Schürger, K (2008)
Extensions of Black-Scholes processes and Benford's law
Stochastic Processes and their Applications 118(7), 1219-1243.
ISSN/ISBN: 0304-4149
DOI: 10.1016/j.spa.2007.07.017
Abstract: Let Z be a stochastic process of the form Z(t)=Z(0)exp(μt+X(t)−t/2) where Z(0)>0, μ are constants, and X is a continuous local martingale having a deterministic quadratic variation ≪X≫ such that ≪X≫_{t}→∞ as t→∞. We show that the mantissa (base b) of Z(t) (denoted by M(b)(Z(t)) converges weakly to Benford’s law as t→∞. Supposing that X satisfies a certain growth condition, we obtain large deviation results for certain functionals (including occupation time) of (M(b)(Z(t))). Similar results are obtained in the discrete-time case. The latter are used to construct a non-parametric test for nonnegative processes (Z(t)) (based on the observation of significant digits of (Z(n))) of the null hypothesis H_{0}(σ_{0}) which says that Z is a general Black–Scholes process having a volatility σ≥σ_{0}>0. Finally it is shown that the mantissa of Brownian motion is not even weakly convergent.
Bibtex:
@article {MR2428715,
AUTHOR = {Sch{\"u}rger, Klaus},
TITLE = {Extensions of {B}lack-{S}choles processes and {B}enford's law},
JOURNAL = {Stochastic Process. Appl.},
FJOURNAL = {Stochastic Processes and their Applications},
VOLUME = {118},
YEAR = {2008},
NUMBER = {7},
PAGES = {1219--1243},
ISSN = {0304-4149},
CODEN = {STOPB7},
MRCLASS = {60F10 (60G44 91B02 91B28)},
MRNUMBER = {2428715 (2009g:60037)},
DOI = {10.1016/j.spa.2007.07.017},
URL = {http://dx.doi.org/10.1016/j.spa.2007.07.017},
}
Reference Type: Journal Article
Subject Area(s): Probability Theory