Then, from (32) applying the

stochastic calculus [27, 29] and (32), we obtain

Therefore, this provides an important method which is used in the proof of the existence of RPLSO, and it follows from

stochastic calculus that the shadowing distance can be determined for any given ([omega], [delta])-pseudoperiodic orbit.

Absolute Continuity Under Time Shift of Trajectories and Related

Stochastic CalculusMishura,

Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol.

Applebaum, Levy Processes and

Stochastic Calculus, vol.

We note that

stochastic calculus on the so-called q-Brownian motion has been considered in [12-14].

Applebaum, Levy Processes and

Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2004.

In Section 11, we enlarge our free

stochastic calculus of Sections 9 and 10 in terms of certain free stochastic processes generated by those of Sections 9 and 10.

Stochastic calculus with jumps is restricted to compound Poisson processes which have only a finite number of jumps on any bounded interval.

Peng, "G-Expectation, G-Brownian motion and related

stochastic calculus of Ito type," in Stochastic Analysis and Applications: The Abel Symposium 2005, vol.

For H [not equal to] 0.5, it is neither a Markov process nor semimartingale, so the classical Ito's calculus cannot be used to define a fully

stochastic calculus for fBm.