Stationary Jump and Diffusion Processes for Planar Directions Obtained by Wrapping

R. Gatto

2023, v.29, Issue 3, 403-433

ABSTRACT

This article presents stochastic processes
with jump or diffusion dynamics for
planar directions, hence on the circle.
The circular jump processes are:
the circular random telegraph signal,
the wrapped versions of the Poisson, the
signed Poisson and of the compound Poisson processes.
The circular diffusion processes are
the wrapped $\alpha$-stable L\'evy processes,
that include the wrapped Wiener process.
The circular jump-diffusion processes obtained by
wrapping a compound Poisson perturbed by an $\alpha$-stable L\'evy process
is also studied.
All these circular processes are weakly stationary.
Their means and autocovariance functions are provided.
Moreover, the trigonometric moments of the one-dimensional distributions
of these processes are obtained.

Keywords: Autocovariance function; $\alpha$-stable process; compound Poisson processes; Fourier series; jump-diffusion process; random telegraph signal; trigonometric moment

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