Fractional Brownian Motion

Appendix 1 Auxiliary Results from Mathematical, Functional and Stochastic Analysis

A1.1. Special functions

DEFINITION A1.1.– The gamma function is defined by the integral

[A1.1]

which converges for any x >0. Here are some basic properties of the gamma function:

1. i) the gamma function Γ(x) is positive, continuous and has continuous derivatives of all orders on (0, ∞);
2. ii) ;
3. iii);
4. iv) the gamma function is logarithmically convex on (0, +∞), i.e. its logarithm is convex on (0, +∞);
5. v) the Euler reflection formula is
   [A1.2]
6. vi) Legendre’s formula is
   [A1.3]
7. vii) Gauss’s formula for logarithmic derivative of gamma function has the following form:

[A1.4]

where γ is the Euler constant,

DEFINITION A1.2.– The beta function, also called the Euler integral of the first kind, is defined by the ...