Normalized fractional Brownian motion (FBM), denoted by B_{H}(t), t\in [0,T], is a Gaussian process with the following covariance \[ EB_{H}(t)B_{H}(s) = (1/2)\{|t|^{2H} + |s|^{2H} - |t - s|^{2H} \},\] t\,s \in [0,T], 0 A theory of single integrals of nonrandom functions w.r.t. FBM is developed and used to study the derivative of FBM. Then a theory of multiple integrals of nonrandom functions w.r.t. FBM is developed. The chaos decomposition of multiple integrals is studied representing a multiple integral as a sum of a finite number of mutually orthogonal random variables and then used to prove strong laws.

Geometric fractional Brownian motion (GFBM), an alternative model that has been proposed for stock price processes, is defined and its properties are studied. An arbitrage opportunity for GFBM is constructed to show that GFBM is not a suitable model for a stock price process and parameter estimation for GFBM is studied.

An approximation to FBM is also studied which uses a particular random walk and converges weakly to FBM.