N. H. Bingham: Modelling and prediction of financial time series
Abstract
Time series have a rich theory even in one dimension, but financial time series are specially interesting in many dimensions, in view of Markowitzian diversification.
In Econometrics, a great deal of the data is in discrete time, but continuous time is also relevant, and allows one to apply calculus and work with dynamic models, an insight due to Bergstrom.
Matters are simplest in the stationary case. But financial time series are typically non-stationary, and one may have to stationarise; there are various ways of proceeding here.
We begin with the classical Kolmogorov Isomorphism Theorem, to pass between the time domain (autocorrelation function, ACF) and the frequency domain (spectral measure), by the Fourier transform. We discuss the partial autocorrelation function (PACF), Verblunsky's theorem (which links the ACF with the more convenient PACF), orthogonal polynomials on the unit circle (OPUC -- the theory involved here) and Szego's theorem (which tells us when the remote past matters).
We discuss ARMA and GARCH models and their continuous-time analogues CARMA and COGARCH. We discuss volatility clustering, one of the stylised facts of mathematical finance. The infinite-dimensional case is sketched. We close with some applications.