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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.

MSC:
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
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