×

Stieltjes integration and stochastic calculus with respect to self-affine functions. (English) Zbl 0769.60047

A continuous function \(f:[0,1]\to R\) is a self-affine function of base \(\alpha\in(0,1)\) if its graph has a fractal-type property, \(\alpha\) being the parameter of the scaling transformation. The purpose of this paper is to present a sort of a deterministic stochastic calculus in the framework of self-affine functions of order \(\alpha=1/2\). Typically, such a function \(f\) has unbounded variation, in particular, the classical Stieltjes integral cannot be defined. However, the authors construct a so-called ‘order-two Stieltjes integral’ of a function \(h(x)=g(x,f(x))\) (where \(g:R^ 2\to R\) is a \(C^ 2\) function) with respect to \(f\). They establish an analogue of the Itô formula (Itô’s formula is the Fundamental Theorem of Stochastic Calculus). In the case when the image of the Lebesgue measure by \(f\) is absolutely continuous w.r.t. the Lebesgue measure, the authors obtain a Tanaka-like formula that provides an expression for the density (i.e. the local times) in terms of an order-two Stieltjes integral. A Stratonovich’s formula is also proven, and the example when \(f\) is the so-called Rudin-Shapiro function is discussed. Previously, H. Föllmer [Lect. Notes Math. 850, 143-150 (1981; Zbl 0461.60074)] and the reviewer [Ann. Probab. 17, No. 4, 1521- 1535 (1989; Zbl 0687.60054)] developed analogues of the stochastic calculus for deterministic functions, using a fixed sequence of partitions. Here, the authors use a second order averaging method, which gives more robustness.
Reviewer: J.Bertoin (Paris)

MSC:

60H05 Stochastic integrals
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Preprint, Delft University of Technology. Report of the Dept. of Technical Mathematics and Informatics No. 89-88., 1989.
[2] J. Bertoin, Sur une intégrale pour les processus à {\(\alpha\)}-variation bornée. Ann. Probab.,17 (1989), 1521–1535. · Zbl 0687.60054 · doi:10.1214/aop/1176991171
[3] A. Fisher, Convex-invariant means and a pathwise central limit theorem. Adv. in Math.,63 (1987), 213–246. · Zbl 0627.60034 · doi:10.1016/0001-8708(87)90054-5
[4] H. Föllmer, Calcul d’Itô sans probabilités. Séminaire de Probabilités XV, Lecture Notes in Math.850, Springer, New York, 1981.
[5] T. Kamae, A characterization of self-affine functions, Japan J. Appl. Math.,3 (1986), 271–280. · Zbl 0646.28005 · doi:10.1007/BF03167102
[6] T. Kamae and M. Keane, A class of deterministic self-affine processes. Japan. J. Appl. Math.,7 (1990), 183–195. · Zbl 0719.60041 · doi:10.1007/BF03167840
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.