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Nonlinearity of arch and stochastic volatility models and Bartlett’s formula

Kokoszka P. S., Politis D. N.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

47--59

We review some notions of linearity of time series and show that ARCH or stochastic volatility (SV) processes are not only non-linear: they are not even weakly linear, i.e., they do not even have a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in the context of data typically modeled as ARCH or SV processes such as financial returns. More surprisingly, we show that even the squares of an ARCH or SV process are not weakly linear. Finally, we discuss an alternative estimator for the variance of sample autocorrelations that is applicable (and consistent) in the context of financial returns data.

Stochastic volatility : approximation and goodness-of-fit test

Gradinaru M., Nourdin I.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

1--19

Let X be the unique solution started from x0 of the stochastic differential equation dXt= θ(t;Xt)dBt +b(t;Xt)dt with B a standard Brownian motion. We consider an approximation of the volatility θ(t;Xt), the drift being considered as a nuisance parameter. The approximation is based on a discrete time observation of X and we study its rate of convergence as a process. A goodness-of-fit test is also constructed.

Lévy processes and self-decomposability in finance

Bingham N. H.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 2

367--378

The main theme of Urbanik’s work was infinite divisibility and its ramifications. The aim of this memorial article is to trace the application of this theme in mathematical finance, one of the main growth areas in contemporary probability theory. We begin in Section 1 with a discussion of the nature of prices. In particular, we focus on whether (or when) prices may be taken as continuous, with a view to using Lévy processes to model the case of prices with jumps. We turn in Section 2 to asset return distributions; prime candidates for modelling here include the normal, hyperbolic and Student t cases. In Section 3, we turn to distributions of type G, in particular, those in which the mixing law is not only infinitely divisible but also self-decomposable (i.e. in the class SD), which includes all three cases above. Then in Section 4 we turn to the dynamic counterpart of this, in which the law of class SD occurs as the limit law of a stochastic process of Ornstein-Uhlenbeck type, with Lévy driving noise. Finally, in Section 5 we discuss stochastic volatility models.