Modified Brownian Motion Approach to Modeling Returns Distribution

Another approach to Brownian motion

For the partial sums Sn = Y1 + . . . + Yn of a centered stationary strongly mixing sequence {Yi} with finite second moment, the well-known sufficient conditions for the central limit theorem are that Var(Sn)/n is slowly varying as n → ∞ and the sequence {S2 n/σn} is uniformly integrable (σn = Var(Sn)). The conditions are checkable under various mixing conditions and they lead to the central lim...

The Patched Brownian Motion Distribution

Spectral distribution of the free unitary Brownian motion: another approach

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any t ∈ (0, 4) a Jordan curve γt around the origin, not intersecting the semi-axis [1,∞[ and whose image under some meromorphic function ht lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and ...

A Dynamical Approach to Fractional Brownian Motion

Herein we develop a dynamical foundation for fractional Brownian Motion. A clear relation is established between the asymptotic behaviour of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the ...

Area Distribution of Elastic Brownian Motion

We calculate the excursion and meander area distributions of the elastic Brownian motion by using the self adjoint extension of the Hamiltonian of the free quantum particle on the half line. We also give some comments on the area of the Brownian motion bridge on the real line with the origin removed. We will stress on the power of self adjoint extension to investigate different possible boundar...