I have a random process which seems to oscillate between extremes in terms of trending but which is locally like a Brownian motion or a fractional Brownian motion.

Is there a name for such a random/stochastic process?
I have a random process which seems to oscillate between extremes in terms of trending but which is locally like a Brownian motion or a fractional Brownian motion.

Is there a name for such a random/stochastic process?
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Looks an awful like a version of a jump-diffusion process, especially since its brownian between oscillations.