Extensions of Black-Scholes processes and Benford's law

Klaus Schürger

Stochastic Processes and their Applications, 2008, vol. 118, issue 7, 1219-1243

Abstract: Let Z be a stochastic process of the form Z(t)=Z(0)exp([mu]t+X(t)- t/2) where Z(0)>0, [mu] are constants, and X is a continuous local martingale having a deterministic quadratic variation such that t-->[infinity] as t-->[infinity]. We show that the mantissa (base b) of Z(t) (denoted by M(b)(Z(t)) converges weakly to Benford's law as t-->[infinity]. Supposing that 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 H0([sigma]0) which says that Z is a general Black-Scholes process having a volatility . Finally it is shown that the mantissa of Brownian motion is not even weakly convergent.

Keywords: Significant; digits; Leading; digits; Benford's; law; Brownian; motion; Exponential; local; martingales; Black-Scholes; processes; Occupation; time; Poisson's; summation; formula; Azuma's; inequality; Large; deviations; Weak; theorems; Strong; theorems; Non-parametric; hypothesis; testing; for; processes (search for similar items in EconPapers)
Date: 2008
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