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Diffusions are continuous time stochastic processess having continuous paths and satisfying the strong Markov property.

I know it is possible to characterize some diffusion processes as solutions to SDEs. However I think I read somewhere that not all diffusions can be written as solutions of an SDE. Is there any reference to confirm the last statement?

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Yes, not all strong Markov processes with continuous sample paths solve an SDE. Here is an example:

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The process $X_t := |B_t|^{1/3}$ is not a semimartingale (see this question) and, hence, it doeesn't solve an SDE. (If it would solve an SDE, it was a semimartingale.) On the other hand, $(X_t)_{t \geq 0}$ is a Markov process (very similar reasoning as here). Since $(X_t)_{t \geq 0}$ has continuous sample paths, this already implies that it has the strong Markov property; this follows from an approximation procedure, see e.g. Theorem A.25 in Brownian Motion - An Introduction to Stochastic Processes by Schilling & Partzsch. In summary, $(X_t)_{t \geq 0}$ is a process with continuous sample paths satisfying the strong Markov property and failing to solve an SDE.

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  • $\begingroup$ Thank you very much for the clear answer! $\endgroup$ – A-B-izi May 3 at 15:40
  • $\begingroup$ @A-B-izi You are welcome. $\endgroup$ – saz May 3 at 17:49

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