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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/

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 H00) which says that Z is a general Black–Scholes process having a volatility σ≥σ0>0. Finally it is shown that the mantissa of Brownian motion is not even weakly convergent.

@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/}, URL = {}, }

Reference Type: Journal Article

Subject Area(s): Probability Theory