# Do there exist diffusions that do not solve any SDE?

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?

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.