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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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