A novel technique of functional Feynman-Kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by Mandelbrot [in Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds., Springer, New York (1972)] and constructed explicitly by Bacry, Delour, and Muzy [Phys. Rev. E 64:026103 (2001)]. The distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments (SDSI). It is shown that the SDSI has two separate layers of complexity that can be captured in a precise way by a pair of functional Feynman-Kac equations for the Laplace transform. Exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule. The expansion of the moments gives a new representation of the Selberg integral.
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G. E. Andrews, R. Askey and R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999).
E. Bacry, J. Delour and J.-F. Muzy, Multifractal random walk. Phys. Rev. E 64:026103 (2001a).
E. Bacry, J. Delour and J.-F. Muzy, Modelling financial time series using multifractal random walks. Physica A 299:84–92 (2001b).
E. Bacry and J.-F. Muzy, Log-infinitely divisible multifractal random walks. Comm. Math. Phys. 236:449–475 (2003).
J. Barral and B. B. Mandelbrot, Multifractal products of cylindrical pulses. Prob. Theory Relat. Fields 124:409–430 (2002).
L. Calvet and A. Fisher, Multifractality in asset returns. Theory and evidence. Rev. Econ. Stat. LXXXIV:381–406 (2002).
P. Chainais, R. Riedi and P. Abry, Infinitely divisible cascades, In: International Symposium on Physics in Signal and Image Processing (Grenoble, France, 2002).
H. Cramer and M. Leadbetter, Stationary and Related Stochastic Processes; Sample Function Properties and Their Applications (Wiley, New York, 1967).
C. de Calan, J. M. Luck, Th. M. Nieuwenhuizen, and D. Petritis, On the distribution of a random variable occuring in 1D disordered systems. J. Phys. A: Math. Gen. 18:501–523 (1985).
H. Geman and M. Yor, Bessel processes, asian options, and perpetuities. Math. Finance 3:349–375 (1993).
I. Gikhman and A. Skorohod, The Theory of Stochastic Processes,vol. I (Springer-Verlag, New York, 1974).
I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Probab. Appl. 5:285–301 (1960).
A. Goldberger, L. Amaral, J. Hausdorff, P. Ivanov, C. Peng and H. Stanley, Fractal dynamics in physiology: Alterations with disease and aging. Proc. Natil. Acad. Sci. USA 99:2466–2472 (2002).
P. Ivanov, L. Amaral, A. Goldberger, S. Havlin, M. Rosenblum, Z. Struzik, and H. Stanley, Multifractality in human heartbeat dynamics. Nature 399:461–465 (1999).
J.-P. Kahane, Sur le chaos multiplicatif. Ann. Sci. Math. Quebec 9:105–150 (1985).
J.-P. Kahane, Positive martingales and random measures. Chi. Ann. Math. 8B:1–12 (1987).
J.-P. Kahane, Produits de poids aléatoires indépendants et applications. In: J. Belair and S. Dubuc (eds.), Fractal Geometry and Analysis (Kluwer, Boston, 1991), p. 277.
O. Khorunzhiy, Limit theorems for sums of products of random variables. Markov Process. Relat. Fields 9:675–686 (2003).
B. B. Mandelbrot, Possible refinement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: M. Rosenblatt and C. Van Atta (eds.), Statistical Models and Turbulence (Lecture Notes in Physics 12, Springer, New York, 1972), p. 333.
B. B. Mandelbrot, Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C.R. Academ. Sci. Paris 278A:289–292 & 355–358 (1974a).
B. B. Mandelbrot, Intermitent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62:331–358 (1974b).
B. B. Mandelbrot, Limit lognormal multifractal measures. In: E. A. Gotsman et al. (eds.), Frontiers of Physics: Landau Memorial Conference (Pergamon, New York, 1990), p. 309.
H. Matsumoto and M. Yor, On Dufresne’s relation between the probability laws of exponential functionals of Brownian motion with different drifts. Adv. Appl. Prob. 35:184–206 (2003).
C. Meneveau and K. R. Sreenivasan, The multifractal nature of the turbulent energy dissipation, J. Fluid Mech. 224:429–484 (1991).
C. Monthus and A. Comtet, On the flux distribution in a one dimensional disordered system. J. Phys. I France 4:635–653 (1994).
J.-F. Muzy and E. Bacry, Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws. Phys. Rev. E 66:056121 (2002).
I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5:571–587 (1999).
D. Ostrovsky, Limit lognormal multifractal as an exponential functional. J. Stat. Phys. 116:1491–1520 (2004).
D. Schertzer and S. Lovejoy, Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades. J. Geophys. Res. 92:9693–9721 (1987).
D. Schertzer, S. Lovejoy, F. Schmitt, Y. Chigirinskaya and D. Marsan Multifractal cascade dynamics and turbulent intermittency. Fractals 5:427–471 (1997).
F. Schmitt and D. Marsan, Stochastic equations generating continuous multiplicative cascades. Eur. J. Phys. B 20:3–6 (2001).
F. Schmitt, A causal multifractal stochastic equation and its statistical properties. Eur. J. Phys. B 34:85–98 (2003).
F. Schmitt, D. Schertzer and S. Lovejoy, Multifractal analysis of foreign exchange data. Appl. Stochastic Models Data Anal. 15:29–53 (1999).
A. Selberg, Remarks on a multiple integral. Norske Mat. Tidsskr. 26:71–78 (1944).
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The author wishes to express gratitude to the Mathematics Department of Lehigh University for generous support during his stay at Lehigh University, where this article was written.
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Ostrovsky, D. Functional Feynman-Kac Equations for Limit Lognormal Multifractals. J Stat Phys 127, 935–965 (2007). https://doi.org/10.1007/s10955-007-9315-z
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DOI: https://doi.org/10.1007/s10955-007-9315-z