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Extensions of Black-Scholes processes and Benford’s law. (English) Zbl 1152.60027
Let \(Z\) be a stochastic process of the form \(Z(t) = Z(0) \exp (\mu t + X(t) - \langle X \rangle_t/2)\) where \(Z(0) > 0\), \(\mu\) are constants, and \(X\) is a continuous local martingale, having a deterministic quadratic variation \(\langle X \rangle\) such that \(\langle X \rangle_t \to \infty\) as \(t \to \infty\). In the paper, it is shown that the mantissa (base \(b\)) of \(Z(t)\), denoted by \(M^{(b)}(Z(t))\), converges weakly to Benford’s law as \(t \to \infty\). Supposing that \(\langle X \rangle\) satisfies a certain growth condition, large deviation results are obtained 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 in the observation of significant digits of \((Z(n))\), of the null hypothesis \(H_0(\sigma_0)\) which says that \(Z\) is a generalized geometric Brownian motion having a volatility \(\sigma \geq \sigma_0 (> 0)\). Finally, it is shown that the mantissa of a Brownian motion is not even weakly convergent.

60F05 Central limit and other weak theorems
60F10 Large deviations
60F15 Strong limit theorems
60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
62M07 Non-Markovian processes: hypothesis testing
91B28 Finance etc. (MSC2000)
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
60E10 Characteristic functions; other transforms
Full Text: DOI
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