On Generalized Berman Function With Drift Function

We evaluate the generalized Berman function involved in the study of the sojourn time of Gaussian processes,given by(?)x?[0,mes(E)],where mes(E)is the Lebesgue measure of a compact set E(?)R,h is a continuous drift function,and B? is a centered fractional Brownian motion(fBm)with Hurst index?/2 ?(0,1].It is closely related to the Piterbarg and Berman constants.It note specifies its explicit expression for ?=1,2 under certain conditions of drift functions.Explicit expressions of B2h(x,E)with typical drift functions are given and several bounds of B0o(x,E)are established as well.Numerical studies are perfomed to illustrate results.This thesis consists in three parts.First,explicit expressions of B?h(x)are established for the standard Brownian motion(B1(t))and Gaussian distribution(B2(t))with strictly positive drift function h(t)=ct?,t?0,c>0,and according to the convexity of h(t)+t2,the property of B?h(x,E)is further studied when ?=2,and the specific expression of B2h(x,E)is calculated by using some special form of drift function h(t)=c|t|?-t2,?>0 and any time interval E=[a,b].The results are strongly determined by the parameters involved,especially in a quite complicated form as the time interval includes zero and the power index ? is away from 1.This motivates the further study on its upper and lower bounds in Part Three.All findings are well illustrated by a small scale of numerical studies for Part Four.