In this dissertation a white noise approach to solving linear partial differential equations is investigated, where all equations are driven by a generalized Gaussian noise. To do so, generalized Gaussian random fields over Hilbert spaces are introduced and later on extended to Gaussian processes. Those processes are characterized through covariance operators which leads to colored noise, a procedure closely connected to the classical white noise analysis.;The definition of random fields acting on random functions, motivated by investigating stochastic PDEs with multiplicative noise, eventually leads to the definition of the stochastic integral through Wick products. Assuming that the solution to a linear stochastic differential equation admits a chaos expansion, the S-system of deterministic PDEs for the coefficients of the expansion is introduced. Computing a finite number of coefficient equations results in an approximate solution, where the error bound of the approximation decreases as the number of coefficients increases.;In the second part of the dissertation the stochastic integration approach is applied to the study of the term structure of interest rates. The instantaneous forward rate process is decomposed into short rate, spread, shape function, and deformation process. The short rate and deformation process are modeled by stochastic differential equations driven by colored noise, because the use of colored noise preserves the observed long range dependence in the term structure.
展开▼
