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?


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.

  • $\begingroup$ 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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