ARCH models have become popular for modeling financial time series. They seem, at first, however, to be incompatible with the option pricing approach of Black, Scholes, Merton et al., because they are discrete-time models and possess too much variability. We show that completeness of the market...

It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval...

We extend results of Maejima (1984) concerning the time that a two-dimensional stationary Gaussian process spends in an elliptical domain. Here: (a) the process may be cross-correlated while the domain is elliptical; (b) the cross-correlations do not vanish asymptotically; (c) a functional limit...

We develop the theory of fractionally differenced ARIMA time series with stable infinite variance innovations establishing conditions for existence and invertibility. We analyze their asymptotic dependence structure by means of the codifference and the covariation, measures of dependence which...

Jointly [alpha]-stable random variables with index 0 [alpha] 2 have only finite moments of order less than [alpha], but their conditional moments can be higher than [alpha]. We provide conditions for this to happen and use the existence of the conditional moments to study the regression...

We develop a pathwise construction of stochastic integrals relative to continuous martingales. The key to the construction is an almost sure approximation technique which associates a sequence of finitely generated filtrations ("skeleton filtrations") and a sequence of simple stochastic...

Let (X1,X2) be an [alpha]-stable random vector with 0 [alpha] 2, not necessarily symmetric. Its distribution is characterized by a finite measure [Gamma] on the unit circle called the spectral measure. It is known that if [Gamma] satisfies some integrability condition then the conditional...

A self-similar process Z(t) has stationary increments and is invariant in law under the transformation Z(i)--c-HZ(ct), c[greater-or-equal, slanted]0. The choice ensures that the increments of Z(t) exhibit a long range positive correlation. Mandelbrot and Van Ness investigated the case where Z(t)...

Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index [alpha] in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that...