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Extensions of Black-Scholes processes and Benford's law

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  • Schürger, Klaus

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.

Suggested Citation

  • Schürger, Klaus, 2008. "Extensions of Black-Scholes processes and Benford's law," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1219-1243, July.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:7:p:1219-1243
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    References listed on IDEAS

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    1. Hill, Theodore P. & Schürger, Klaus, 2005. "Regularity of digits and significant digits of random variables," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1723-1743, October.
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