Limit Lognormal Multifractal as an Exponential Functional

Abstract

The limiting distribution of the limit lognormal multifractal, first introduced by Mandelbrot (Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds., Lecture Notes in Physics 12, Springer, New York, 1972, p. 333) and constructed explicitly by Bacry et al. (Phys. Rev. E 64, 026103 (2001)), is investigated using its Laplace transform. A partial differential equation for the Laplace transform is derived and it is shown that multifractality alone does not determine the limiting distribution. The increments of the limit multifractal process are strongly stochastically dependent. The precise nature of this stochastic dependence structure of increments (SDSI) is the determining characteristic of the limiting distribution. The SDSI of the limit process is quantified by means of two integro-differential relations obtained by renormalization in the sense of Leipnik (J. Aust. Math. Soc. B 32, 327–347 (1991)). One is interpreted as a counterpart of the star equation of Mandelbrot and the other is shown to be an analogue of the classical Girsanov theorem. In the weak intermittency limit an approximate single-variable equation for the Laplace transform is obtained and successfully tested numerically by simulation.