Fractional Brownian Motion in Finance

This thesis deals with the stochastic integral with respect to Gaussian processes, which can be expressed in the form Bt = t 0 K(t, s)dWs. Here W stands for a Brownian motion and K for a square integrable Volterra kernel. Such processes generalize fractional Brownian motion. Since these processes are not semimartin- gales, Itô calculus cannot be used and other methods must be employed to define the stochastic integral with respect to these proceses. Two ways are considered in this thesis. If both the integrand and the process B are regular enough, it is possible to define the integral in the pathwise sense as a generalization of Lebesgue-Stieltjes integral. The other method uses the methods of Malliavin cal- culus and defines the integral as an adjoint operator to the Malliavin derivative. As an application, the stochastic differential equation dSt = µStdt + σStdBt, which is used to model price of a stock, is solved. Implications of such a model are briefly discussed.