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I feel it always necessitates a certain amount of work before reaching the conclusion that the transform of a diffusion by a function is not a Markovian diffusion.

I was wondering if there were known necessary conditions over the transform (and/or maybe the diffusion process), that should be satisfied for the transform to stay a Markovian diffusion.

Hopefully such conditions would give a "methodology" (or even better an explicit calculus) to discard some transforms to be Markov diffusion processes.

Let's formalize the problem (or a simplified version of it). So given a Itô-diffusion (let's stay one-dimensional for the moment) $X_t$ obeying an SDE such as :

$dX_t=b(X_t)dt+a(X_t)dW_t$ (with enough regularity so that $X_t$ exists, and $W_t$ is 1-dim Brownian Motion)

And a function $f$, what are the conditions over $f$ so that $f(X_t)$ stays a diffusion, i.e. can be written as :

$df(X_t)=B(f(X_t))dt+A(f(X_t))dW_t$ ?

Reference or proofs are welcome and so is extension to multivariate case.

Best regards

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    @Ilya : Yes you are right that's a typo, thank's for pointing that out but it's too late to edit the comment. Thank's for your interest in the topic and best regards.2012-01-26

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