2018/02/25 by P. Chigansky, Chigansky, P., Marina Kleptsyna +3 · 1 citation
Economics, Econometrics and Finance · #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Functional Analysis (math.FA) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1802.09045
openalex publication_date 2018/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm--Liouville theory for differential operators. This includes the Brownian motion and a whole class of processes, which derive from it by means of linear transformations. The more general eigenproblem for the \em fractional Brownian motion (f.B.m.) is not solvable in closed form, but the exact asymptotics of its eigenvalues and eigenfunctions can be obtained, using a method based on analytic properties of the Laplace transform. In this paper we consider two processes closely related to the f.B.m.: the fractional Ornstein--Uhlenbeck process and the integrated fractional Brownian motion. While both derive from the f.B.m. by simple linear transformations, the corresponding eigenproblems turn out to be much more complex and their asymptotic structure exhibits new effects.