Application of the spectral theory and perturbation theory to the study of Ornstein-Uhlenbeck processes

Abstract

The theoretical bases of this paper are the theory of spectral analysis and the theory of singular and regular perturbations. We obtain an approximate price of Ornstein-Uhlenbeck double barrier options with multidimensional stochastic diffusion as expansion in eigenfunctions using infinitesimal generators of a $(l+r+1)$-dimensional diffusion in Hilbert spaces. The theorem of accuracy estimation of options prices approximation is established. We also obtain explicit formulas for derivatives price based on the expansion in eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations.

Keywords

• spectral theory
• singular perturbation theory
• regular perturbation theory
• sturm-liouville theory
• infinitesimal generator
• multidimensional diffusion