2007/02/22 by Julien Barral, Benoit Mandelbrot, Benoît B. Mandelbrot +2
Economics, Econometrics and Finance · Mathematics · #28A78 #28A80 #60F10 #60G57 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:28A78 #msc:28A80 #msc:60F10 #msc:60G57
paper · pdf · doi:10.48550/arxiv.math/0702644
23 pages, 6 figures
arxiv created 2007/02/22 · openalex publication_date 2007/02/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The original density is 1 for t∈ (0,1), b is an integer base (b≥ 2%), and p∈ (0,1) is a parameter. The first construction stage divides the unit interval into b subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of (1% )/(2)+(p)/(2) and 1/2-(p)/(2). It is shown that the resulting density can be renormalized so that, as n→ ∞ (n being the number of iterations) the signed measure converges in some sense to a non-degenerate limit. If H=1+logb p>1/2, hence p>b^-1/% 2, renormalization creates a martingale, the convergence is strong, and the limit shares the Hölder and Hausdorff properties of the fractional Brownian motion of exponent H. If H≤ 1/2, hence p≤ b^-1/2%, this martingale does not converge. However, a different normalization can be applied, for H≤ 1/2 to the martingale itself and for H>% 1/2 to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of H and of b. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.