Feb 24, 2017 at 9:10

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