# Necessary condition(s) for transforms of Markov diffusion to stay Markov diffusions

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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For the 1-dim case it seems to be related to the injectivity of certain functions (which appear after the application of Ito formula for $f$). For multi-dim case the story is more complicated, I don't think that local conditions like inverse function theorem will be a right answer. Unfortunately, I haven't met any literature - but the question is nice, so +1 and good luck. –  Ilya Jan 25 '12 at 17:47
@Ilya : Well even for 1-dim case injectivity of $f$ is not necessary, take for example $B^2_t$, it is a Markov process but $f(x)=x^2$ is not injective, so it really seems tricky. A known sufficient condition is that the law of $X_t|x$ is the same for every $x$ solution of the equation $f^{-1}(y)=x$ (for every $y$ in the range of $f$), so IMO it also has to involve in some way a "symmetry" argument. Regarding Itô, I found that $B(f(X_t))=1/2\partial_{x^2}f(X_t).a^2(X_t)+b(X_t)$ and $A(f(X_t))=\partial_{x}f(X_t)a(f(X_t))$ but this didn't lead me very far. Best regards. –  TheBridge Jan 25 '12 at 21:25
I remember, that for GBM $f(x) = x^2$ is also a Markov process - so with injectivity I was rather thinking of sufficient conditions. By the way, in the result of Ito formula you wrote in the comment, the drift term seems to be incorrect - the right one should be $$B(f(X_t)) = b(X_t)\partial_xf(X_t)+\frac12a^2(X_t)\partial_{x^2}f(X_t)$$ –  Ilya Jan 26 '12 at 7:07
@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. –  TheBridge Jan 26 '12 at 8:06