In this thesis we consider Gaussian moving average processes X(t) which have the form
X(t)=_int_0^t \phi(t-s)dw_s,
where w_t is a Wiener process. Especially, we are interested in two cases
\phi_1(x)=\mathbbm{1}_[0,a](x) +\mathbbm{1}_(a,1)(x) and \phi_2(x)=x^\beta.
We shall call φ the kernel of the process. The motivation of this consideration lies on the dependence of increments of a random process. The increments of a Wiener process are independent, hence when we build a model with it, we assume that the increments of
model are uncorrelated. However, it seems far from reality when we need a stochastic process with correlated (positively or negatively) increments.
We will see that the random process M_t having \phi_1 as the kernel shares many properties with a Wiener process but it has positively dependent increments. This process is a toy model of the random process C_t which has
\phi_2 as the kernel. C_t can be understood as a fractional integration of order \beta+1 of white noise. We study properties of C_t. As an application, we
consider a stochastic di_x000B_fferential equation involving M_t and discuss how to fi_x000C_nd parameters of M_t from sampling.
